5-1 Solving Inequalities by Addition and Subtraction. Solve each inequality. Then graph the solution set on a number line. 1.

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1 5-1 Solving Inequalities by Addition and Subtraction Solve each inequality Then graph the solution set on a number line 1 x 3 > 7 The solution set is {p p 7} 5 10 > n 1 The solution set is {x x > 10} The solution set is {n n < 11} 6 k + 24 > y The solution set is {k k > 29} The solution set is {y y 2} 7 8r + 6 < 9r 3 g + 6 < 2 The solution set is {r r > 6} The solution set is {g g < 4} 8 8n 7n p + 4 The solution set is {n n 3} The solution set is {p p 7} 5 10 > n 1 The solution set is {n n < 11} Define a variable, write an inequality, and solve each problem Check your solution 9 Twice a number increased by 4 is at least 10 more than the number Let n = the number Page 1

2 The solution set is {n n 3} 5-1 Solving Inequalities by Addition and Subtraction Define a variable, write an inequality, and solve each problem Check your solution 9 Twice a number increased by 4 is at least 10 more than the number The inequality is true when n is greater than or equal to 6, so the solution checks 10 Three more than a number is less than twice the number Let n = the number Let n = the number The solution set is {n n 6} The solution set is {n n > 3} To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3 To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6 The inequality is true when n is greater than 3, so the solution checks The inequality is true when n is greater than or equal to 6, so the solution checks 10 Three more than a number is less than twice the number Let n = the number 11 AMUSEMENT A thrill ride swings passengers back and forth, a little higher each time up to 137 feet Suppose the height of the swing after 30 seconds is 45 feet How much higher will the ride swing? The ride will swing no higher than 92 more feet Solve each inequality Then graph the solution set on a number line 12 m 4 < 3 esolutions - Powered by Cognero The Manual solution set is {n n > 3} To check substitute three different values into the original inequality: 3, a number less than 3, and a Page 2 The solution set is {m m < 7}

3 5-1 Solving Inequalities by Addition and Subtraction The ride will swing no higher than 92 more feet Solve each inequality Then graph the solution set on a number line 12 m 4 < 3 The solution set is {b b 2} > 18 + r The solution set is {m m < 7} The solution set is {r r < 5} c 1 13 p 6 3 The solution set is {c c 4} The solution set is {p p 9} q r 8 7 The solution set is {r r 15} 15 t 3 > 8 The solution set is {q q 7} m 15 The solution set is {t t > 5} The solution set is {m m 4} 16 b h 26 < 4 The solution set is {b b 2} The solution set is {h h < 30} >Manual 18 + r- Powered by Cognero esolutions 22 8 r 14 Page 3

4 The solution set is {h h < 30} 5-1 Solving Inequalities by Addition and Subtraction 22 8 r 14 The solution set is {z z 4} 26 w 5 2w The solution set is {r r 22} The solution set is {w w 5} 23 7 > 20 + c 27 3y + 6 2y The solution set is {c c < 27} The solution set is {y y 6} 24 2a 4 + a 28 6x + 5 7x The solution set is {a a 4} The solution set is {x x 5} 25 z + 4 2z a < 3a The solution set is {z z 4} The solution set is {a a > 9} 26 w 5 2w Define a variable, write an inequality, and solve each problem Check your solution 30 Twice a number is more than the sum of that number and 9 Let n = the number The solution set is {w w 5} Page 4

5 The solution set is {a a > 9} 5-1 Solving Inequalities by Addition and Subtraction The inequality is true when n is greater than 9, so the solution checks Define a variable, write an inequality, and solve each problem Check your solution 30 Twice a number is more than the sum of that number and 9 31 The sum of twice a number and 5 is at most 3 less than the number Let n = the number Let n = the number The solution set is {n n >9} To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9 The solution set is {n n 8} To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8 The inequality is true when n is greater than 9, so the solution checks 31 The sum of twice a number and 5 is at most 3 less than the number Let n = the number The inequality is true when n is equal to or less than 8, so the solution checks 32 The sum of three times a number and 4 is at least twice the number plus 8 Let n = the number The solution set is {n n 8} To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8 The solution set is {n n 12} Page 5 To check substitute three different values into the

6 5-1 Solving Inequalities Addition and The inequality is true by when n is equal tosubtraction or less than 8, so the solution checks 32 The sum of three times a number and 4 is at least twice the number plus 8 The inequality is true when n is greater than or equal to 12, so the solution checks 33 Six times a number decreased by 8 is less than five times the number plus 21 Let n = the number Let n = the number The solution set is {n n 12} The solution set is {n n < 29} To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12 To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29 The inequality is true when n is greater than or equal to 12, so the solution checks The inequality is true when n is less than 29, so the solution checks 33 Six times a number decreased by 8 is less than five times the number plus 21 Let n = the number Define a variable, write an inequality, and solve each problem Then interpret your solution 34 FINANCIAL LITERACY Keisha is babysitting at $8 per hour to earn money for a car So far she has saved $1300 The car that Keisha wants to buy costs at least $5440 How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs The solution set is {n n < 29} To check substitute three different values into the original inequality: 29, a number less than 29, and a Page 6

7 5-1 Solving Inequalities Addition The inequality is true by when n is lessand thansubtraction 29, so the solution checks Define a variable, write an inequality, and solve each problem Then interpret your solution 34 FINANCIAL LITERACY Keisha is babysitting at $8 per hour to earn money for a car So far she has saved $1300 The car that Keisha wants to buy costs at least $5440 How much money does Keisha still need to earn to buy the car? The solution set is {n n > 41} So, there were originally more than 41 songs on the digital media player 37 TEMPERATURE The water temperature in a swimming pool increased 4 F this morning The temperature is now less than 81 F What was the water temperature this morning? Let t = the water temperature in the morning Let b = the amount of money that Keisha still needs The solution set is {t t < 77} So, the water temperature in the morning was originally less than 77 F The solution set is {b b 4140} So, Keisha needs to earn at least $4140 to buy the car 35 TECHNOLOGY A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet Of those, about 16 million said they use the Internet at school How many teens that are online do not use the Internet at school? 38 BASKETBALL A player s goal was to score at least 150 points this season So far, she has scored 123 points If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal Let n = the number, in millions, of online teens that do not use the Internet at school The solution set is {n n > 5} So, at least 5 million teens use the Internet somewhere other than school 36 MUSIC A DJ added 20 more songs to his digital media player, making the total more than 61 How many songs were originally on the player? Let n = the original number of songs on the player The solution set is {p p 27} So, the player must score at least 27 points to reach her goal for the season 39 SPAS Samantha received a $75 gift card for a local day spa for her birthday She plans to get a hair cut and a manicure How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card The solution set is {n n > 41} So, there were originally more than 41 songs on the digital media player 37 TEMPERATURE The water temperature in a swimming pool increased 4 F this morning The temperature is now less than 81 F What was the water temperature this morning? The solution set is {m m 17} So, after her visit to the spa, there will be no more than $17 left on Samantha s gift card Page 7 40 VOLUNTEER Kono knows that he can only volunteer up to 25 hours per week If he has

8 The solution set is {p p 27} So, the player must 5-1 Solving bytoaddition score at Inequalities least 27 points reach herand goalsubtraction for the season 39 SPAS Samantha received a $75 gift card for a local day spa for her birthday She plans to get a hair cut and a manicure How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card The solution set is So, Kono can volunteer at most 19 hours and 25 minutes more this week Solve each inequality Check your solution, and then graph it on a number line 41 c + ( 14) 23 The solution set is {c c 37} To check substitute three different values into the original inequality: 37, a number less than 37, and a number greater than 37 The solution set is {m m 17} So, after her visit to the spa, there will be no more than $17 left on Samantha s gift card 40 VOLUNTEER Kono knows that he can only volunteer up to 25 hours per week If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer The inequality is true when c is equal to or greater than 37, so the solution checks 42 91g + 45 < 101g The solution set is So, Kono can volunteer at most 19 hours and 25 minutes more this week Solve each inequality Check your solution, and then graph it on a number line 41 c + ( 14) 23 The solution set is {g g > 45} To check substitute three different values into the original inequality: 45, a number less than 45, and a number greater than 45 Page 8

9 The inequality is true when g is greater than 45, so the solution checks than 37, so the solution checks 5-1 Solving Inequalities by Addition and Subtraction 42 91g + 45 < 101g 43 The solution set is {g g > 45} To check substitute three different values into the original inequality: 45, a number less than 45, and a number greater than 45 The solution set is To check substitute three values into the original inequality:, a number less than number greater than, and a The inequality is true when g is greater than 45, so the solution checks The inequality is true when k is greater than, so the solution checks Page 9

10 , so the solution checks Solving Inequalities by Addition and Subtraction 45 MULTIPLE REPRESENTATIONS In this problem, you will explore multiplication and division in inequalities The solution set is To check substitute three different values into the original inequality: number greater than, a number less than, and a a GEOMETRIC Suppose a balance has 12 pounds on the left side and 18 pounds on the right side Draw a picture to represent this situation b NUMERICAL Write an inequality to represent the situation c TABULAR Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance Create a second table showing the result of reducing the weight on each side of the balance by a factor of,, or Include a column in each table for the inequality representing each situation d VERBAL Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality a b 12 lb < 18 lb c x2 x3 x4 The inequality is true when p is equal to or less than, so the solution checks x x x 45 MULTIPLE REPRESENTATIONS In this problem, you will explore multiplication and division in esolutions Manual - Powered by Cognero inequalities < < < < < < < 6 3 < d If a true inequality is multiplied by a positive number, the resulting inequality is also true If a true Page 10 inequality is divided by a positive number, the resulting inequality is also true

11 x 3 < x 5-1 Solving Inequalities by Addition and Subtraction d If a true inequality is multiplied by a positive number, the resulting inequality is also true If a true inequality is divided by a positive number, the resulting inequality is also true So,? = 2 51 m +? 43 If m , then complete each inequality 46 m? So,? = REASONING Compare and contrast the graphs of a < 4 and a 4 So,? = m +? 27 In both graphs, the line is darkened to the left of 4 In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph In the graph of a 4, there is a dot at 4 to indicate that 4 is included in the graph 53 CHALLENGE Suppose b > d + So,? = m 5? So,? = 12 and d + > a + 2 Order a, b, c, and d from least to greatest Since, b > d Since, a > c Since 49 m? 14, c + 1 < a 4,, d > a So, c < a < d < b 54 OPEN ENDED Write three linear inequalities that are equivalent to y < 3 So,? = 3 50 m 19? So,? = 2 51 m +? 43 So,? = REASONING Compare and contrast the graphs of To write equivalent inequalities, add or subtract the same values from both sides of the inequality Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0 55 WRITING IN MATH Summarize the process of solving and graphing linear inequalities Solving linear inequalities is similar to solving linear equations You must isolate the variable on one side of the inequality To graph, if the problem is a less than or a greater than inequality, an open circle is used Otherwise a dot is used If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right Page WRITING IN MATH Explain why x 2 > 5 has the same solution set as x > 7

12 To write equivalent inequalities, add or subtract the same values from bothbysides of the inequality 5-1 Solving Inequalities Addition and Subtraction Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0 55 WRITING IN MATH Summarize the process of solving and graphing linear inequalities Solving linear inequalities is similar to solving linear equations You must isolate the variable on one side of the inequality To graph, if the problem is a less than or a greater than inequality, an open circle is used Otherwise a dot is used If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right 56 WRITING IN MATH Explain why x 2 > 5 has the same solution set as x > 7 The inequalities are equivalent By adding 2 to each side of the first inequality, you get the second inequality 57 Which equation represents the relationship shown? x y A y = 7x 8 B y = 7x + 8 C y = 8x 7 D y = 8x + 7 The inequalities are equivalent By adding 2 to each side of the first inequality, you get the second inequality 57 Which equation represents the relationship shown? x y A y = 7x 8 B y = 7x + 8 C y = 8x 7 D y = 8x + 7 Substitute values of x and y in the equations Choice A does not work for (1, 1) Choice B does not work for (1, 1) Choice D does not work for (1, 1) Choice C works for all values Substitute values of x and y in the equations Choice A does not work for (1, 1) Choice B does not work for (1, 1) etc So, the correct choice is C Choice D does not work for (1, 1) 58 What is the solution set of the inequality 7 + x < 5? F G H J Choice C works for all values Page 12 Subtracting 7 from each side of the inequality results

13 has plus three more The expression that represents how much money Francisco has would be etc 5-1 Solving Inequalities by Addition and Subtraction So, the correct choice is C 58 What is the solution set of the inequality 7 + x < 5? F G H So, the correct choice is B 60 GRIDDED RESPONSE The mean score for 10 students on the chemistry final exam was 178 However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score What should the mean score be? J The mean is the total of the scores divided by the number of scores So, working backwards, multiply the mean, 178, by the number of scores, 10 Then, add the 10 point mistake back to the total of the scores and divide by 10 again Subtracting 7 from each side of the inequality results in x < 2 So, the correct choice is H 59 Francisco has $3 more than the number of dollars that Kayla has Which expression represents how much money Francisco has? A So the mean score should be 179 B Find the inverse of each function 61 f (x) = 7x 28 C D Let k represent the number of dollars Kayla has Francisco has one-fourth the number of dollars Kayla has plus three more The expression that represents how much money Francisco has would be So, the correct choice is B 60 GRIDDED RESPONSE The mean score for 10 students on the chemistry final exam was 178 However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score What should the mean score be? 62 The mean is the total of the scores divided by the number of scores So, working backwards, multiply the mean, 178, by the number of scores, 10 Then, add the 10 point mistake back to the total of the scores and divide by 10 again 63 So the mean score should be 179 Find the inverse of each function 61 f (x) = 7x 28 Page 13

14 Write the equation in slope-intercept form 5-1 Solving Inequalities by Addition and Subtraction 66 ( 3, 1), y = 3x The slope of the line with equation y = 3x + 7 is 3 The slope of the perpendicular line is the opposite reciprocal of 3, or Find the y-intercept 64 Write the equation in slope-intercept form 67 (1, 3), y = Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation 65 ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1 The slope of the perpendicular line is the opposite reciprocal of 1, or 1 Find the y-intercept +4 The slope of the line with equation y = + 4 is The slope of the perpendicular line is the opposite reciprocal of, or 2 Find the y-intercept Write the equation in slope-intercept form Write the equation in slope-intercept form 68 ( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form 66 ( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3 The slope of the perpendicular line is the opposite reciprocal of 3, or Find the y-intercept Page 14

15 Write the equation in slope-intercept form Write the equation in slope-intercept form 5-1 Solving Inequalities by Addition and Subtraction 69 TRAVEL On an island cruise in Hawaii, each passenger is given a lei A crew member hands out 3 red, 3 blue, and 3 green leis in that order If this pattern is repeated, what color lei will the 50th person receive? 68 ( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form Let each group of 3 people with the same color lei be one term in the pattern (n) So the constant change d = 3 The slope of the line with equation 2x 5y = 3 is The slope of the perpendicular line is the opposite reciprocal of, or Find the y-intercept 50 3 = 16 R2 Write the equation in slope-intercept form 69 TRAVEL On an island cruise in Hawaii, each passenger is given a lei A crew member hands out 3 red, 3 blue, and 3 green leis in that order If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n) So the constant change d = 3 th So, n needs to be in the 17 group of 3 Substitute 17 for n in the equations for each color to see which color equals 50 r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 th So, the 50 person would receive a blue lei Find the nth term of each arithmetic sequence described 70 a 1 = 52, d = 12, n = , 7, 5, 3, for n = 18 Page 15

16 5-1 Solving Inequalities by Addition and Subtraction So, if you work 30 hours, your pay is $ , 7, 5, 3, for n = 18 Solve each equation 74 8y = , 1, 15, 2, for n = p = JOBS Refer to the time card shown Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours 76 3a = Let p = your pay per hour 78 Let y = your total pay and x = the number of hours worked So, the direct variation equation relating your pay to the hours worked is y = 7x Solve for x = So, if you work 30 hours, your pay is $210 Solve each- Powered equation esolutions Manual by Cognero 74 8y = 56 Page 16

17 5-1 Solving Inequalities by Addition and Subtraction Page 17

18 The solution set is {r r 8} 5-2 Solving Inequalities by Multiplication and Division 1 FUNDRAISING The Jefferson Band Boosters raised more than $5500 from sales of their $15 band DVD Define a variable, and write an inequality to represent the number of DVDs they sold Solve the inequality and interpret your solution 4 Let d = the number of DVDs sold The solution set is {c c 42} This means that the band sold at least 367 DVDs Solve each inequality Graph the solution on a number line 5 2 The solution set is {h h < 10} The solution set is {n n < 60} 6 9t > The solution set is {t t > 12} The solution set is {r r 8} 7 84 < 7v 4 The solution set is {v v > 12} Page 1

19 The solution set is {z z 8} The solution set is {t t > 12} 5-2 Solving Inequalities by Multiplication and Division 7 84 < 7v Define a variable, write an inequality, and solve each problem Then interpret your solution 10 CELL PHONE PLAN Mario purchases a prepaid phone plan for $50 at $013 per minute How many minutes can Mario talk on this plan? Let m = the number of minutes that Mario can talk The solution set is {v v > 12} x So, Mario can talk up to 384 minutes 11 FINANCIAL LITERACY Rodrigo needs at least $560 to pay for his spring break expenses, and he is saving $25 from each of his weekly paychecks How long will it be before he can pay for his trip? The solution set is Let p = the number of pay periods for which Rodrigo will need to save z So, Rodrigo will need to save for 23 weeks Solve each inequality Graph the solution on a number line The solution set is {z z 8} 12 Define a variable, write an inequality, and solve each problem Then interpret your solution 10 CELL PHONE PLAN Mario purchases a prepaid phone plan for $50 at $013 per minute How many minutes can Mario talk on this plan? The solution set is {m m 68} Let m = the number of minutes that Mario can talk 13 Page 2

20 The solution set is {m m 68} The solution set is {d d 68} 5-2 Solving Inequalities by Multiplication and Division The solution set is {a a < 40} The solution set is {x x 20} The solution set is {c c > 121} The solution set is {f f < 432} The solution set is {d d 68} The solution set is {h h >21} The solution set is {x x 20} Page 3

21 The solution set is {h h >21} The solution set is {p p 16} 5-2 Solving Inequalities by Multiplication and Division 22 4r < The solution set is {r r < 16} The solution set is {j j 16} > 2y 20 The solution set is {y y > 16} The solution set is {n n 108} < 26t 21 6p 96 The solution set is {t t > 1} The solution set is {p p 16} 25 6v > r < 64 The solution set is {v v < 12} The solution set is {r r < 16} z Page 4

22 The solution set is {v v < 12} The solution set is 5-2 Solving Inequalities by Multiplication and Division 29 7f > z The solution set is {z z 11} The solution set is 27 4b 3 30 CHEERLEADING To remain on the cheerleading squad, Lakita must attend at least of the study table sessions offered She attends 15 sessions If Lakita met the requirements, what is the maximum number of study table sessions? The solution set is Let s represent the number of sessions 28 2d < 5 At most, 25 study table sessions are offered The solution set is 31 BRACELETS How many bracelets can Caitlin buy for herself and her friends if she wants to spend no more than $22? 29 7f > 5 The solution set is Let b represent the number of bracelets Caitlin can buy no more than 4 bracelets 30 CHEERLEADING To remain on the cheerleading 32 CHARITY The National Honor Society at Page 5 Pleasantville High School wants to raise at least $500 for a local charity Each student earns $050 for every quarter of a mile walked in a walk-a-thon

23 5-2 Solving Inequalities by Multiplication and Division Caitlin can buy no more than 4 bracelets Jan can buy no more than 76 gallons of gasoline Match each inequality to the graph of its solution a 32 CHARITY The National Honor Society at Pleasantville High School wants to raise at least $500 for a local charity Each student earns $050 for every quarter of a mile walked in a walk-a-thon How many miles will the students need to walk? b Let m represent the number of quarter miles c d 35 The students needs to walk at least 1000 quarter miles or at least 250 miles 33 MUSEUM The American history classes are planning a trip to a local museum Admission is $8 per person Determine how many people can go for $260 Let p represent the number of people The point is at 135, so the correct choice is b 36 25j 8 No more than 32 people can go to the museum 34 GASOLINE If gasoline costs $315 per gallon, how many gallons of gasoline, to the nearest tenth, can Jan buy for $24? Let g represent the number of gallons of gasoline The point is at 032, so the correct choice is a 37 36p < 45 The point is at 125, so the correct choice is d Jan can buy no more than 76 gallons of gasoline Match each inequality to the graph of its solution a < 5t b Page 6 The point is at 046, so the correct choice is c c

24 5-2 Solving Inequalities by Multiplication and Division The point is at 125, so the correct choice is d So, there are more than 165 people employed at the west branch 41 MULTIPLE The equation for the volume of a < 5t pyramid is the area of the base times the height a GEOMETRIC Draw a pyramid with a square base b cm long and a height of h cm b NUMERICAL Suppose the pyramid has a 3 The point is at 046, so the correct choice is c 39 CANDY Fewer than 42 employees at a factory stated that they preferred fudge over fruit candy This is about two thirds of the employees How many employees are there? Let e represent the number of employees volume of 72 cm Write an equation to find the height c TABULAR Create a table showing the value of h when b = 1, 3, 6, 9, and 12 d NUMERICAL Write an inequality for the possible lengths of b such that b < h Write an inequality for the possible lengths of h such that b > h a So, there are fewer than 63 employees b 40 TRAVEL A certain travel agency employs more than 275 people at all of its branches Approximately three fifths of all the people are employed at the west branch How many people work at the west branch? Let p represent the number of people So, there are more than 165 people employed at the west branch 41 MULTIPLE The equation for the volume of a pyramid is the area of the base times the height a GEOMETRIC Draw a pyramid with a square base b cm long and a height of h cm b NUMERICAL Suppose the pyramid has a 3 volume of 72 cm Write an equation to find the height c TABULAR Create a table showing the value of h when b = 1, 3, 6, 9, and 12 esolutions Manual - Powered bywrite Cognero d NUMERICAL an inequality for the possible lengths of b such that b < h Write an inequality for the possible lengths of h such that b > c b h d b < h when 0 < b < 6; b > h when 6 < h 42 ERROR ANALYSIS Taro and Jamie are solving 6d 84 Is either of them correct? Explain your reasoning Page 7

25 c b h Solving Inequalities by Multiplication and Division d b < h when 0 < b < 6; b > h when 6 < h 42 ERROR ANALYSIS Taro and Jamie are solving 6d 84 Is either of them correct? Explain your reasoning 2 44 CHALLENGE Determine whether x > 1 and x > 1 are equivalent Explain Not equivalent; the inequalities do not have the same 2 solution sets If x = 2, then x = 4 and 4 > 1 The 2 solution set of x > 1 contains the subsets of real numbers given by x > 1 and by x < 1 Taro is correct Jamie reversed the inequality sign when dividing by 6 The inequality sign should only be reversed when dividing or multiplying by a negative number 43 CHALLENGE Solve each inequality for x Assume that a > 0 a b c a When dividing by a negative number, reverse the inequality symbol 45 REASONING Explain whether the statement If a > b, then is sometimes, always, or never true Sometimes Consider three cases Case I: a > 0 and b > 0 Choose a = 3 and b = 2 Then 3 > 2 but Case II: a < 0 and b < 0 Choose a = 2 and b = 3 Then 2 > 3, but Case III: a > 0 and b < 0 Choose a = 2 and b = 3 Then 2 > 3, and b The statement is only true when a > 0 and b < 0 46 OPEN ENDED Create a real world situation to represent the inequality Student answers may vary Sample answer: The c temperature never exceeds 47 WRITING IN MATH How are solving linear inequalities and linear equations similar? different? 2 44 CHALLENGE Determine whether x > 1 and x > 1 are equivalent Explain Not equivalent; the inequalities do not have the same 2 solution sets If x = 2, then x = 4 and 4 > 1 The 2 esolutions Manual Cognero the subsets of real solution set- Powered of x > 1bycontains numbers given by x > 1 and by x < 1 Sample answer: The same processes are used when solving linear inequalities and equations that involve addition, subtraction, multiplication, or division by a positive number However, when a linear inequality is multiplied or divided by a negative number, the inequality symbol must change directions so thatpage the 8 inequality remains true

26 Student answers may vary Sample answer: The 5-2 Solving Inequalities by Multiplication and Division temperature never exceeds 47 WRITING IN MATH How are solving linear inequalities and linear equations similar? different? Sample answer: The same processes are used when solving linear inequalities and equations that involve addition, subtraction, multiplication, or division by a positive number However, when a linear inequality is multiplied or divided by a negative number, the inequality symbol must change directions so that the inequality remains true The rate 009 is the rate to be multiplied by each minute So choices A and B can be eliminated If he does not want to spend more than $250, then choice D is correct 49 SHORT RESPONSE Find the value of x Let y represent the height of the smaller triangle For example: Inequality: If y = 8, then Equation: So, x is 10 inches 48 Juan s long-distance phone company charges 9 for each minute Which inequality can be used to find how long he can talk to a friend if he does not want to spend more than $250 on the call? A m B m C 009m 250 D 009m 250 The rate 009 is the rate to be multiplied by each minute So choices A and B can be eliminated If he does not want to spend more than $250, then choice D is correct 49 SHORT RESPONSE Find the value of x 50 What is the greatest rate of decrease of this function? F 5 G 3 H 2 J 1 The greatest rate of decrease is when the slope is the steepest and negative This means that the line will slope down from left to right So, the steepest line is between ( 1, 7) and (0, 2) Let y represent the height of the smaller triangle Page 9

27 5-2 Solving Inequalities by Multiplication and Division So, x is 10 inches 50 What is the greatest rate of decrease of this function? So, the correct choice is F 51 What is the value of x if 4x 3 = 2x? A 2 B C D 2 F 5 G 3 H 2 J 1 The greatest rate of decrease is when the slope is the steepest and negative This means that the line will slope down from left to right So, the steepest line is between ( 1, 7) and (0, 2) So, the correct choice is C Solve each inequality Check your solution, and then graph it on a number line a < 6a So, the correct choice is F 51 What is the value of x if 4x 3 = 2x? A 2 B {a a > 4} C D y y So, the correct choice is C Solve each inequality Check your solution, and then graph it on a number line Page b > 12b

28 {a a > 4} 5-2 Solving Inequalities by Multiplication and Division 53 2y y b > 12b Find the inverse of each function HOME DECOR Pam is having blinds installed at her home The cost c of installation for any number of blinds b can be described by c = b Graph the equation and determine how much it would cost if Pam has 8 blinds installed Evaluate c = b for b = 8 Page 11

29 5-2 Solving Inequalities by Multiplication and Division So, it would cost Pam $77 to have 8 blinds installed 58 HOME DECOR Pam is having blinds installed at her home The cost c of installation for any number of blinds b can be described by c = b Graph the equation and determine how much it would cost if Pam has 8 blinds installed 59 RESCUE A boater radioed for a helicopter to pick up a sick crew member At that time, the boat and helicopter were at the positions shown How long will it take to reach the boat? Let t represent the time Evaluate c = b for b = 8 It should take the helicopter 2 hours to reach the boat So, it would cost Pam $77 to have 8 blinds installed 59 RESCUE A boater radioed for a helicopter to pick up a sick crew member At that time, the boat and helicopter were at the positions shown How long will it take to reach the boat? Solve each open sentence 60 x + 3 = 10 Case1: Case2: Let t represent the time { 13, 7} 61 2x 8 = 6 Case1: It should take the helicopter 2 hours to reach the boat Solve each open sentence 60 x + 3 = 10 Page 12 Case2:

30 5-2 Solving { 13, 7}Inequalities by Multiplication and Division 61 2x 8 = x 7 = 9 + 4x Case1: Case2: x = 4x 8 {1, 7} 62 3x + 1 = 2 means the distance between 3x and 1 is 2 Since distance cannot be negative, the solution is the empty set Solve each equation 63 4y + 11 = (6w 3) = 3w x 7 = 9 + 4x 67 Page 13

31 5-2 Solving Inequalities by Multiplication and Division Page 14

32 5-3 Solving Multi-Step Inequalities 1 CANOEING If four people plan to use the canoe with 60 pounds of supplies, write and solve an inequality to find the allowable average weight per person The canoe holds a maximum of 800 pounds The weight of 4 people plus 60 pounds is to be in the canoe Let n represent the average weight of each person Write an inequality The solution set is {x x 28} So, 2 is the greatest number of CDs Rita can buy Solve each inequality Graph the solution on a number line 3 6h The solution set is {h h 7} 4 3 r+9 The solution set is {n n 185} So, the average allowable weight per person is at most 185 pounds 2 SHOPPING Rita is ordering a movie for $1195 and a few CDs She has $50 to spend Shipping and sales tax will be $10 If each CD costs $999, write and solve an inequality to find the greatest number of CDs that she can buy Rita has $50 to spend on an $1195 movie plus a $10 fee plus an unknown number of $999 CDs Let x represent the maximum number of CDs she can buy Write an inequality The solution set is {r r 18} 5 3x + 7 > 43 The solution set is {x x 28} So, 2 is the greatest number of CDs Rita can buy The solution set is {x x < 12} Solve each inequality Graph the solution on a number line 3 6h m 17 < 6m + 25 Page 1

33 The solution set is {x x < 12} 5-3 Solving Multi-Step Inequalities 6 4m 17 < 6m + 25 So, the solution checks 8 Negative three times a number plus 4 is less than five times the number plus eight Let n = the number The solution set is {m m > 21} Define a variable, write an inequality, and solve each problem Then check your solution 7 Four times a number minus 6 is greater than eight plus two times the number Let n = the number The solution set is To check this answer, substitute a number greater than into the original inequality Let n = 0 So, the solution checks The solution set is {n n > 7} To check this answer, substitute a number greater than 7 into the original inequality Let n = 8 Solve each inequality Graph the solution on a number line 9 6 3(5v 2) So, the solution checks 8 Negative three times a number plus 4 is less than five times the number plus eight The solution set is {v v 0} Let n = the number To check this answer, substitute a number greater than or equal to 0 into the original inequality LetPage v=2 1

34 5-3 Solving Multi-Step Inequalities So, the solution checks Solve each inequality Graph the solution on a number line 9 6 3(5v 2) So, the solution checks 10 5(g + 4) > 3(g 4) The solution set is {v v 0} To check this answer, substitute a number greater than or equal to 0 into the original inequality Let v = 1 So, the solution checks The solution set is {g g < 1} To check this answer, substitute a number less than 1 into the original inequality Let g = 2 So, the solution checks x 9 + 2(1 4x) 10 5(g + 4) > 3(g 4) Since the inequality results in a false statement, the solution set is the empty set, The solution set is {g g < 1} To check this answer, substitute a number less than 1 into the original inequality Let g = 2 Solve each inequality Graph the solution on a number line 12 5b 1 11 Page 3

35 Since the inequality results in a false statement, the solution set is the empty set, The solution set is {a a < 3} 5-3 Solving Multi-Step Inequalities Solve each inequality Graph the solution on a number line 12 5b m+7 The solution set is {b b 2} The solution set is {m m 40} > a > 6 The solution set is {a a < 3} 14 9 m+7 The solution set is {w w > 56} 16 a The solution set is {a a 1} The solution set is {m m 40} Page < 7 10w > 6

36 The solution set is {w w > 56} The solution set is {z z 9} 5-3 Solving Multi-Step Inequalities 16 a p + 6 < 12 The solution set is {a a 1} < 7 10w The solution set is 20 3b b The solution set is {w w < 3} The solution set is {b b 1} 21 15h + 30 < 10h 45 The solution set is {z z 9} 19 p + 6 < 12 The solution set is {h h < 15} Page 5

37 The solution set is {b b 1} 5-3 Solving Multi-Step Inequalities 21 15h + 30 < 10h 45 So, the solution checks 23 Two thirds of a number added to six is at least twenty-two Let n = the number The solution set is {h h < 15} Define a variable, write an inequality, and solve each problem Check your solution 22 Three fourths of a number decreased by nine is at least forty-two The solution set is {n n 24} To check this answer, substitute a number greater than or equal to 24 into the original inequality Let n = 24 Let n = the number So, the solution checks 24 Seven tenths of a number plus 14 is less than fortynine Let n = the number The solution set is {n n 68} To check this answer, substitute a number greater than or equal to 68 into the original inequality Let n = 72 So, the solution checks 23 Two thirds of a number added to six is at least twenty-two Let n = the number The solution set is {n n < 50} To check this answer, substitute a number less than 50 into the original inequality Let n = 10 Page 6

38 5-3 Solving Multi-Step Inequalities So, the solution checks 24 Seven tenths of a number plus 14 is less than fortynine Let n = the number The solution set is {n n < 50} To check this answer, substitute a number less than 50 into the original inequality Let n = 10 So, the solution checks 25 Eight times a number minus twenty-seven is no more than the negative of that number plus eighteen Let n = the number The solution set is {n n 5} To check this answer, substitute a number less than or equal to 5 into the original inequality Let n = 1 So, the solution checks 26 Ten is no more than 4 times the sum of twice a number and three So, the solution checks 25 Eight times a number minus twenty-seven is no more than the negative of that number plus eighteen Let n = the number Let n = the number The solution set is {n n 5} To check this answer, substitute a number less than or equal to 5 into the original inequality Let n = 1 The solution set is To check this answer, substitute a number greater than or equal to = 1 into the original inequality Let n Page 7

39 5-3 Solving Multi-Step Inequalities So, the solution checks 26 Ten is no more than 4 times the sum of twice a number and three Let n = the number 27 Three times the sum of a number and seven is greater than five times the number less thirteen Let n = the number The solution set is To check this answer, substitute a number greater than or equal to So, the solution checks The solution set is {n n < 17} To check this answer, substitute a number less than 17 into the original inequality Let n = 1 into the original inequality Let n = 1 So, the solution checks So, the solution checks 27 Three times the sum of a number and seven is greater than five times the number less thirteen Let n = the number 28 The sum of nine times a number and fifteen is less than or equal to the sum of twenty-four and ten times the number Let n = the number The solution set is {n n 9} To check this answer, substitute a number greater than or equal to 9 into the original inequality Let n = 1 The solution set is {n n < 17} To check this answer, substitute a number less than Page 8

40 5-3 Solving Multi-Step Inequalities So, the solution checks 28 The sum of nine times a number and fifteen is less than or equal to the sum of twenty-four and ten times the number So, the solution checks Solve each inequality Check your solution 29 3(7n + 3) < 6n Let n = the number The solution set is {n n 9} To check this answer, substitute a number greater than or equal to 9 into the original inequality Let n = 1 The solution set is To check this answer, substitute a number greater than into the original inequality Let n = 1 So, the solution checks Solve each inequality Check your solution 29 3(7n + 3) < 6n So, the solution checks (a 7) + 9 The solution set is To check this answer, substitute a number greater than into the original inequality Let n = 1 The solution set is {a a 11} To check this answer, substitute a number less than or equal to 11 into the original inequality Let n = 7 Page 9

41 5-3 Solving Multi-Step Inequalities So, the solution checks Since all values of x make the inequality true, all real numbers are the solution {b b is a real number} (a 7) t 2(t + 3) + 2 The solution set is {a a 11} To check this answer, substitute a number less than or equal to 11 into the original inequality Let n = 7 The solution set is {t t 1} To check this answer, substitute a number greater than or equal to 1 into the original inequality Let t = 1 So, the solution checks 31 2y + 4 > 2(3 + y) So, the solution checks 34 8a + 2(1 5a) 20 Since the inequality results in a false statement, the solution set is the empty set, ø 32 3(2 b) < 10 3(b 6) The solution set is {a a 9} To check this answer, substitute a number greater than or equal to 9 into the original inequality Let a = 1 Since all values of x make the inequality true, all real numbers are the solution {b b is a real number} t 2(t + 3) + 2 So, the solution checks Page 10 Define a variable, write an inequality, and solve each problem Then interpret your solution

42 5-3 Solving Multi-Step Inequalities So, the solution checks 34 8a + 2(1 5a) 20 The solution set is {s s > 375,000} The amount of sales needed to have an annual income greater than $65,000 must be more than $375, ANIMALS Keith s dog weighs 90 pounds A healthy weight for his dog would be less than 75 pounds If Keith s dog can lose an average of 125 pounds per week on a certain diet, after how long will the dog reach a healthy weight? Let w = the number of weeks The solution set is {a a 9} To check this answer, substitute a number greater than or equal to 9 into the original inequality Let a = 1 The solution set is {w w > 12} It will take more than 12 weeks for the dog to reach a healthy weight 37 Solve 6(m 3) > 5(2m + 4) Show each step and justify your work So, the solution checks Define a variable, write an inequality, and solve each problem Then interpret your solution 35 CARS A car salesperson is paid a base salary of $35,000 a year plus 8% of sales What are the sales needed to have an annual income greater than $65,000? Let s = the amount of sales made 6(m 3) > 5(2m + 4) 6m 18 > 10m m 18 6m > 10m m 18 > 4m > 4m > 4m 95 > m Original inequality Distributive Property Subtract 6m from each side Simplify Subtract 20 from each side Simplify Divide each side by 4 Simplify The solution set is {m m < 95} 38 Solve 8(a 2) 10(a + 2) Show each step and justify your work The solution set is {s s > 375,000} The amount of sales needed to have an annual income greater than $65,000 must be more than $375, ANIMALS Keith s dog weighs 90 pounds A healthy weight for his dog would be less than 75 pounds If Keith s dog can lose an average of 125 pounds per week on a certain diet, after how long will the dog reach a healthy weight? 8(a 2) 10(a + 2) 8a 16 10a a 16 8a 10a a 16 2a a a 18 a Original inequality Distributive Property Subtract 8a from each side Simplify Subtract 20 from each side Simplify Divide each side by 2 Simplify The solution set is{a a 18} Page 11

43 The solution set is {t t 187} 95 > m Simplify 5-3 Solving Multi-Step Inequalities The solution set is {m m < 95} 38 Solve 8(a 2) 10(a + 2) Show each step and justify your work 8(a 2) 10(a + 2) 8a 16 10a a 16 8a 10a a 16 2a a a 18 a b 40 ICE CREAM Benito has $6 to spend A sundae costs $325 plus $065 per topping Write and solve an inequality to find how many toppings he can order Original inequality Distributive Property Subtract 8a from each side Simplify Subtract 20 from each side Simplify Divide each side by 2 Let t = the number of toppings Benito can order Simplify Benito can order 4 or fewer toppings The solution set is{a a 18} 39 MUSICAL A high school drama club is performing a musical to benefit a local charity Tickets are $5 each They also received donations of $565 They want to raise at least $1500 a Write an inequality that describes this situation Then solve the inequality b Graph the solution a Let t = the number of tickets that need to be sold 41 SCIENCE The normal body temperature of a camel is 977 F in the morning If it has had no water by noon, its body temperature can be greater than 104 F a Write an inequality that represents a camel s body temperature at noon if the camel had no water b If C represents degrees Celsius, then Write and solve an inequality to find the camel s body temperature at noon in degrees Celsius a t > 104 b The solution set is {t t 187} b The camel s body temperature at noon is greater than 40 C 40 ICE CREAM Benito has $6 to spend A sundae costs $325 plus $065 per topping Write and solve an inequality to find how many toppings he can order Let t = the number of toppings Benito can order 42 NUMBER THEORY Find all sets of three consecutive positive even integers with a sum no greater than 36 Page 12

44 Create sets of 4 positive odd integers that start with numbers less than 75 1, 3, 5, 7; 3, 5, 7, 9; 5, 7, 9, 11; 7, 9, 11, Solving Multi-Step Inequalities The camel s body temperature at noon is greater than 40 C 42 NUMBER THEORY Find all sets of three consecutive positive even integers with a sum no greater than 36 Solve each inequality Check your solution 44 2(x 4) 2 + 3(x 6) The solution set is {x x 8} To check this answer, substitute a number greater than or equal to 8 into the original inequality Let x = 9 Create sets of 3 positive even integers that start with numbers less than or equal to 10 2, 4, 6; 4, 6, 8; 6, 8, 10; 8, 10, 12; 10, 12, NUMBER THEORY Find all sets of four consecutive positive odd integers whose sum is less than 42 So, the solution checks 45 5x + 2 Create sets of 4 positive odd integers that start with numbers less than 75 1, 3, 5, 7; 3, 5, 7, 9; 5, 7, 9, 11; 7, 9, 11, 13 Solve each inequality Check your solution 44 2(x 4) 2 + 3(x 6) The solution set is The solution set is {x x 8} To check this answer, substitute a number greater than or equal to 8 into the original inequality Let x = 9 To check this answer, substitute a number greater than or equal to 5 into the original inequality Let x = Page 13

45 5-3 Solving Multi-Step Inequalities So, the solution checks 45 So, the solution checks 46 56z + 15 < 25z 47 5x + 2 The solution set is {z z < 2} To check this answer, substitute a number less than 2 into the original inequality Let z = 10 The solution set is To check this answer, substitute a number greater than or equal to into the original inequality Let x = So, the solution checks 47 07(2m 5) So, the solution checks The solution set is {m m 18} To check this answer, substitute a number greater than or equal to 18 into the original inequality Let m = z + 15 < 25z 47 So, the solution checks The solution set is {z z < 2} To check this answer, substitute a number less than 2 into the -original esolutions Manual Powered inequality by Cognero Let z = 10 GRAPHING CALCULATOR Use a graphing calculator to solve each inequality 48 3x + 7 > 4x + 9 Page 14 Use a graphing calculator to solve the inequality

46 [ 5, 15] scl: 1 by [ 2,100] scl: 10 At x = 8, Y1 = Y2 or 13x 11 = 7x + 37 When x 8, 13x 11 7x + 37 Thus the solution set is {x x 8} 5-3 Solving Multi-Step Inequalities So, the solution checks GRAPHING CALCULATOR Use a graphing calculator to solve each inequality 48 3x + 7 > 4x (x 3) < 3(2x + 2) Use a graphing calculator to solve the inequality Let Y1 = 2(x 3) and Y2 = 3(2x + 2) Use the intersect function from the CALC menu to find the intersection of the two graphs Use a graphing calculator to solve the inequality Let Y1 = 2x + 7 and Y2 = 4x + 9 Use the intersect function from the CALC menu to find the intersection of the two graphs [ 5, 5] scl: 1 by [ 17, 3] scl: 2 At x = 3, Y1 = Y2 or 2(x 3)= 3(2x + 2) When x > [ 7, 3] scl: 1 by [ 5, 5] scl: 1 3, 2(x 3)< 3(2x + 2) Thus the solution set is {x x > 3} At x = 2, Y1 = Y2 or 2x + 7 = 4x + 9 When x < 2, 2x + 7 > 4x + 9 Thus the solution set is {x x < 2} x 11 7x + 37 x 9 < 2x Use a graphing calculator to solve the inequality Use a graphing calculator to solve the inequality Let Y1 = 13x 11and Y2 = 7x + 37 Use the intersect function from the CALC menu to find the intersection of the two graphs Let Y1 = x 9 and Y2 = 2x Use the intersect function from the CALC menu to find the intersection of the two graphs [ 9, 1] scl: 1 by [ 18, 2] scl: 2 [ 5, 15] scl: 1 by [ 2,100] scl: 10 At x = 6, Y1 = Y2 or x 9 = 2x When x > 6, At x = 8, Y1 = Y2 or 13x 11 = 7x + 37 When x 8, x 9 < 2x Thus the solution set is {x x > 6} 13x 11 7x + 37 Thus the solution set is {x x 8} 50 2(x 3) < 3(2x + 2) Use a graphing calculator to solve the inequality Let Y1 = 2(x 3) and Y2 = 3(2x + 2) Use the intersect function from the CALC menu to find the esolutions Manual - Powered by Cognero intersection of the two graphs 52 Use a graphing calculator to solve the inequality Let Y1 = and Y2 = x 22 Use the Page 15 intersect function from the CALC menu to find the intersection of the two graphs

47 At x = 6, Y1 = Y2 or x x 9 = 2x When x > 6, 5-3 Solving Inequalities x 9Multi-Step < 2x Thus the solution set is {x x > 6} 52 is 53 Use a graphing calculator to solve the inequality Let Y1 = and Y2 = x 22 Use the intersect function from the CALC menu to find the intersection of the two graphs, Thus the solution set (4x + 3) x+2 Use a graphing calculator to solve the inequality Let Y1 = (4x + 3) and Y2 = x + 2 Use the intersect function from the CALC menu to find the intersection of the two graphs [ 10, 10] scl: 1 by [ 10, 10] scl: 1 [ 35, 5] scl: 2 by [ 55, 5] scl: 4, Y1 = Y2 or At x = x, is 53 When Thus the solution set x+2 Use a graphing calculator to solve the inequality Let Y1 = (4x + 3) and Y2 = x + 2 Use the intersect function from the CALC menu to find the intersection of the two graphs [ 10, 10] scl: 1 by [ 10, 10] scl: 1 At x = 15, Y1 = Y2 or 15, (4x + 3) 15, (4x + 3) (4x + 3) = x + 2 When x x + 2 Thus the solution set is {x x 15} (4x + 3) At x = 15, Y1 = Y2 or (4x + 3) = x + 2 When x 54 MULTIPLE REPRESENTATIONS In this problem, you will solve compound inequalities A number x is greater than 4, and the same number is less than 9 a NUMERICAL Write two separate inequalities for the statement b GRAPHICAL Graph the solution set for the first inequality in red Graph the solution set for the second inequality in blue Highlight the portion of the graph in which the red and blue overlap c TABULAR Make a table using ten points from your number line, including points from each section Use one column for each inequality and a third column titled Both are True Complete the table by writing true or false d VERBAL Describe the relationship between the colored regions of the graph and the chart e LOGICAL Make a prediction of what the graph of 4 < x < 9 looks like a x > 4; x < 9 x + 2 Thus the solution set is {x x 15} 54 MULTIPLE REPRESENTATIONS In this problem, you will solve compound inequalities A number x is greater than 4, and the same number is b c Page 16

48 At x = 15, Y1 = Y2 or (4x + 3) = x + 2 When x 15, (4x + 3) x + 2 Thus the solution set is 5-3 Solving Multi-Step Inequalities {x x 15} 54 MULTIPLE REPRESENTATIONS In this problem, you will solve compound inequalities A number x is greater than 4, and the same number is less than 9 a NUMERICAL Write two separate inequalities for the statement b GRAPHICAL Graph the solution set for the first inequality in red Graph the solution set for the second inequality in blue Highlight the portion of the graph in which the red and blue overlap c TABULAR Make a table using ten points from your number line, including points from each section Use one column for each inequality and a third column titled Both are True Complete the table by writing true or false d VERBAL Describe the relationship between the colored regions of the graph and the chart e LOGICAL Make a prediction of what the graph of 4 < x < 9 looks like a x > 4; x < 9 b c statement are in the red section The points that make both statements true are in the highlighted section e The graph would be the highlighted section of the number line 55 REASONING Explain how you could solve 3p without multiplying or dividing each side by a negative number Add 3p and 2 to each side The inequality becomes 9 3p Then divide each side by 3 to get 3 p 56 CHALLENGE If ax + b < ax + c is true for all values of x, what will be the solution of ax + b > ax + c? Explain how you know The solution of ax + b > ax + c would be the empty set, ø Subtracting ax from each side of ax + b < ax + c leaves b < c If the original inequality has infinitely many solutions, then b must be a real number that is less than c Subtracting ax from each side of the second inequality ax + b > ax + c leaves b > c If b is always less than c, then this inequality has no solutions 57 CHALLENGE Solve each inequality for x Assume that a > 0 a b c a b d The points that make x > 4 a true statement are in the blue section The points that make x < 9 a true statement are in the red section The points that make both statements true are in the highlighted section e The graph would be the highlighted section of the number line 55 REASONING Explain how you could solve 3p without multiplying or dividing each side by a negative number c Page 17

49 infinitely many solutions, then b must be a real number that is less than c Subtracting ax from each side of the second inequality ax + b > ax + c leaves b > c IfMulti-Step b is always Inequalities less than c, then this inequality 5-3 Solving has no solutions 57 CHALLENGE Solve each inequality for x Assume that a > 0 a b 58 WHICH ONE DOESN T BELONG? Name the inequality that does not belong Explain c a b 4y + 9 > 3 does not belong It is the only inequality that does not have a solution set of {y y > 3} c 59 WRITING IN MATH Explain when the solution set of an inequality will be the empty set or the set of all real numbers Show an example of each 58 WHICH ONE DOESN T BELONG? Name the inequality that does not belong Explain Sample answer: The solution set for an inequality that results in a false statement is the empty set, as in 12 < 15 The solution set for an inequality in which any value of x results in a true statement is all real numbers, as in What is the solution set of the inequality 4t + 2 < 8t (6t 10)? A {t t < 65} B {t t > 65} C {t t < 4} D {t t > 4} 4y + 9 > 3 does not belong It is the only inequality that does not have a solution set of {y y > 3} Page 18

50 Sample answer: The solution set for an inequality that results in a false statement is the empty set, as in 12 < 15 The solution set for an inequality in which any value of Multi-Step x results in ainequalities true statement is all real 5-3 Solving numbers, as in What is the solution set of the inequality 4t + 2 < 8t (6t 10)? A {t t < 65} B {t t > 65} C {t t < 4} D {t t > 4} The solution set is {t t < 4}, so the correct choice is C 61 GEOMETRY The section of Liberty Ave between 5th St and King Ave is temporarily closed Traffic is being detoured right on 5th St, left on King Ave and then back on Liberty Ave How long is the closed section of Liberty Ave? F 100 ft G 120 ft H 144 ft J 180 ft The solution set is {t t < 4}, so the correct choice is C 61 GEOMETRY The section of Liberty Ave between 5th St and King Ave is temporarily closed Traffic is being detoured right on 5th St, left on King Ave and then back on Liberty Ave How long is the closed section of Liberty Ave? The closed section of Liberty Ave is 120 feet, so the correct choice is G 62 SHORT RESPONSE Rhiannon is paid $52 for working 4 hours At this rate, how many hours will it take her to earn $845? F 100 ft G 120 ft H 144 ft J 180 ft Find the rate Solve for the number of hours using r = 13 It will take Rhiannon 65 hours of work to earn $ GEOMETRY Classify the triangle The closed section of Liberty Ave is 120 feet, so the correct choice is G 62 SHORT RESPONSE Rhiannon is paid $52 for working 4 hours At this rate, how many hours will it take her to earn $845? Page 19 A right

51 5-3 Solving Multi-Step Inequalities It will take Rhiannon 65 hours of work to earn $ GEOMETRY Classify the triangle So, the solution checks 65 12b > 48 A B C D right parallel obtuse equilateral The solution set is{b b > 4} To check this answer, substitute a number greater than 4 into the original inequality Let b = 0 All three sides of the triangle are equal in length, so it is equilateral The correct choice is D Solve each inequality Check your solution (Lesson 5-2) 64 5 So, the solution checks 66 t 30 The solution set is {y y 10} To check this answer, substitute a number less than or equal to 10 into the original inequality Let y = 30 The solution set is {t t 45} To check this answer, substitute a number greater than or equal to 45 into the original inequality Let t = 48 So, the solution checks 65 12b > 48 So, the solution checks Solve each inequality Check your solution, and graph it on a number line 67 6 h > 8 The solution set is{b b > 4} To check this answer, substitute a number greater than 4 into the original inequality Let b = 0 Page 20

52 5-3 Solving Multi-Step Inequalities So, the solution checks Solve each inequality Check your solution, and graph it on a number line 67 6 h > 8 So, the solution checks m The solution set is {h h < 14} To check this answer, substitute a number less than 14 into the original inequality Let h = 10 The solution set is {m m 1} To check this answer, substitute a number greater than or equal to 1 into the original inequality Let m = 4 So, the solution checks So, the solution checks 68 p 9 < 2 Solve each equation by graphing Verify your answer algebraically 70 2x 7 = 4x + 9 The solution set is {p p < 11} To check this answer, substitute a number less than 11 into the original inequality Let p = 10 So, the solution checks m x = 7x 11 Page 21

53 So, the solution checks 5-3 Solving Multi-Step Inequalities Solve each equation by graphing Verify your answer algebraically 70 2x 7 = 4x x = 7x (x 3) = 5x x = 7x (x 3) = 5x + 12 Page THEME PARKS In a recent year, 709 million people visited the top 5 theme parks in North America That represents an increase of

54 5-3 Solving Multi-Step Inequalities 73 THEME PARKS In a recent year, 709 million people visited the top 5 theme parks in North America That represents an increase of about 114% in the number of visitors from the prior year About how many people visited the top 5 theme parks in North America in the prior year? 76 f (c + 3) 77 COSMETOLOGY On average, a barber received a tip of $4 for each of 12 haircuts Write and evaluate an expression to determine the total amount that she earned The total amount the hair stylist earned is $40740 About 701 million people visited theme parks in North America in the previous year 2 If f (x) = 4x 3 and g(x) = 2x + 5, find each value 74 f ( 2) Graph each set of numbers on a number line 78 { 4, 2, 2, 4} Draw a solid dot for each number in the set 79 { 3, 0, 1, 5} Draw a solid dot for each number in the set 75 g(2) 5 80 {integers less than 3} Draw a solid dot for several representative numbers in the set 81 {integers greater than or equal to 2} 76 f (c + 3) 77 COSMETOLOGY On average, a barber received Draw a solid dot for several representative numbers in the set 82 {integers between 3 and 4} Page 23

55 Draw a solid dot for several representative numbers in the set 5-3 Solving Multi-Step Inequalities 82 {integers between 3 and 4} Draw a solid dot for each number in the set 83 {integers less than 1} Draw a solid dot for several representative numbers in the set Page 24

56 Notice that the two inequalities overlap at a > 5, so the solution set is {a a > 5} To graph the solution set, graph a > Solving Compound Inequalities Solve each compound inequality Then graph the solution set 1 4 p 8 and p g + 4 < 7 and and The solution set is {g 2 g < 3} To graph the solution set, graph 2 g and graph g < 3 Then find the intersection The solution set is {p 12 p 16} To graph the solution set, graph 12 p and graph p 16 Then find the intersection 2 r + 6 < 8 or r 3 > 10 or The solution set is {r r < 14 or r > 7}, Notice that the graphs do not intersect To graph the solution set, graph r < 14 and graph r > 7 Then find the union 3 4a or a > 5 or 5 BIKES The recommended air pressure for the tires of a mountain bike is at least 35 pounds per square inch (psi), but no more than 80 pounds per square inch If a bike s tires have 24 pounds per square inch, what is the recommended range of air that should be put into the tires? Let x be the air pressure The phrase at least means the same as greater than or equal to The phrase no more than means the same as less than or equal to The word but indicates that the problem represents an intersection and x x and x 11 x 56 So, an inequality that represents the range of recommended air pressure for tires is 11 psi x 56 psi Solve each compound inequality Then graph the solution set 6 f 6 < 5 and f 4 2 and Notice that the two inequalities overlap at a > 5, so the solution set is {a a > 5} To graph the solution set, graph a > 5 The solution set is {f 6 f < 11} To graph the solution set, graph 6 f and graph f < 11 Then find the intersection 7 n and n g + 4 < 7 and and Page 1 The solution set is {n 12 n 7}

57 The solution set is {f 6 f < 11} To graph the solution set, graph 6 f and graph f < 11 Then find the intersection 5-4 Solving Compound Inequalities 7 n and n Notice that the graphs do not intersect To graph the solution set, graph t 1 and t < 1 Then find the union 10 5 < 3p and and The solution set is {n 12 n 7} To graph the solution set, graph 12 n and graph n 7 Then find the intersection The solution set is {p 4 < p 5} To graph the solution set, graph 4 < p and graph p 5 Then find the intersection 8 y 1 7 or y + 3 < 1 or c + 4 < 18 and The solution set is {y y 8 or y < 4} Notice that the graphs do not intersect To graph the solution set, graph y 8 and graph y < 4 Then find the union The solution set is {c 1 c < 2} To graph the solution set, graph 1 c and graph c < 2 Then find the intersection 9 t or t 9 < 10 or 12 5h 4 6 and 7h + 11 < 32 and The solution set is {t t 1 or t < 1} Notice that the graphs do not intersect To graph the solution set, graph t 1 and t < 1 Then find the union 10 5 < 3p and The solution set is {h 2 h < 3} To graph the solution set, graph 2 h and graph h < 3 Then find the intersection m 2 or 5 3m 13 The solution set is {p 4 < p 5} To graph solution graph 4 < p and graph p esolutions Manualthe - Powered by set, Cognero 5 Then find the intersection or Page 2

58 The solution set is {h 2 h < 3} To graph the solution set, graph 2 h and graph h < 3 Then find the intersection 5-4 Solving Compound Inequalities m 2 or 5 3m 13 or Notice that the two inequalities overlap at y < 3, so the solution set is {y y < 3} To graph the solution set, graph y < 3 16 SPEED The posted speed limit on an interstate highway is shown Write an inequality that represents the sign Graph the inequality Notice that the two inequalities overlap and all real numbers are solutions The solution set is {m m is a real number} To graph the solution set, graph all points 14 4a and 10 < 6a 14 and Sample answer: Let r = rate of speed The lowest speed you can go is 40 mph, while the highest speed is 70 mph Therefore, the inequality is 40 r NUMBER THEORY Find all sets of two consecutive positive odd integers with a sum that is at least 8 and less than 24 Sample answer: Let x = the smaller of two consecutive odd numbers, then 8 2x Notice that the two inequalities do not overlap So, the solution set is empty, ø The graph is also empty List every combination of numbers in which the smaller number is 3 x 11 3, 5; 5, 7; 7, 9; 9, 11; 11, y or 3y + 4 < 5 or Write a compound inequality for each graph 18 Notice that the two inequalities overlap at y < 3, so the solution set is {y y < 3} To graph the solution set, graph y < 3 16 SPEED The posted speed limit on an interstate highway is shown Write an inequality that represents the sign Graph the inequality This graph represents an intersection Both endpoints are closed circles which include the endpoints The compound inequality is 1 x 4 19 This graph represents an intersection The left endpoint is an open circle which represents greater than The right endpoint is a closed circle which represents less than or equal to The compound Page 3 inequality is 3 < x 2

59 This graph represents an intersection Both endpoints are closed circles which include the endpoints The 5-4 Solving Compound compound inequality isinequalities 1 x 4 19 This graph represents an intersection The left endpoint is an open circle which represents greater than The right endpoint is a closed circle which represents less than or equal to The compound inequality is 3 < x 2 The graphs do not intersect, so it represents a union The endpoint on the right is an open endpoint, which represents greater than The endpoint on the left is only a point The inequalities are x -3 or x > 0 Solve each compound inequality Then graph the solution set 24 3b + 2 < 5b 6 2b + 9 and 20 The graphs do not intersect, so it represents a union The endpoint on the left is an open endpoint, which represents less than The endpoint on the right is a closed endpoint, which represents greater than or equal to The inequalities are x < 0 or x 3 The solution set is {b 4 < b 5} To graph the solution set, graph 4 < b and graph b 5 Then find the intersection 21 The graphs do not intersect, so it represents a union Both endpoints are open, so the endpoints are not included The inequalities are x < 4 or x a + 3 6a 1 > 3a 10 and 22 The graphs do not intersect, so it represents a union The endpoint on the left is a closed endpoint, which represents less than or equal to The endpoint on the right is only a point The inequalities are x 3 or x 6 The solution set is To graph the solution set, graph 23 4 < b and grap Then find the intersection The graphs do not intersect, so it represents a union The endpoint on the right is an open endpoint, which represents greater than The endpoint on the left is only a point The inequalities are x -3 or x > m 7 < 17m or 6m > 36 or Solve each compound inequality Then graph the solution set 24 3b + 2 < 5b 6 2b + 9 and The solution set is {m m < 6 or m > 1} Notice that the graphs do not intersect To graphpage the 4 solution set, graph m < 6 and graph m > 1 Then find the union

60 To graph the solution set, graph 4 < b and grap Then find the intersection 5-4 Solving Compound Inequalities 26 10m 7 < 17m or 6m > 36 or The solution set is {m m < 6 or m > 1} Notice that the graphs do not intersect To graph the solution set, graph m < 6 and graph m > 1 Then find the union 27 5n 1 < 16 or 3n 1 < 8 or The solution set is {n n < 3 or n > 3} To graph the solution set, graph all points except for 3 28 COUPON Juanita has a coupon for 10% off any digital camera at a local electronics store She is looking at digital cameras that range in price from $100 to $250 a How much are the cameras after the coupon is used? b If the tax amount is 65%, how much should Juanita expect to spend? a The least expensive camera is $100 The 10% coupon is worth or $10 After the coupon is used, the least expensive camera is $100 $10 or $90 The most expensive camera is $250 The 10% coupon is worth or $25 After the coupon is used, the most expensive camera is $250 $25 or $225 The cameras are between $90 and $225 inclusive b The most expensive camera is $90 after the coupon, so add 65% tax Notice that the two inequalities overlap, but the point 3 is not included The solution set is {n n < 3 or n > 3} To graph the solution set, graph all points except for 3 28 COUPON Juanita has a coupon for 10% off any digital camera at a local electronics store She is looking at digital cameras that range in price from $100 to $250 a How much are the cameras after the coupon is used? b If the tax amount is 65%, how much should Juanita expect to spend? a The least expensive camera is $100 The 10% coupon is worth or $10 After the coupon is used, the least expensive camera is $100 $10 or $90 The most expensive camera is $250 The 10% coupon is worth or $25 AfterManual the coupon is by used, the most expensive camera esolutions - Powered Cognero is $250 $25 or $225 So, the least expensive camera will cost $9585 The most expensive camera is $225 after the coupon, so add 65% tax So the most expensive camera will cost $23963 Juanita should expect to spend between $9585 and $23963 inclusive Define a variable, write an inequality, and solve each problem Then check your solution 29 Eight less than a number is no more than 14 and no less than 5 Let n = the number Page 5 The solution set is {n 13 n 22} To check this answer, substitute a number greater than or equal to 13 and less than or equal to 22 into

61 So the most expensive camera will cost $ Solving Compound Juanita should expectinequalities to spend between $9585 and $23963 inclusive Define a variable, write an inequality, and solve each problem Then check your solution 29 Eight less than a number is no more than 14 and no less than 5 Let n = the number The solution set is {n 13 n 22} To check this answer, substitute a number greater than or equal to 13 and less than or equal to 22 into the original inequality Let n = 15 So, the solution checks So, the solution checks 31 The product of 5 and a number is greater than 35 or less than 10 Let n = the number or The solution set is {n n < 7 or n > 2} To check this answer, substitute a number less than 7 into the original inequality Let n = 8 So, the solution checks Now check the second inequality To check this, substitute a number that is greater than 2 into the original inequality Let n = 0, 30 The sum of 3 times a number and 4 is between 8 and 10 Let n = the number So, the solution checks 32 One half a number is greater than 0 and less than or equal to 1 Let n = the number The solution set is {n 4 < n < 2} To check this answer, substitute a number greater than 4 and less than 2 into the original inequality Let n = 0 The solution set is {n 0 < n 2} To check this answer, substitute a number greater than 0 and less than or equal to 2 into the original inequality Let n = 1 So, the solution checks 31 The product of 5 and a number is greater than 35 or less than 10 Page 6

62 5-4 Solving Compound Inequalities So, the solution checks 32 One half a number is greater than 0 and less than or equal to 1 Let n = the number The solution set is {n 0 < n 2} To check this answer, substitute a number greater than 0 and less than or equal to 2 into the original inequality Let n = 1 So, the solution checks 33 SNAKES Most snakes live where the temperature ranges from 75 F to 90 F inclusive Write an inequality to represent temperatures where snakes will not thrive Snakes can live in temperatures between 75 and 90 This means that snakes do not typically live in temperatures lower than 75 or higher than 90 Let t represent the temperatures So, the inequalities can be written as t < 75 or t > FUNDRAISING Yumas is selling gift cards to raise money for a class trip He can earn prizes depending on how many cards he sells So far, he has sold 34 cards How many more does he need to sell to earn a prize in category 4? Yumas has sold 34 cards The lowest number of Snakes can live in temperatures between 75 and 90 This means that snakes do not typically live in temperatures lower than 75 or higher than 90 Let t represent the temperatures So, the inequalities can be written as t < 75 or t > FUNDRAISING Yumas is selling gift cards to raise money for a class trip He can earn prizes depending on how many cards he sells So far, he has sold 34 cards How many more does he need to sell to earn a prize in category 4? Yumas has sold 34 cards The lowest number of cards he can sell to get a category 4 prize is 46, so he needs to sell at least 12 more cards The maximum number of cards he can sell and still get a category 4 prize is 60, which means he needs to sell at most 26 cards This means that he needs to sell between 12 and 26 inclusive 35 TURTLES Atlantic sea turtle eggs that incubate below 23 C or above 33 C rarely hatch Write the temperature requirements in two ways: as a pair of simple inequalities, and as a compound inequality Let t represent the temperature cannot be below 23 is represented by 23 t cannot be above 33 is represented by t 33 This compound inequality can be expressed in two ways: 23 t t and t GEOMETRY The Triangle Inequality Theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side a Write and solve three inequalities to express the relationships among the measures of the sides of the triangle shown b What are four possible lengths for the third side Page 7 of the triangle? c Write a compound inequality for the possible values of x

63 cannot be above 33 is represented by t 33 This compound inequality can be expressed in two ways: 23 t Compound Solving Inequalities 23 t and t GEOMETRY The Triangle Inequality Theorem states that the sum of the measures of any two sides of a triangle is greater than the measure of the third side a Write and solve three inequalities to express the relationships among the measures of the sides of the triangle shown number; therefore x must be greater than 0 The second inequality states that x must be greater than 5, which overlaps with the first inequality, while the third inequality states that x must be less than 13 Therefore, the compound inequality is 5 < x < HURRICANES The Saffir Simpson Hurricane Scale rates hurricanes on a scale from 1 to 5 based on their wind speed a Write a compound inequality for the wind speeds of a category 3 and a category 4 hurricane b What is the intersection of the two graphs of the inequalities you found in part a? b What are four possible lengths for the third side of the triangle? c Write a compound inequality for the possible values of x a 1st inequality: 2nd inequality: 3rd inequality: b Sample answer: Every side of a triangle must be positive, so the inequality x > 5 can be disregarded The other two inequalities show that the third side of the triangle must be greater than 5, but less than 13 Four possibilities are 6, 9, 10, and 11 c The first inequality states that x must be greater than 5; however a length may not be a negative number; therefore x must be greater than 0 The second inequality states that x must be greater than 5, which overlaps with the first inequality, while the third inequality states that x must be less than 13 Therefore, the compound inequality is 5 < x < HURRICANES The Saffir Simpson Hurricane Scale rates hurricanes on a scale from 1 to 5 based on their wind speed a Write a compound inequality for the wind speeds of a category 3 and a category 4 hurricane a Let x represent the wind speed For a category 3: 111 x 130 For a category 4: 131 x 155 b The union of the two graphs is where either of the graphs are So the solution is {x 111 x 155} The intersection is where it overlaps However, these graphs do not overlap, so it is an empty set or 38 MULTIPLE REPRESENTATIONS In this problem, you will investigate measurements The absolute error of a measurement is equal to one half the unit of measure The relative error of a measure is the ratio of the absolute error to the expected measure a TABULAR Copy and complete the table b ANALYTICAL You measured a length of 128 centimeters Compute the absolute error and then write the range of possible measures c LOGICAL To what precision would you have to measure a length in centimeters to have an absolute error of less than 005 centimeters? d ANALYTICAL To find the relative error of an area or volume calculation, add the relative errors of each linear measure If the measures of the sides of a rectangular box are 65 centimeters, 72 centimeters, and 1025 centimeters, what is the relative error of the volume of the box? Page 8

64 area or volume calculation, add the relative errors of each linear measure If the measures of the sides of a rectangular box are 65 centimeters, 72 centimeters, and 1025 centimeters, what is the 5-4 Solving Compound Inequalities relative error of the volume of the box? Neither of them are correct Chloe did not add 5 to 3, and Jonas did not add 5 to 7 They each only added the 5 to one side of the compound inequality, not both 40 CHALLENGE Solve each inequality for x Assume a is constant and a > 0 a b or a b The absolute error is one half the unit measure, a cm; The smallest measurement would be the length minus the absolute error, =1275 The largest measurement would be the length plus the absolute error, =1285 c 005 is the absolute error for 01 ( ) Since it has to be less than 01, the precision must be in the hundredths or to the nearest hundredths place d The relative error of the first linear measure b is The relative error of the second linear measure is or The relative error of the third linear measure is To find the relative error of the volume, add these together: = ERROR ANALYSIS Chloe and Jonas are solving the 3 < 2x 5 < 7 Is either of them correct? Explain your reasoning Neither of them are correct Chloe did not add 5 to 3, and Jonas did not add 5 to 7 They each only added the 5 to one side of the compound inequality, not both 40 CHALLENGE Solve each inequality for x Assume a is Manual constant and a by > 0 esolutions - Powered Cognero a b or 41 OPEN ENDED Create an example of a compound inequality containing or that has infinitely many solutions Answers may vary Sample answer: x 4 or x 4 42 CHALLENGE Determine whether the following statement is always, sometimes, or never true Explain The graph of a compound inequality that involves an or statement is bounded on the left and right by two values of x Sometimes; the graph of x > 2 or x < 5 includes the entire number line 43 WRITING IN MATH Give an example of a Page 9 compound inequality you might encounter at an amusement park Does the example represent an

65 and right by two values of x Sometimes; the graphinequalities of x > 2 or x < 5 includes the 5-4 Solving Compound entire number line So, the correct choice is C 43 WRITING IN MATH Give an example of a compound inequality you might encounter at an amusement park Does the example represent an intersection or a union? 45 GEOMETRY What is the surface area of the rectangular solid? Sample answer: The speed at which a roller coaster runs while staying on the track could represent a compound inequality that is an intersection 44 What is the solution set of the inequality 7 < x + 2 < 4? A {x 5 < x < 6} B {x 5 < x < 2} C {x 9 < x < 2} D {x 9 < x < 6} 2 F 2496 cm G 2784 cm2 2 H 3136 cm J 3712 cm2 The surface area of the rectangular solid can be calculated by finding the area of each of the sides There are 4 sides with these dimensions, so the So, the correct choice is C 2 surface area of the four sides is 4(464) = 1856 cm Another side has an area of 45 GEOMETRY What is the surface area of the rectangular solid? There are two sides with these dimensions, so the 2 surface area of the two sides is 2(64) = 128 cm 2 F 2496 cm G 2784 cm2 2 H 3136 cm J 3712 cm2 2 The total surface area is = 3136 cm So, the correct choice is H 46 GRIDDED RESPONSE What is the next term in the sequence? The surface area of the rectangular solid can be calculated by finding the area of each of the sides The pattern is to add 5 to the numerator and add 3 to the denominator So, the next term should be There are 4 sides with these dimensions, so the 2 surface area of the four sides is 4(464) = 1856 cm Another side has an area of There are two sides with these dimensions, so the 47 After paying a $15 membership fee, members of a video club can rent movies for $2 Nonmembers can rent movies for $4 What is the least number of movies which must be rented for it to be less expensive for members? Page 10 A 9 B 8 C 7

66 The pattern is to add 5 to the numerator and add 3 to the denominator So, the next term should be 5-4 Solving Compound Inequalities 47 After paying a $15 membership fee, members of a video club can rent movies for $2 Nonmembers can rent movies for $4 What is the least number of movies which must be rented for it to be less expensive for members? A 9 B 8 C 7 D 6 So, she will need to babysit at least 22 hours to earn $300 this month 49 MAGAZINES Carlos has earned more than $260 selling magazine subscriptions Each subscription was sold for $12 How many did Carlos sell? Let m represent the number of magazine subscriptions Let m represent the number of movies So, Carlos sold at least 22 subscriptions 50 PUNCH Raquel is mixing lemon lime soda and a fruit juice blend that is 45% juice If she uses 3 quarts of soda, how many quarts of fruit juice must be added to produce punch that is 30% juice? Let x represent the number of quarts of fruit juice Raquel must add Set up an equation where the amount of juice in the mixture is equal to the the amount of juice in the soda plus the amount of juice in the blend This means that they must rent more than 75 movies, so choices C and D can be eliminated The question asks for the least number of movies that can be rented, so the correct choice is B 48 BABYSITTING Marilyn earns $150 per month delivering newspapers plus $7 an hour babysitting If she wants to earn at least $300 this month, how many hours will she have to babysit? Let h represent the number of hours that Marilyn must babysit So, Raquel needs to add 6 quarts of fruit juice Solve each proportion If necessary, round to the nearest hundredth So, she will need to babysit at least 22 hours to earn $300 this month 49 MAGAZINES Carlos has earned more than $260 selling magazine subscriptions Each subscription was sold for $12 How many did Carlos sell? Let m represent the number of magazine subscriptions esolutions Manual - Powered by Cognero 51 Page 11

67 5-4 Solving Compound Inequalities So, Raquel needs to add 6 quarts of fruit juice Solve each proportion If necessary, round to the nearest hundredth Determine whether each relation is a function Explain 57 Page 12 In a function, there is exactly one output for every

68 =5+0 =5 5-4 Solving Compound Inequalities Determine whether each relation is a function Explain 2 =4 Subtract Additive Identity = In a function, there is exactly one output for every input 2 is paired with 5, 6 is paired with 0, 10 is paired with 5, and 7 is paired with 0, so this relation is a function Subtract = =1 Multiplicative Inverse In a function, there is exactly one output for every input 5 is paired with 10, 2 is paired with 7, 3 is paired with 5, and 2 is paired with 3 Notice that 2 is paired with both 7 and 3 This relation is not a function Multiply Subtract Multiply 59 In a function, there is exactly one output for every input 4 is paired with 11, 2 is paired with 7, 1 is paired with 3, and 4 is paired with 1 Notice that 4 is paired with both 11 and 1 This relation is not a function 60 Multiplicative Inverse Add Solve each equation 64 4p 2 = 6 In a function, there is exactly one output for every input 2 is paired with 7, 5 is paired with 3, 7 is paired with 6, and 10 is paired with 7 So, this relation is a function Evaluate each expression = 5p + 3 = 5 + (4 4) =5+0 =5 2 2 =4 Subtract Additive Identity Page = 1 +

69 5-4 Solving Compound Inequalities = 5p = = = a 8 = y 16 = = 11k = 4c 8 69 = 17 esolutions Manual - Powered by Cognero Page 14

70 Because absolute values are positive, all values of c + 2 will be greater than 2 The solution set is {c c is a real number} 5-5 Inequalities Involving Absolute Value Solve each inequality Then graph the solution set 1 a 5 < 3 Case 1 a 5 is positive and 5 n Case 1 n + 5 is positive or Case 2 n + 5 is negative Case 2 a 5 is negative The solution set is {n n 8 or n 2} The solution set is {a 2 < a < 8} 6 p 2 8 Case 1 p 2 is positive 2 u + 3 < 7 Case 1 u + 3 is positive and or Case 2 p 2 is negative Case 2 u + 3 is negative The solution set is {p p 6 or p 10} The solution set is {u 10 < u < 4} 7 FINANCIAL LITERACY Jerome bought stock in $7085 The stock price ranged from within $075 of Find the range of prices for which the stock could tra 3 t t + 4 cannot be negative So it is not possible for t + 4 to be less than or equal to 2 Therefore, there is no solution, and the solution set is the empty set, Ø The graph is also empty m Case 1 m 7085 is positive 4 c + 2 > 2 Because absolute values are positive, all values of c + 2 will be greater than 2 The solution set is {c c is a real number} 5 n Case 1 n + 5 is or and Case 2 m 7085 is negative Page 1 Case 2 n + 5 is

71 The solution set is {p p 6 or p 10} 5-5 Inequalities Involving Absolute Value 7 FINANCIAL LITERACY Jerome bought stock in $7085 The stock price ranged from within $075 of Find the range of prices for which the stock could tra m The solution set is {x 24 < x < 8} 9 r Case 1 r + 1 is positive and Case 2 r + 1 is negative Case 1 m 7085 is positive and The solution set is {r 3 r 1} c 4} Case 2 m 7085 is negative 10 2c 1 7 Case 1 2c 1 is positive The solution set is {m 7010 m 7160} Solve each inequality Then graph the solution set 8 x + 8 < 16 Case 1 x + 8 is positive and Case 2 x + 8 is negative The solution set is {x 24 < x < 8} 9 r + 1 Case 2 2c 1 is negative The solution set is {c h 3 < 12 2 and Case 1 r + 1 is positive and Case 2 r + 1 is negative Case 1 3h 3 is positive Page 2

72 The solution set is {c 3 c 4} 5-5 Inequalities Involving Absolute Value + 5 to be less than 8 Therefore, there is no solution, and the solution set is the empty set, Ø The graph is also empty 14 r + 2 > h 3 < 12 Case 1 r + 2 is positive Case 1 3h 3 is positive or Case 2 r + 2 is negative The solution set is {r r < 8 or r > 4} and Case 2 3h 3 is negative 15 k 4 > 3 Case 1 k 4 is positive or Case 2 k 4 is negative The solution set is {h 3 < h < 5} The solution set is {k k < 1 or k > 7} 12 m + 4 < h 3 m + 4 cannot be negative So it is not possible for m + 4 to be less than 2 Therefore, there is no solution, and the solution set is the empty set, Ø The graph is also empty 9 Case 1 2h 3 is positive or Case 2 2h 3 is negative 13 w + 5 < 8 w + 5 cannot be negative So it is not possible for w + 5 to be less than 8 Therefore, there is no solution, and the solution set is the empty set, Ø The graph is also empty 14 r + 2 > 6 or Case 1 r -+Powered esolutions Manual 2 is by Cognero positive The solution set is {h h 3 or h 6} 17 4p Case 2 r + 2 is negative Case 1 4p + 2 is positive Page 3

73 Because absolute values are positive, all values of 2c 3 will be greater than 4 The solution set is {c c is a real number} The solution set is {h h 3 or h 6} 5-5 Inequalities Involving Absolute Value 17 4p Case 1 4p + 2 is positive 20 SCUBA DIVING The pressure of a scuba tank should be within 500 pounds per square inch (psi) of 2500 psi Write the range of optimum pressures The range of optimum pressures for scuba tanks is between 2500 psi plus 500 psi and 2500 psi minus 500 psi The optimum range can be expressed as {p 2000 p 3000} Solve each inequality Then graph the solution set 21 4n Case 1 4n + 3 is positive or Case 2 4p + 2 is negative The solution set is {p p 3 or p 2} or Case 2 4n + 3 is negative The solution set is 18 5v + 3 > 9 Because absolute values are positive, all values of 5v + 3 will be greater than 9 The solution set is {v v is a real number} 22 5t 2 6 Case 1 5t 2 is positive and Case 2 5t 2 is negative 19 2c 3 > 4 Because absolute values are positive, all values of 2c 3 will be greater than 4 The solution set is {c c is a real number} 20 SCUBA DIVING The pressure of a scuba tank should be within 500 pounds per square inch (psi) of 2500 psi Write the range of optimum pressures The range of optimum pressures for scuba tanks is between 2500 psi plus 500 psi and 2500 psi minus 500 psi The optimum range can be expressed as {p The solution set is Page 4 23

74 The solution set is The solution set is 5-5 Inequalities Involving Absolute Value 22 5t Case 1 5t 2 is positive and Case 2 5t 2 is negative Case 1 The solution set is is positive and 23 Case 2 is negative Case 1 is positive and Case 2 The solution set is is negative 24 Case 1 is positive Page 5

75 The solution set is The solution set is {p p 14 or p 22} 5-5 Inequalities Involving Absolute Value Case 1 is positive cannot be negative So it is not possible for to be less than or equal to 5 Therefore, there is no solution, and the solution set is the empty set, Ø The graph is also empty 26 Because absolute values are positive, all values of will be greater than 7 or The solution set is {g g is a real number} Case 2 is negative 27 6r 4 < 8 Case 1 6r 4 is positive and Case 2 6r 4 is negative The solution set is {p p 14 or p 22} 25 cannot be negative So it is not possible for to be less than or equal to 5 Therefore, there is no solution, and the solution set is the empty The solution set is Page 6

76 will be greater than 7 The solution set is The solution set is {g g is a real number} 5-5 Inequalities Involving Absolute Value 28 3p 7 > r 4 < 8 Case 1 6r 4 is positive Case 1 3p 7 is positive and or Case 2 6r 4 is negative The solution set is Case 2 3p 7 is negative The solution set is 28 3p 7 > 5 Case 1 3p 7 is positive 29 h + 15 < 3 Case 1 h +15 is positive and or Case 2 h + 15 is negative Case 2 3p 7 is negative Page 7 The solution set is {h 15 < h < 45}

77 The solution set is downloads is between $10 plus $3 and $10 minus $3 The range can be expressed as {m 7 m 13} b 5-5 Inequalities Involving Absolute Value 29 h + 15 < 3 Case 1 h +15 is positive 31 CHEMISTRY Water can be present in our atmosphere as a solid, liquid, or gas Water freezes at 32 F and vaporizes at 212 F a Write the range of temperatures in which water is not a liquid b Graph this range c Write the absolute value inequality that describes this situation a The range of temperatures that water is not a liquid include 90 below 122 and 90 above 122 The range can be expressed as {t t < 32 or t > 212} and Case 2 h + 15 is negative b c The difference between the temperature of water that is not a liquid and 122 should be more than 90 The absolute value inequality that describes this situation is t 122 > 90 Write an open sentence involving absolute value for each graph The solution set is {h 15 < h < 45} 32 Since the endpoints are circles at 2 and 2 and the shaded area is between the circles, the sign for this sentence is less than 30 MUSIC DOWNLOADS Kareem is allowed to download $10 worth of music each month This month he has spent within $3 of his allowance a What is the range of money he has spent on music downloads this month? b Graph the range of the money that he spent a The range of money he has spent on music downloads is between $10 plus $3 and $10 minus $3 The range can be expressed as {m 7 m 13} b 31 CHEMISTRY Water can be present in our atmosphere as a solid, liquid, or gas Water freezes at 32 F and vaporizes at 212 F a Write the range of temperatures in which water is not a liquid b Graph this range c Write the absolute value inequality that describes this situation a The range of temperatures that water is not a liquid include 90 below 122 and 90 above 122 The Manual range -can be expressed esolutions Powered by Cognero as {t t < 32 or t > 212} b The midpoint between 2 and 2 is 0 The distance between the midpoint and each endpoint is 2 So, the equation is or 33 Since the endpoints are dots at 5 and 3 and the shaded area is between the dots, the sign for this sentence is less than or equal to The midpoint between 5 and 3 is 1 The distance between the midpoint and each endpoint is 4 So, the equation is or 34 Since the endpoints are dots at 3 and 1 and the shaded areas are outside of the dots, the sign for this Page 8 sentence is greater than or equal to The midpoint between 3 and 1 is 1 The distance

78 sentence is less than or equal to The midpoint between 5 and 3 is 1 The distance between the midpoint andabsolute each endpoint 5-5 Inequalities Involving Valueis 4 So, the equation is or The range of scores for Ginger s game is between 52 plus 5 strokes and 52 minus 5 strokes The range can be expressed as {g 47 g 57} Express each statement using an inequality involving absolute value Do not solve 34 Since the endpoints are dots at 3 and 1 and the shaded areas are outside of the dots, the sign for this sentence is greater than or equal to The midpoint between 3 and 1 is 1 The distance between the midpoint and each endpoint is 2 So, the equation is or 38 The ph of a swimming pool must be within 03 of a ph of 75 Since within implies that the ph can be up to 03 of a ph away from the midpoint of 75 in either direction, the symbol to use is less than or equal to In other words, the distance between the ph level p, and the midpoint 75, is less than or equal to Since the endpoints are circles at 1 and 10 and the shaded areas are outside of the circles, the sign for this sentence is greater than The midpoint between 1 and 10 is 55 The distance between the midpoint and each endpoint is 45 So, the equation is or 36 ANIMALS A sheep s normal body temperature is 39 C However, a healthy sheep may have body temperatures 1 C above or below this temperature What is the range of body temperatures for a sheep? The range of normal sheep body temperature is between 39 C plus 1 C and 39 C minus 1 C The range can be expressed as {t 38 t 40} 37 MINIATURE GOLF Ginger s score was within 5 strokes of her average score of 52 Determine the range of scores for Ginger s game The range of scores for Ginger s game is between 52 plus 5 strokes and 52 minus 5 strokes The range can be expressed as {g 47 g 57} Express each statement using an inequality involving absolute value Do not solve 38 The ph of a swimming pool must be within 03 of a ph of 75 Since within implies that the ph can be up to 03 of a ph away from the midpoint of 75 in either direction, the symbol to use is less than or equal to In other words, the distance between the ph level p, and the midpoint 75, is less than or equal to 03 So the inequality is p The temperature inside a refrigerator should be within 15 degrees of 38 F Since within implies that the temperature can be up to 15 degrees away from the midpoint of 38 degrees in either direction, the symbol to use is less than or equal to In other words, the distance between the temperature t, and the midpoint 38, is less than or equal to 15 So the inequality is t Ramona s bowling score was within 6 points of her average score of 98 Since within implies that Ramona s score can be up to 6 points away from her average of 98 points in either direction, the symbol to use is less than or equal to In other words, the distance between the bowling score b, and her average score 98, is less than or equal to 6 So the inequality is b The cruise control of a car should keep the speed within 3 miles per hour of 55 Page 9 Since within implies that the cruise control speed can be up to 3 miles per hour away from the

79 In other words, the distance between the bowling score b, and her average score 98, is less than or equal to Inequalities Involving Absolute Value So the inequality is b 98 6 a b Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0 41 The cruise control of a car should keep the speed within 3 miles per hour of 55 Since within implies that the cruise control speed can be up to 3 miles per hour away from the average of 55 miles per hour in either direction, the symbol to use is less than or equal to In other words, the distance between the cruise control speed c, and the average 55, is less than or equal to 3 Make a table of values x f (x) So the inequality is c MULTIPLE REPRESENTATIONS In this problem, you will investigate the graphs of absolute value inequalities on a coordinate plane a TABULAR Copy and complete the table Substitute the x and f (x) values for each point into each inequality Mark whether the resulting statement is true or false c b GRAPHICAL Graph f (x) = x 1 c GRAPHICAL Plot each point from the table that made f (x) x 1 a true statement, on the graph in red Plot each point that made f (x) x 1 a true statement, in blue d LOGICAL Make a conjecture about what the graphs about f (x) x 1 and f (x) x 1 look like Complete the table with other points to verify your conjecture e GRAPHICAL Use what you discovered to graph f (x) x 3 d f (x) x 1 would have infinitely many points plotted above the graph of f (x) = x 1 The result would look like shading above the graph f(x) x 1 would have infinitely many points plotted below the graph of f (x) = x 1 The result would look like shading below the graph a b Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0 Page 10

80 f(x) x 1 would have infinitely many points plotted belowinvolving the graph of f (x) = x Value 1 The result 5-5 Inequalities Absolute would look like shading below the graph 43 ERROR ANALYSIS Lucita sketched a graph of her solution to 2a 3 > 1 Is she correct? Explain your reasoning Sample answer: Lucita s graph is not correct She forgot to change the direction of the inequality sign for the negative case of the absolute value The arrow from 1 should point to the left e f (x) x 3 Since f (x) cannot be negative, the minimum point of the graph is where f (x) = 0 44 REASONING The graph of an absolute value inequality is sometimes, always, or never the union of two graphs Explain Sometimes; the graph could be the intersection of two graphs, the empty set, or all real numbers So the minimum point of the graph would occur at x = 3 Using a table of values we can graph more of f (x) = x 3 Since only points above the graph make the statement true, the region above the graph would be shaded Because the inequality symbol is, the boundary is included in the solution This is shown as a solid line x f (x) CHALLENGE Demonstrate why the solution of t > 0 is not all real numbers Explain your reasoning Sample answer: If t = 0, then the absolute value is equal to 0, not greater than 0 46 OPEN ENDED Write an absolute value inequality to represent a real-world situation Interpret the solution Sample answer: The range of normal body temperature of a healthy human varies plus or minus 14 from 986 Write the inequality to show the normal range of human body temperature t 986 < 14 Case 1 t 986 is positive and 43 ERROR ANALYSIS Lucita sketched a graph of her solution to 2a 3 > 1 Is she correct? Explain your reasoning Sample answer: Lucita s graph is not correct She forgot to change the direction of the inequality sign for the negative case of the absolute value The Case 2 t 986 is negative Page 11

81 > 0 is not all real numbers Explain your reasoning Sample answer: If t = 0, Absolute then the absolute 5-5 Inequalities Involving Value value is equal to 0, not greater than 0 46 OPEN ENDED Write an absolute value inequality to represent a real-world situation Interpret the solution Sample answer: The range of normal body temperature of a healthy human varies plus or minus 14 from 986 Write the inequality to show the normal range of human body temperature t 986 < 14 sentence uses and, and if the inequality symbol is > or, the compound sentence uses or To solve, if x < n, then set up and solve the inequalities x < n and x > n, and if x > n, then set up and solve the inequalities x > n or x < n 48 The formula for acceleration in a circle is Which of the following shows the equation solved for r? A r=v B r= 2 C r = av D r= Case 1 t 986 is positive and Case 2 t 986 is negative So, the correct choice is B The solution set is {t 972 < t < 100} The normal body temperature of a healthy human is between 972 and WRITING IN MATH Explain how to determine whether an absolute value inequality uses a compound inequality with and or a compound inequality with or Then summarize how to solve absolute value inequalities Sample answer: When an absolute value is on the left and the inequality symbol is < or, the compound sentence uses and, and if the inequality symbol is > or, the compound sentence uses or To solve, if x < n, then set up and solve the inequalities x < n and x > n, and if x > n, then set up and solve the inequalities x > n or x < n 49 An engraver charges a $3 set-up fee and $025 per word Which table shows the total price p for w words? F w p 15 $3 20 $ $ $775 G w p $675 $7 $725 $750 w p $375 $5 $625 $850 w p $675 $8 $925 H J 48 The formula for acceleration in a circle is Which of the following shows the equation solved for r? A r=v Page 12

82 $5 $625 $850 Find the total number of purchases The probability of a notebook being picked is 5-5 Inequalities Involving Absolute Value J w p 15 $ $8 25 $ $1050 p = w 51 Solve for n 2n 3 = 5 A { 4, 1} B { 1, 4} C {1, 1} D {4, 4} Case 1 2n 3 is positive Case 2 2n 3 is negative Substitute the w values into the equation Choices F and H have the wrong value of p for w = 15 The solution set is { 1, 4} So the correct choice is B Choice G has the wrong value of p for w = 20 So, the correct choice is J 50 SHORT RESPONSE The table shows the items in stock at the school store the first day of class What is the probability that an item chosen at random was a notebook? Item Number Purchased pencil 57 pen 38 eraser 6 folder 25 notebook 18 Find the total number of purchases Solve each compound inequality Then graph the solution set 52 b + 3 < 11 and b + 2 > 3 and The solution set is {b 5 < b < 8} t 4 8 First, express 6 2t 4 8 using and Then solve each inequality and The probability of a notebook being picked is 51 Solve for n 2n 3 = 5 A { 4, 1} B { 1, 4} C {1, 1} D {4, 4} Case 1 2n 3 Case 2 2n 3 is The solution set is {t 5 t 6} 54 2c 3 5 or 3c Page 13

83 The solution set is {t 5 t 6} 5-5 Inequalities Involving Absolute Value 1st angle = = 35 3rd angle = 2( ) = 120 So the three angles are 25, 35, and c 3 5 or 3c or The solution set is {c c 4 or c 4} 56 FINANCIAL LITERACY Jackson s bank charges him a monthly service fee of $6 for his checking account and $2 for each out-of-network ATM withdrawal Jackson s account balance is $87 Write and solve an inequality to find how many out-ofnetwork ATM withdrawals of $20 Jackson can make without overdrawing his account First set up the inequality The service fee plus the product of the number of withdrawals and the amount of each withdrawal is less than or equal to the amount in the account 55 GEOMETRY One angle of a triangle measures 10 more than the second The measure of the third angle is twice the sum of the measure of the first two angles Find the measure of each angle Let a = the measurement of the second angle 1st angle: a rd angle: 2(2nd angle + 1st angle) = 2[a + (a + 10)] Jackson can make up to 3 withdrawals Solve each equation Then check your solution 57 c 7 = 11 Solve Check The measurement of the second angle is 25, so solve for the other two angles 1st angle = = 35 3rd angle = 2( ) = 120 So the three angles are 25, 35, and FINANCIAL LITERACY Jackson s bank charges him a monthly service fee of $6 for his checking account and $2 for each out-of-network ATM withdrawal Jackson s account balance is $87 Write and solve an inequality to find how many out-ofnetwork ATM withdrawals of $20 Jackson can make without overdrawing his account FirstManual set up- the inequality The service fee plus the esolutions Powered by Cognero product of the number of withdrawals and the amount of each withdrawal is less than or equal to 58 2w = 24 Solve Check Page p = 11

84 Check 5-5 Inequalities Involving Absolute Value 58 2w = 24 Solve Graph each equation 61 y = 4x 1 The slope is 4 and the y-intercept is 1 To graph the equation, plot the y-intercept (0, 1) Then move up 4 units and right 1 unit Plot the point Draw a line through the two points Check p = 11 Solve 62 y x = 3 Write the equation in slope-intercept form Check 60 = 20 The slope is 1 and the y-intercept is 3 To graph the equation, plot the y-intercept (0, 3) Then move up 1 unit and right 1 unit Plot the point Draw a line through the two points Solve Check 63 2x y = 4 Write the equation in slope-intercept form Graph each equation 61 y = 4x 1 The Manual slope is 4 and the y-intercept is 1 To graph the esolutions - Powered by Cognero equation, plot the y-intercept (0, 1) Then move up 4 units and right 1 unit Plot the point Draw a line Page 15 The slope is 2 and the y-intercept is 4 To graph the

85 5-5 Inequalities Involving Absolute Value 63 2x y = 4 Write the equation in slope-intercept form The slope is 2 and the y-intercept is 4 To graph the equation, plot the y-intercept (0, 4) Then move down 2 units and left 1 unit Plot the point Draw a line through the two points 64 3y + 2x = 6 Write the equation in slope-intercept form The slope is and the y-intercept is 2 To graph the equation, plot the y-intercept (0, 2) Then move down 2 units and right 3 units Plot the point Draw a line through the two points 64 3y + 2x = 6 Write the equation in slope-intercept form The slope is and the y-intercept is 2 To graph the equation, plot the y-intercept (0, 2) Then move down 2 units and right 3 units Plot the point Draw a line through the two points 65 4y = 4x 16 Write the equation in slope-intercept form The slope is 1 and the y-intercept is 4 To graph the equation, plot the y-intercept (0, 4) Then move up 1 unit and right 1 unit Plot the point Draw a line through the two points Page y 2x = 8

86 5-5 Inequalities Involving Absolute Value 65 4y = 4x 16 Write the equation in slope-intercept form The slope is 1 and the y-intercept is 4 To graph the equation, plot the y-intercept (0, 4) Then move up 1 unit and right 1 unit Plot the point Draw a line through the two points 66 2y 2x = 8 Write the equation in slope-intercept form The slope is 1 and the y-intercept is 4 To graph the equation, plot the y-intercept (0, 4) Then move up 1 unit and right 1 unit Plot the point Draw a line through the two points 66 2y 2x = 8 Write the equation in slope-intercept form 67 9 = 3x y Write the equation in slope-intercept form The slope is 1 and the y-intercept is 4 To graph the equation, plot the y-intercept (0, 4) Then move up 1 unit and right 1 unit Plot the point Draw a line through the two points The slope is 3 and the y-intercept is 9 To graph the equation, plot the y-intercept (0, 9) Then move down 3 units and right 1 unit Plot the point Draw a line through the two points 67 9 = 3x y Page 17

87 5-5 Inequalities Involving Absolute Value 67 9 = 3x y Write the equation in slope-intercept form = 5y 2x Write the equation in slope-intercept form The slope is 3 and the y-intercept is 9 To graph the equation, plot the y-intercept (0, 9) Then move down 3 units and right 1 unit Plot the point Draw a line through the two points The slope is and the y-intercept is 2 To graph the equation, plot the y-intercept (0, 2) Then move up 2 units and right 5 units Plot the point Draw a line through the two points = 5y 2x Write the equation in slope-intercept form The slope is and the y-intercept is 2 To graph the equation, plot the y-intercept (0, 2) Then move up 2 units and right 5 units Plot the point Draw a line through the two points Page 18

88 5-6 Graphing Inequalities in Two Variables Graph each inequality 1 y > x + 3 y>x+3 Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point 2 y 8 y 8 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point 0 8 Since 0 is greater than or equal to 8, shade the half plane that contains (0, 0) Since 0 is not greater than 3, shade the half plane that does not contain (0, 0) 3 x + y > 1 2 y 8 y 8 Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point 0 8 Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is greater than or equal to 8, shade the half plane that contains (0, 0) Since 0 is not greater than 1, shade the half plane that does not contain (0, 0) 3 x + y > 1 Page 1

89 5-6 Graphing Inequalities in Two Variables 3 x + y > 1 Solve for y in terms of x 4 y x 6 y x 6 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is not less than or equal to 6, shade the half plane that does not contain (0, 0) Since 0 is not greater than 1, shade the half plane that does not contain (0, 0) 5 y < 2x 4 y < 2x 4 4 y x 6 y x 6 Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is not less than 4, shade the half plane that does not contain (0, 0) Since 0 is not less than or equal to 6, shade the half plane that does not contain (0, 0) Page 2 6 x y 4

90 5-6 Graphing Inequalities in Two Variables 5 y < 2x 4 y < 2x 4 6 x y 4 Solve for y in terms of x Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is not less than 4, shade the half plane that does not contain (0, 0) Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is less than or equal to 4, shade the half plane that contains (0, 0) 6 x y 4 Solve for y in terms of x Use a graph to solve each inequality 7 7x + 1 < 15 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and get 0 on a side by itself Since 0 is less than or equal to 4, shade the half plane that contains (0, 0) Because the inequality involves <, graph x = 2 using Page 3 a dashed line Choose (0, 0) as a test point in the original inequality

91 5-6 Graphing Inequalities in Two Variables Use a graph to solve each inequality 7 7x + 1 < 15 Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and get 0 on a side by itself Because the inequality involves <, graph x = 2 using a dashed line Choose (0, 0) as a test point in the original inequality 8 3x 2 11 Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and get 0 on a side by itself Because the inequality involves, graph x = with a solid line Choose (0, 0) as a test point in the original inequality Since 1 is less than 15, shade the half plane that contains the point (0, 0) Notice that the x intercept of the graph is at 2 Since the half plane to the left of the x intercept is shaded, the solution is x < 2 Since 2 is not greater than or equal to 11, shade the half plane that does not contain the point (0, 0) Notice that the x intercept of the graph is at Since the half plane to the left of the x intercept is shaded, the solution is 8 3x 2 11 Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and get 0 on a side by itself 9 3y 5 34 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and Page 4 solve for y

92 5-6 Graphing Inequalities in Two Variables 9 3y y 21 > 1 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for y Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and solve for y Because the inequality involves, graph y = 13 using a solid line Choose (0, 0) as a test point in the original inequality Because the inequality involves >, graph using a dashed line Choose (0, 0) as a test point in the original inequality Since 5 is less than or equal 34, shade the half plane that contains the point (0, 0) Notice that the y intercept of the graph is at 13 Since the half plane below the y intercept is shaded, the solution is y 13 Since 21 not is greater than 1, shade the half plane that does not contain the point (0, 0) Notice that the y intercept of the graph is at 5 Since the half plane above the y intercept is shaded, the solution is 10 4y 21 > 1 Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and solve for y 11 FINANCIAL LITERACY The surf shop has a weekly overhead of $2300 a Write an inequality to represent the number of skimboards and longboards the shop sells each week to make a profit Page 5 b How many skimboards and longboards must the shop sell each week to make a profit?

93 11 FINANCIAL LITERACY The surf shop has a weekly overhead of $ Graphing Inequalities in Two Variables a Write an inequality to represent the number of skimboards and longboards the shop sells each week to make a profit b How many skimboards and longboards must the shop sell each week to make a profit? One possible solution would be to sell 1 skim board Calculate how many longboards must be sold when just 1 skim boards are sold So, the shop would have to sell 1 skim board and 4 longboard surf boards a Let x represent the skimboards and y represent the longboards 115x + 685y 2300 b To graph the inequality, first solve for x Graph each inequality 12 y < x 3 y<x 3 Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Graph then use the test point (0, 0) to determine the shading The point (0, 0) results in which Since 0 is not less than 3, shade the half plane that does not contain (0, 0) is false Shade the side that does not include the origin 13 y > x + 12 One possible solution would be to sell 1 skim board Calculate how many longboards must be sold when just 1 skim boards are sold y > x + 12 Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Page 6 Since 0 is not less than 12, shade the half plane that

94 5-6 Graphing Inequalities in Two Variables 13 y > x y 3x 1 y > x + 12 Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is not less than 12, shade the half plane that does not contain (0, 0) Since 0 is greater than or equal to 1, shade the half plane that contains (0, 0) 14 y 3x 1 15 y 4x + 12 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is greater than or equal to 1, shade the half plane that contains (0, 0) Since 0 is less than or equal to 12, shade the half plane that contains (0, 0) 15 y 4x x + 3y > 12 Solve for y in terms of x Page 7

95 5-6 Graphing Inequalities in Two Variables 16 6x + 3y > 12 Solve for y in terms of x Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is not greater than 4, shade the half plane that does not contain (0, 0) 17 2x + 2y < 18 Solve for y in terms of x Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is less than 9, shade the half plane that contains (0, 0) 18 5x + y > x + 2y < 18 Solve for y in terms of x Solve for y in terms of x Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Page 8 Since 0 is not greater than 10, shade the half plane

96 5-6 Graphing Inequalities in Two Variables 18 5x + y > x + y < 3 Solve for y in terms of x Solve for y in terms of x Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is not greater than 10, shade the half plane that does not contain (0, 0) Since 0 is not less than 3, shade the half plane that does not contain (0, 0) 19 2x + y < x + y 4 Solve for y in terms of x Solve for y in terms of x Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is not less than 3, shade the half plane that does not contain (0, 0) Since 0 is greater than or equal to 4, shade the half Page 9 plane that contains (0, 0)

97 5-6 Graphing Inequalities in Two Variables 20 2x + y x + y 6 Solve for y in terms of x Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is greater than or equal to 4, shade the half plane that contains (0, 0) Since 0 is less than or equal to 6, shade the half plane that contains (0, 0) 22 10x + 2y x + y 6 Solve for y in terms of x Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is less than by or Cognero equal to esolutions Manual - Powered plane that contains (0, 0) 6, shade the half Page 10

98 5-6 Graphing Inequalities in Two Variables 22 10x + 2y x + 8y 48 Solve for y in terms of x Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is less than or equal to 7, shade the half plane that contains (0, 0) Since 0 is greater than or equal to 6, shade the half plane that contains (0, 0) 23 24x + 8y 48 Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test pointmanual - Powered by Cognero esolutions Use a graph to solve each inequality 24 10x 8 < 22 Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and solve for x Page 11 Because the inequality involves <, graph x = 3 using a dashed line

99 5-6 Graphing Inequalities in Two Variables Use a graph to solve each inequality 24 10x 8 < 22 Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and solve for x Because the inequality involves <, graph x = 3 using a dashed line Choose (0, 0) as a test point in the original inequality 25 20x 5 > 35 Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and solve for x Because the inequality involves >, graph x = 2 using a dashed line Choose (0, 0) as a test point in the original equation Since 8 is less than 22, shade the half plane that contains the point (0, 0) Notice that the x intercept of the graph is at 3 Since the half plane to the left of the x intercept is shaded, the solution is x < 3 Since 5 is not greater than 35, shade the half plane that does not contain the point (0, 0) Notice that the x intercept of the graph is at 2 Since the half plane to the right of the x intercept is shaded, the solution is x > y x 5 > 35 Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and solve for x Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and solve for y Page 12

100 5-6 Graphing Inequalities in Two Variables 26 4y y Graph the boundary, which is the related function Replace the inequality sign with an equal sign, and solve for y Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for y Because the inequality involves, graph y = 25 using a solid line Choose (0, 0) as a test point in the original equation Because the inequality involves, graph y = 5 using a solid line Choose (0, 0) as a test point in the original equation Since 77 is not greater than or equal 23, shade the half plane that does not contain the point (0, 0) Notice that the y intercept of the graph is at 25 Since the half plane above the y intercept is shaded, the solution is y 25 Since 8 is less than or equal 33, shade the half plane that contains the point (0, 0) Notice that the y intercept of the graph is at 5 Since the half plane under the y intercept is shaded, the solution is y y Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for y 28 35x + 25 < 6 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x Page 13

101 Notice that the y intercept of the graph is at 5 Since the half plane under the y intercept is shaded, the solution is yinequalities Graphing in Two Variables 28 35x + 25 < 6 Since the half plane to the left of the x intercept is shaded, the solution is 29 14x 12 > 31 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x Because the inequality involves <, graph x = Because the inequality involves >, graph x = using a dashed line Choose (0, 0) as a test point in the original inequality using a dashed line Choose (0, 0) as a test point in the original equation Since 25 is not less than 6, shade the half plane that does not contain the point (0, 0) Since 12 is greater than 31, shade the half plane that contains the point (0, 0) Notice that the x intercept of the graph is at Notice that the x intercept of the graph is at Since the half plane to the left of the x intercept is Since the half plane to the right of the x intercept is shaded, the solution is shaded, the solution is 29 14x 12 > 31 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x 30 DECORATING Sybrina is decorating her bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14, while a set of bed linens costs $60 Page 14 a Write an inequality for this situation b How many gallons of paint and bed linen sets can Sybrina buy and stay within her budget?

102 shaded, the solution is DECORATING 30Graphing Sybrina is decorating her 5-6 Inequalities in Two Variables bedroom She has $300 to spend on paint and bed linens A gallon of paint costs $14, while a set of bed linens costs $60 a Write an inequality for this situation b How many gallons of paint and bed linen sets can Sybrina buy and stay within her budget? a Let x represent the number of gallons of paint and y represent the number of sets of bed linens 14x + 60y 300 b To graph the inequality, solve for x first Thus when 5 gallons of paint are purchased, 3 bed linen sets can be purchased Use a graph to solve each inequality 31 3x + 2 < 0 First, graph the boundary, which is the related equation Replace the inequality sign with an equals sign and solve for x Because the inequality involves <, graph with a dashed line Graph Then use the test point (0, 0) to determine the shading The point (0, 0) results in 0 5 which is Choose (0, 0) as a test point in the original inequality true Shade the side that includes the origin Since 2 is not less than 0, shade the half plane that does not contain the point (0, 0) One possible solution would be to buy 5 gallons of paint Calculate how many bed linen sets must be purchased when 5 gallons of paint are bought Notice that the x intercept of the graph is at Since the half plane to the left of the x intercept is shaded, the solution is 32 4x 1 > 3 Thus when 5 gallons of paint are purchased, 3 bed linen sets can be purchased esolutions - Powered by Cognero Use Manual a graph to solve each inequality 31 3x + 2 < 0 First, graph the boundary, which is the related equation Replace the inequality sign with an equals sign and solve for x Page 15

103 Notice that the x intercept of the graph is at Since the half plane to the left of the x intercept is shaded, theinequalities solution is in Two Variables 5-6 Graphing 32 4x 1 > 3 Notice that the x intercept of the graph is at 1 Since the half plane to the right of the x intercept is shaded, the solution is x > x 8 4 First, graph the boundary, which is the related equation Replace the inequality sign with an equals sign and solve for x Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x Because the inequality involves >, graph x = 1 with a dashed line Choose (0, 0) as a test point in the original inequality Because the inequality involves, graph x = using a solid line Choose (0, 0) as a test point in the original inequality Since 1 is not greater than 3, shade the half plane that does not contain the point (0, 0) Since 8 is not greater than or equal to 4, shade the half plane that does not contain the point (0, 0) Notice that the x intercept of the graph is at 1 Since the half plane to the right of the x intercept is shaded, the solution is x > x 8 4 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x Notice that the x intercept of the graph is at Since the half plane to the left of the x intercept is shaded, the solution is 34 5x + 1 < 3 Page 16 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x

104 Since the half plane to the left of the x intercept is shaded, the solution is 5-6 Graphing Inequalities in Two Variables Since the half plane to the right of the x intercept is shaded, the solution is 35 7x + 13 < x + 1 < 3 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x Because the inequality involves <, graph x = Because the inequality involves <, graph x = using a dashed line Choose (0, 0) as a test point in the original inequality a dashed line Choose (0, 0) as a test point in the original inequality Since 13 is not less than 10, shade the half plane that does not contain the point (0, 0) Since 1 is less than 3, shade the half plane that contains the point (0, 0) Notice that the x intercept of the graph is at Notice that the x intercept of the graph is at Since the half plane to the right of the x intercept is shaded, the solution is using Since the half plane to the right of the x intercept is shaded, the solution is 36 4x x + 13 < 10 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x Graph the boundary, which is the related function Page 17 Replace the inequality sign with an equal sign and solve for x

105 Since the half plane to the right of the x intercept is shaded, the solution is shaded, the solution is 5-6 Graphing Inequalities in Two Variables 36 4x 4 6 Graph the boundary, which is the related function Replace the inequality sign with an equal sign and solve for x 37 SOCCER The girls soccer team wants to raise $2000 to buy new goals How many of each item must they sell to buy the goals? a Write an inequality that represents this situation b Graph this inequality c Make a table of values that shows at least five possible solutions d Plot the solutions from part c a Let x represent the number of hot dogs and let y represent the number of sodas x + 125y 2000 Because the inequality involves, graph x = using a solid line Choose (0, 0) as a test point in the original inequality Since 4 is not less than or equal to 6, shade the half plane that does not contain the point (0, 0) Notice that the x intercept of the graph is at Since the half plane to the left of the x intercept is shaded, the solution is b c Sample answer: x y d Sample points should be in the shaded region of the graph in part b Graph each inequality Determine which of the ordered pairs are part of the solution set for each inequality 38 y 6; {(0, 4), ( 2, 7), (4, 8), ( 4, 8), (1, 6)} y 6 SOCCER team wants to raise $2000 to buy new goals How many of each item must they sell to buy the goals? The girls soccer 37 esolutions Manual - Powered by Cognero Because the inequality involves, graph the Page 18 boundary using a solid line Choose (0, 0) as a test point

106 d Sample Inequalities points shouldin betwo in thevariables shaded region of 5-6 Graphing the graph in part b Graph each inequality Determine which of the ordered pairs are part of the solution set for each inequality 38 y 6; {(0, 4), ( 2, 7), (4, 8), ( 4, 8), (1, 6)} y 6 Because ( 2, 7), (4, 8) and (1, 6) are in the shaded half plane or on the boundary line, they are part of the solution set 39 x < 4; {(2, 1), ( 3, 0), (0, 3), ( 5, 5), ( 4, 2)} y < 4 Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point 0 0 Since 0 is not less than 4, shade the half plane that does not contains (0, 0) 6 4 Since 0 is not greater than or equal to 6, shade the half plane that does not contain (0, 0) To determine which of the ordered pairs are part of the solution set, plot them on the graph To determine which of the ordered pairs are part of the solution set, plot them on the graph Because ( 5, 5) is in the shaded half plane, it is part of the solution set 40 2x 3y 1; {(2, 3), (3, 1), (0, 0), (0, 1), (5, 3)} Because ( 2, 7), (4, 8) and (1, 6) are in the shaded half plane or on the boundary line, they are part of the solution set Solve for y in terms of x 39 x < 4; {(2, 1), ( 3, 0), (0, 3), ( 5, 5), ( 4, 2)} y < 4 Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test Page 19

107 40 2x 3y 1; {(2, 3), (3, 1), (0, 0), (0, 1), (5, 3)} 5-6 Graphing Inequalities in Two Variables Solve for y in terms of x Because (2, 3), (0, 0), and (5, 3) are in the shaded half plane or on the boundary line, they are part of the solution set 41 5x + 7y 10; {( 2, 2), (1, 1), (1, 1), (2, 5), (6, 0)} Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is greater than or equal to half plane that contains (0, 0), shade the Since 0 is not greater than or equal to half plane that does not contain (0, 0), shade the To determine which of the ordered pairs are part of the solution set, plot them on the graph To determine which of the ordered pairs are part of the solution set, plot them on the graph Because (2, 3), (0, 0), and (5, 3) are in the shaded line, they are part of the solution set half plane on thebyboundary esolutions Manual -or Powered Cognero Page 20

108 To determine which of the ordered pairs are part of 5-6 Graphing Inequalities in Two Variables the solution set, plot them on the graph To determine which of the ordered pairs are part of the solution set, plot them on the graph Because (1, 1), (2, 5), and (6, 0) are in the shaded half plane or on the boundary line, they are part of the solution set 42 3x + 5y < 10; {(3, 1), (1, 1), (0, 8), ( 2, 0), (0, 2)} Solve for y in terms of x Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is less than 10, shade the half plane that contains (0, 0) To determine whichbyofcognero the ordered esolutions Manual - Powered pairs are part of the solution set, plot them on the graph Because (3, 1), (1, 1), and (2, 0) are in the shaded half plane, they are part of the solution set 43 2x 2y 4; {(0, 0), (0, 7), (7, 5), (5, 3), (2, 5)} Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is not greater than or equal to 4, shade the half plane that does not contain (0, 0) Page 21 To determine which of the ordered pairs are part of the solution set, plot them on the graph

109 Since 0 is not greater than equal to 4, shade the 5-6 Graphing Inequalities in or Two Variables half plane that does not contain (0, 0) To determine which of the ordered pairs are part of the solution set, plot them on the graph Because (7, 5), (5, 3), and (2, 5) are in the shaded half plane or on the boundary line, they are part of the solution set 44 RECYCLING Mr Jones would like to spend no more than $3750 per week on recycling A curbside recycling service will remove up to 50 pounds of plastic bottles and paper products per week They charge $025 per pound of plastic and $075 per pound of paper products a Write an inequality that describes the number of pounds of each product that can be included in the curbside service b Write an inequality that describes Mr Jones weekly cost for the service if he stays within his budget c Graph an inequality for the weekly costs for the service a Let x represent the pounds of plastic and y represent the pounds of paper products They will remove up to 50 pounds, so use x + y 50 b The sum of the costs for paper and plastic products must be less than or equal to $3750 per week 025x + 075y 3750 c Because (7, 5), (5, 3), and (2, 5) are in the shaded half plane or on the boundary line, they are part of the solution set 44 RECYCLING Mr Jones would like to spend no more than $3750 per week on recycling A curbside recycling service will remove up to 50 pounds of plastic bottles and paper products per week They charge $025 per pound of plastic and $075 per pound of paper products a Write an inequality that describes the number of pounds of each product that can be included in the curbside service b Write an inequality that describes Mr Jones weekly cost for the service if he stays within his budget c Graph an inequality for the weekly costs for the service a Let x represent the pounds of plastic and y represent the pounds of paper products They will remove up to 50 pounds, so use x + y 50 b The sum of the costs for paper and plastic products must be less than or equal to $3750 per week 025x + 075y MULTIPLE REPRESENTATIONS Use inequalities A and B to investigate graphing compound inequalities on a coordinate plane A 7(y + 6) 21x + 14 B 3y 3x 12 a NUMERICAL Solve each inequality for y b GRAPHICAL Graph both inequalities on one graph Shade the half plane that makes A true in red Shade the half plane that makes B true in blue c VERBAL What does the overlapping region represent? a For equation A: Page 22

110 5-6 Graphing Inequalities in Two Variables 45 MULTIPLE REPRESENTATIONS Use inequalities A and B to investigate graphing compound inequalities on a coordinate plane A 7(y + 6) 21x + 14 B 3y 3x 12 a NUMERICAL Solve each inequality for y b GRAPHICAL Graph both inequalities on one graph Shade the half plane that makes A true in red Shade the half plane that makes B true in blue c VERBAL What does the overlapping region represent? c The overlapping region represents the solutions that make both A and B true 46 ERROR ANALYSIS Reiko and Kristin are solving by graphing Is either of them correct? Explain your reasoning a For equation A: For equation B: Kristin used a test point located on the line and shaded the incorrect half-plane Reiko is correct b 47 CHALLENGE Write a linear inequality for which ( 1, 2), (1, 1), and (3, 4) are solutions but (0, 1) is not c The overlapping region represents the solutions that make both A and B true 46 ERROR ANALYSIS Reiko and Kristin are solving by graphing Is either of them correct? We want to find a linear inequality for which three points are solutions and one point is not a solution The best way to find this is to plot the points on the coordinate plane and visualize the line that would be satisfactory Plot all of the points and label (1,1) with an X to separate it from the others Explain your reasoning Page 23

111 Kristin used a test point in located the line and 5-6 Graphing Inequalities Two on Variables shaded the incorrect half-plane Reiko is correct 47 CHALLENGE Write a linear inequality for which ( 1, 2), (1, 1), and (3, 4) are solutions but (0, 1) is not We want to find a linear inequality for which three points are solutions and one point is not a solution The best way to find this is to plot the points on the coordinate plane and visualize the line that would be satisfactory Plot all of the points and label (1,1) with an X to separate it from the others Using points ( 1, 2) and (0, 1), you can determine the equation for the line to be The points on the line are a part of the solution and we are shading below the line, so the inequality is 48 REASONING Explain why a point on the boundary should not be used as a test point The test point is used to determine which half plane is the solution set for the inequality A test point on the boundary does not show which half plane contains the points that make the inequality true 49 OPEN ENDED Write a two-variable inequality with a restricted domain and range to represent a real-world situation Give the domain and range, and explain why they are restricted Sample answer: The inequality y > 10x + 45 represents the cost of a monthly smartphone data plan with a flat rate of $45 for the first 2 GB of data used, plus $10 per each additional GB of data used Both the domain and range are nonnegative real numbers because the GB used, and the total cost cannot be negative From the graph, it looks like the three points that are included in the solution almost form a straight line It looks like a good line to use would be the one that includes ( 1, 2) and (0, 1) It also looks like point X would be above this line and the other point would be below it Therefore, the linear inequality should be close to this line and the shading should be below it Using points ( 1, 2) and (0, 1), you can determine the equation for the line to be The points on the line are a part of the solution and we are shading below the line, so the inequality is 50 WRITING IN MATH Summarize the steps to graph an inequality in two variables Sample answer: First solve the inequality for y Then change the inequality sign to an equal sign and graph the boundary If < or > is used, the boundary is not included in the graph and the line is dashed Otherwise, the boundary is included and the line is solid Then choose a test point not on the boundary Substitute the coordinates of the test point into the original inequality If the result makes the inequality true, then shade the half-plane that includes the test point If the result makes the inequality false, shade the half-plane that does not include the test point Lastly, check your solution by choosing a test point that is in the half-plane that is not shaded This second test point should make the inequality false if the solution is correct 51 What is the domain of this function? 48 REASONING Explain why a point on the boundary should not be used as a test point determine which half plane is The Manual test point is used esolutions - Powered by to Cognero the solution set for the inequality A test point on the boundary does not show which half plane contains Page 24 A {x 0 x 3} B

112 the half-plane that does not include the test point Lastly, check your solution by choosing a test point that is in the half-plane that is not shaded This 5-6 Graphing Two second testinequalities point should in make thevariables inequality false if the solution is correct 51 What is the domain of this function? Use the slope and either of the two points to find the y intercept Write the equation in slope intercept form A B C D {x 0 x 3} {x 0 x 9} {y 0 y 9} {y 0 y 3} The domain of a function is the x values, which means that you can delete choices C and D This graph shows values from 0 to 9, so the correct choice is B b The y intercept is 390 This represents the number of trees in the arboretum before any have lost their leaves c To determine when the arboretum will close, let y = 0 52 EXTENDED RESPONSE An arboretum will close for the winter when all of the trees have lost their leaves The table shows the number of trees each day that still have leaves a Write an equation that represents the number of trees with leaves y after d days b Find the y-intercept What does it mean in the context of this problem? c After how many days will the arboretum close? Explain how you got your answer a Choose two points (5, 325) and (20, 130) Find the slope of the line containing the given points The arboretum will close after 30 days 53 Which inequality best represents the statement below? A jar contains 832 gumballs Ebony s guess was within 46 pieces F g G g H g J g The guess was within 46, means less than or equal to 46, which eliminates choices H and J For g , g is one solution, which makes g 786 The number of gumballs cannot be negative, so G is eliminated The correct choice is F Use the slope and either of the two points to find the y intercept esolutions Manual - Powered by Cognero 54 GEOMETRY If the rectangular prism has a 3 volume of 10,080 cm, what is the value of x? Page 25

113 46, g is one solution, which makes g 786 The number of gumballs cannot be negative, so G is eliminated 5-6 Graphing Inequalities in Two Variables The correct choice is F 54 GEOMETRY If the rectangular prism has a So the solution set is {y y > 6 or y < 2} 56 t volume of 10,080 cm, what is the value of x? A B C D Case 1: Case 2: Volume of a rectangular prism is equal to length times width times height So the solution set is {t 1 t 11} d < 4 This problem has an absolute value equal to a negative number That means that the distance between 3 and d is equal to 4, but distances cannot be negative, so there is no solution So the solution set is an empty set, Ø So, the correct choice is D Solve each compound inequality 58 4c 4 < 8c 16 < 6c 6 Solve each open sentence 55 y 2 > 4 Case 1: Case 2: and So the solution set is {y y > 6 or y < 2} 56 t 6 5 Case 1: So, the solution set is {c 3 < c < 5} Page 26 Case 2: 59

114 This problem has an absolute value equal to a negative number That means that the distance between 3 and d is equal to 4, but distances cannot 5-6 Graphing Inequalities insolution Two Variables be negative, so there is no So the solution set is an empty set, Ø Solve each compound inequality 58 4c 4 < 8c 16 < 6c 6 So, the solution set is {p 4 < p < 10} 60 05n 7 or 25n or Notice the two inequalities overlap so that every point is a solution So, the solution set is {n n is any real number} and Write an equation of the line that passes through each pair of points 61 (1, 3) and (2, 5) Find the slope of the line containing the given points So, the solution set is {c 3 < c < 5} 59 Use the slope and either of the two points to find the y intercept and Write the equation in slope intercept form So, the solution set is {p 4 < p < 10} 60 05n 7 or 25n or 62 ( 2, 4) and ( 7, 3) Find the slope of the line containing the given points Page 27

115 Write the equation in slope intercept form Write the equation in slope intercept form 5-6 Graphing Inequalities in Two Variables 62 ( 2, 4) and ( 7, 3) 63 ( 6, 8) and ( 8, 5) Find the slope of the line containing the given points Find the slope of the line containing the given points Use the slope and either of the two points to find the y intercept Use the slope and either of the two points to find the y intercept Write the equation in slope intercept form Write the equation in slope intercept form 64 FITNESS The table shows the maximum heart rate to maintain during aerobic activities Write an equation in function notation for the relation Determine what would be the maximum heart rate to maintain in aerobic training for an 80-year-old 63 ( 6, 8) and ( 8, 5) Find the slope of the line containing the given points Choose two points: (20, 175) and (70, 130) Find the slope of the line containing the given points esolutions - Powered by Cognero Use Manual the slope and either of the y intercept two points to find the Page 28

116 Write the equation in slope intercept form 5-6 Graphing Inequalities in Two Variables 64 FITNESS The table shows the maximum heart rate to maintain during aerobic activities Write an equation in function notation for the relation Determine what would be the maximum heart rate to maintain in aerobic training for an 80-year-old The maximum heart rate for an 80-year-old is 121 beats/min 65 WORK The formula is used to find keyboarding speeds In the formula, s represents the speed in words per minute, w the number of words typed, r the number of errors, and m the number of minutes typed Solve for r Choose two points: (20, 175) and (70, 130) Find the slope of the line containing the given points Use the slope and either of the two points to find the y intercept Write the equation in function notation To determine what the maximum heart rate is for an 80 year old, substitute 80 in for x The maximum heart rate for an 80-year-old is 121 beats/min 65 WORK The formula is used to find keyboarding speeds In the formula, s represents the speed in words per minute, w the number of words typed, r the number of errors, and m the number of Page 29

117 The solution set is {t t 21} Mid-Chapter Quiz Solve each inequality Then graph it on a number line 1 x 8 > 4 The solution set is {x x > 12} 2 m CONCERTS Lupe s allowance for the month is $60 She wants to go to a concert for which a ticket costs $45 a Write and solve an inequality that shows how much money she can spend that month after buying a concert ticket b She spends $999 on music downloads and $2 on lunch in the cafeteria Write and solve an inequality that shows how much she can spend after these purchases and the concert ticket a Let m be the amount Lupe can spend after buying a concert ticket Write an inequality Because she can spend at most $60, the inequality symbol is The solution set is {m m 4} Lupe can spend $15 after buying a concert ticket 3 p 4 < 7 b Let m be the amount Lupe can spend after downloading music, buying lunch, and buying a concert ticket Write an inequality Because she can spend at most $60, the inequality symbol is The solution set is {p p < 3} 4 12 t 9 Lupe can spend $301 after downloading music, buying lunch, and buying a concert ticket Define a variable, write an inequality, and solve each problem Check your solution 6 The sum of a number and 2 is no more than 6 The solution set is {t t 21} 5 CONCERTS Lupe s allowance for the month is $60 She wants to go to a concert for which a ticket costs $45 a Write and solve an inequality that shows how much money she can spend that month after buying a concert ticket b She spends $999 on music downloads and $2 on lunch in the cafeteria Write and solve an inequality esolutions Powered by Cognero that Manual shows- how much she can spend after these purchases and the concert ticket Sample answer: Let x = the number The word sum means addition, and the phrase no more than means the same as less than or equal to The solution set is {x x 8} Page 1 Check the endpoint with 8 and the direction with a

118 Lupe can spend Mid-Chapter Quiz $301 after downloading music, buying lunch, and buying a concert ticket Define a variable, write an inequality, and solve each problem Check your solution 6 The sum of a number and 2 is no more than 6 Sample answer: Let x = the number The word sum means addition, and the phrase no more than means the same as less than or equal to 7 A number decreased by 4 is more than 1 Sample answer: Let x = the number The phrase decreased by means subtraction, and the phrase more than means greater than The solution set is {x x > 3} The solution set is {x x 8} Check the endpoint with 3 and the direction with a value greater than 3 Check the endpoint with 8 and the direction with a value less than 8 8 Twice a number increased by 3 is less than the number decreased by 4 7 A number decreased by 4 is more than 1 Sample answer: Let x = the number The phrase decreased by means subtraction, and the phrase more than means greater than Sample answer: Let x = the number The word twice means to multiply by 2, and the phrase decreased by means to subtract The solution set is {x x > 3} The solution set is {x x < 7} Check the endpoint with 3 and the direction with a value greater than 3 Check the endpoint with -7 and the direction with a value less than -7 Page 2

119 So, Jane needs to save no more than $38 The correct choice is choice C Mid-Chapter Quiz 8 Twice a number increased by 3 is less than the number decreased by 4 Sample answer: Let x = the number The word twice means to multiply by 2, and the phrase decreased by means to subtract Solve each inequality Check your solution 10 The solution set is {y y 15} To check this answer, substitute a number greater than or equal to 15 into the original inequality If y = The solution set is {x x < 7} Check the endpoint with -7 and the direction with a value less than -7 18, then or 6 5, so the solution checks 11 4 < 9 MULTIPLE CHOICE Jane is saving money to buy a new cell phone that costs no more than $90 So far, she has saved $52 How much more money does Jane need to save? A $38 B more than $38 C no more than $38 D at least $38 Let c be the amount Jane still needs to save Write an inequality The phrase no more than means the same as less than or equal to The solution set is {c c > 20} To check this answer, substitute a number greater than 20 into the original inequality If c = 25, then or 5 > 4, so the solution checks 12 8x > 24 The solution set is {x x < 3} To check this answer, substitute a number less than 3 into the original inequality If x = 4, then 8( 4) or 32 > 24, so the solution checks 13 2m 10 So, Jane needs to save no more than $38 The correct choice is choice C Solve each inequality Check your solution 10 Page 3 The solution set is {m m 5}

120 The solution set is {x x < 3} To check this answer, substitute a number less than original inequality If x = 4, then 8( 4) Mid-Chapter 3 into thequiz or 32 > 24, so the solution checks 13 2m 10 The solution set is {a a 5} To check this answer, substitute a number less than or equal to 5 into the original inequality If a = 2, then 9(2) or 18 45, so the solution checks 16 > 3 The solution set is {m m 5} To check this answer, substitute a number less than or equal to 5 into the original inequality If m = 6, then 2( 6) or 12 10, so the solution checks 14 The solution set is {w w > 18} To check this answer, substitute a number greater than 18 into the original inequality If w = 12, then < or 2 > 3, so the solution checks 17 < 2 The solution set is To check this answer, substitute a number less than into the original inequality If x = 1, then, so The solution set is {k k < 14} To check this answer, substitute a number less than 14 into the original inequality If k = 21, then or 3 < 2, so the solution checks the solution checks 15 9a 45 The solution set is {a a 5} To check this answer, substitute a number less than or equal to 5 into the original inequality If a = 2, then 9(2) or 18 45, so the solution checks ANIMALS The world s heaviest flying bird is the great bustard A male bustard can be up to 4 feet long and weigh up to 40 pounds a Write inequalities to describe the ranges of lengths and weights of male bustards b Male bustards are usually about four times as heavy as females Write and solve an inequality that describes the range of weights of female bustards a Let be the length of a male bustard, and let w be the weight of a male bustard The phrase up to means the same as less than Both the length and weight of a bustard must be more than zero > 3 0< <4 0 < w < 40 b Let w be the weight of a female bustard The weight of a male bustard can be represented by 4w Page 4

121 The solution set is {k k < 14} To check this answer, substitute a number less than 14 into the original inequality If k = 21, then Mid-Chapter < 2, so the solution checks or 3Quiz 18 ANIMALS The world s heaviest flying bird is the great bustard A male bustard can be up to 4 feet long and weigh up to 40 pounds a Write inequalities to describe the ranges of lengths and weights of male bustards b Male bustards are usually about four times as heavy as females Write and solve an inequality that describes the range of weights of female bustards a Let be the length of a male bustard, and let w be the weight of a male bustard The phrase up to means the same as less than Both the length and weight of a bustard must be more than zero 0< <4 0 < w < GARDENING Bill is building a fence around a square garden to keep deer out He has 60 feet of fencing Find the maximum length of a side of the garden Let x be the side length of Bill s square garden The perimeter of the garden can be represented by 4x Because Bill has only 60 feet of fencing, the perimeter must be less than or equal to 60 feet b Let w be the weight of a female bustard The weight of a male bustard can be represented by 4w So, the garden can be at most 15 feet on each side Solve each inequality Check your solution 20 4a 2 > GARDENING Bill is building a fence around a square garden to keep deer out He has 60 feet of fencing Find the maximum length of a side of the garden Let x be the side length of Bill s square garden The perimeter of the garden can be represented by 4x Because Bill has only 60 feet of fencing, the perimeter must be less than or equal to 60 feet The solution set is {a a > 4} To check this answer, substitute a number greater than 4 into the original inequality Let a = 5 So, the solution checks 21 2x x 10 So, the garden can be at most 15 feet on each side Solve each inequality Check your solution 20 4a 2 > 14 Page 5

122 Mid-Chapter Quiz checks So, the solution 21 2x x 10 So, the solution checks The solution set is {x x 7} To check this answer, substitute a number greater than or equal to 7 into the original inequality Let x = 10 The solution set is {d d 16} To check this answer, substitute a number greater than or equal to 16 into the original inequality Let d = 12 So, the solution checks 22 p + 4 < 9 So, the solution checks 24 2(4b + 1) < 3b + 8 The solution set is {p p > 13} To check this answer, substitute a number greater than 13 into the original inequality Let p = 15 So, the solution checks 23 The solution set is {b b > 2} To check this answer, substitute a number greater than 2 into the original inequality Let b = Page 6

123 Mid-Chapter Quiz So, the solution checks 24 2(4b + 1) < 3b + 8 So, the solution checks Define a variable, write an inequality, and solve each problem Check your solution 25 Three times a number increased by 8 is no more than the number decreased by 4 Sample answer: Let x = the number Times means multiplication, increased by means addition, no more than means less than or equal to, and decreased by means subtraction The solution set is {b b > 2} To check this answer, substitute a number greater than 2 into the original inequality Let b = 0 The solution set is {x x 6} To check this answer, substitute a number less than or equal to 6 into the original inequality Let x = 7 So, the solution checks Define a variable, write an inequality, and solve each problem Check your solution 25 Three times a number increased by 8 is no more than the number decreased by 4 Sample answer: Let x = the number Times means multiplication, increased by means addition, no more than means less than or equal to, and decreased by means subtraction The solution set is {x x 6} To check this answer, substitute a number less than or equal to - 6 into the original inequality Let x = 7 esolutions Manual Powered by Cognero So, the solution checks 26 Two thirds of a number plus 5 is greater than 17 Sample answer: Let x = the number Of means multiplication The solution set is {x x > 18} Page 7 To check this answer, substitute a number greater than 18 into the original inequality Let x = 21

124 Mid-Chapter Quiz checks So, the solution So, the solution checks 26 Two thirds of a number plus 5 is greater than 17 Sample answer: Let x = the number Of means multiplication 27 MULTIPLE CHOICE Shoe rental costs $2, and each game bowled costs $3 How many games can Kyle bowl without spending more than $15? F 2 G 3 H 4 J 5 Let g be the number of games Kyle bowls The solution set is {x x > 18} To check this answer, substitute a number greater than 18 into the original inequality Let x = 21 The solution set is Because Kyle cannot bowl one-third of a game, the most games he can bowl is 4 So, the correct choice is H So, the solution checks 27 MULTIPLE CHOICE Shoe rental costs $2, and each game bowled costs $3 How many games can Kyle bowl without spending more than $15? F 2 G 3 H 4 J 5 Let g be the number of games Kyle bowls The solution set is Page 8

125 Drew needs to add more than 27, or at least 28, comic books to have a larger collection than Connor So, the correct choice is C Practice Test - Chapter 5 Solve each inequality Then graph it on a number line 1 x 9 < 4 Solve each inequality Graph the solution on a number line 4 The solution set is {x x < 5} The solution set is {h h > 15} 2 6p 5p 3 5 7w 42 The solution set is {p p 3} 3 MULTIPLE CHOICE Drew currently has 31 comic books in his collection His friend Connor has 58 comic books How many more comic books does Drew need to add to his collection in order to have a larger collection than Connor? A no more than 21 B 27 C at least 28 D more than 30 The solution set is {w w 6} 6 Let c = the number of comic books Drew needs to add to his collection The solution set is {t t 36} Drew needs to add more than 27, or at least 28, comic books to have a larger collection than Connor So, the correct choice is C Solve each inequality Graph the solution on a number line 7 9m < 36 4 Page 1 The solution set is {m m > 4}

126 The solution set is {t t 36} The solution set is {g g 48} Practice Test - Chapter 5 7 9m < (x 4) > 5x 13 The solution set is {m m > 4} 8 3c 7 < 11 The solution set is {x x < 3} 11 ZOO The 8th grade science class is going to the zoo The class can spend up to $300 on admission The solution set is {c c < 6} a Write an inequality for this situation b If there are 32 students in the class and 1 adult will attend for every 8 students, can the entire class go to the zoo? a Let s = the number of students and a = the number of adults 8s + 10a < 300 The solution set is {g g 48} b The number of adults attending is 32 8, or 4 Solve the inequality for s = 32 and a = (x 4) > 5x 13 The $296 cost is less than the $300 limit, so the entire Page 2 class can go to the zoo Solve each compound inequality Then graph

127 The solution set is {x x < 3} The solution set is {y y < 5 or y > 14} Notice that the graphs do not intersect To graph the solution set, graph y < 5 and graph y > 14 Practice Test - Chapter 5 11 ZOO The 8th grade science class is going to the zoo The class can spend up to $300 on admission a Write an inequality for this situation b If there are 32 students in the class and 1 adult will attend for every 8 students, can the entire class go to the zoo? h 5 13 First, express 11 2h 5 13 using and Then solve each inequality and a Let s = the number of students and a = the number of adults 8s + 10a < 300 b The number of adults attending is 32 8, or 4 Solve the inequality for s = 32 and a = 4 The solution set is {h 3 h 9} To graph the solution set, graph h 3 and graph h 9 Then find the intersection The $296 cost is less than the $300 limit, so the entire class can go to the zoo Solve each compound inequality Then graph the solution set 12 y 8 < 3 or y + 5 > z 2 > 5 and 7z + 4 < 17 First, solve each inequality or and The solution set is {y y < 5 or y > 14} Notice that the graphs do not intersect To graph the solution set, graph y < 5 and graph y > h 5 13 esolutions - Powered First,Manual express 11 by2hcognero 5 13 using and Then solve each inequality Next, find the intersection of z > 1 and z < 3Page 3 To find the intersection, graph z > 1 and z < 3

128 The solution set is {h 3 h 9} To graph the solution set, graph h 3 and graph h 9 Then find the intersection Practice Test - Chapter z 2 > 5 and 7z + 4 < 17 First, solve each inequality and Notice that the graphs do not intersect So, there are no numbers that satisfy both inequalities Thus, the solution is the empty set, or Ø Define a variable, write an inequality, and solve the problem Check your solution 15 The difference of a number and 4 is no more than 8 Let x = the number The solution set is {x x 12} Check the endpoint with 12 and the direction with a value less than 12 Next, find the intersection of z > 1 and z < 3 To find the intersection, graph z > 1 and z < 3 Notice that the graphs do not intersect So, there are no numbers that satisfy both inequalities Thus, the solution is the empty set, or Ø Define a variable, write an inequality, and solve the problem Check your solution 15 The difference of a number and 4 is no more than 8 So, the solution checks 16 Nine times a number decreased by four is at least twenty-three Let x = the number Let x = the number The solution set is {x x 12} The solution set is {x x 3} Check the endpoint with 12 and the direction with a value less than 12 Check the endpoint with 3 and the direction with a value greater than 3 Page 4

129 Practice Test - Chapter So, the solution checks5 16 Nine times a number decreased by four is at least twenty-three Let x = the number F and J are incorrect Because the points are both solid, choice H is incorrect x 2 or x 3 is the compound inequality for the graph shown So, the correct choice is G Solve each inequality Then graph the solution set 18 p 5 < 3 Case 1 p 5 is positive The solution set is {x x 3} Check the endpoint with 3 and the direction with a value greater than 3 and Case 2 p 5 is negative The solution set is {p 2 < p < 8} 19 2f Case 1 2f + 7 is positive So, the solution checks 17 MULTIPLE CHOICE Write a compound inequality for the graph shown or Case 2 2f + 7 is negative F 2 x < 3 G x 2 or x 3 H x < 2 or x 3 J 2 < x 3 Because there is no intersection in the graph, choices F and J are incorrect Because the points are both solid, choice H is incorrect x 2 or x 3 is the compound inequality for the graph shown So, the correct choice is G Solve each inequality Then graph the solution set 18 p 5 < 3 esolutions Manual - Powered by Cognero Case 1 p 5 is positive and Case 2 p 5 is negative The solution set is {f ƒ 14 or ƒ 7} 20 4m Case 1 4m + 3 is positive Page 5

130 The solution set is } The solution set is {f ƒ 14 or ƒ 7} Practice Test - Chapter m Case 1 4m + 3 is positive Case 1 is positive and Case 2 4m + 3 is negative or Case 2 The solution set is is negative } The solution set is {x x < 17 or x > 23} 21 Case 1 is positive 22 RETAIL A sporting goods store is offering a $15 coupon on any pair of shoes a The most and least expensive pairs of shoes are $14995 and $2495 What is the range of costs for customers with coupons? b When buying a pair of $10995 shoes, you can use a coupon or a 15% discount Which option is best? a and or Case 2 - Powered is negative esolutions Manual by Cognero The range of prices for customers with coupons is Page p b = 9495

131 The solution set is {x x < 17 or x > 23} Practice Test - Chapter 5 22 RETAIL A sporting goods store is offering a $15 coupon on any pair of shoes a The most and least expensive pairs of shoes are $14995 and $2495 What is the range of costs for customers with coupons? b When buying a pair of $10995 shoes, you can use a coupon or a 15% discount Which option is best? 24 2x + 3y 12 Solve for y in terms of x a and The range of prices for customers with coupons is 995 p Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point b = (015) = 9346 The 15% discount is the best option Graph each inequality 23 y < 4x 1 y < 4x 1 Since 0 is not greater than or equal to 4, shade the half-plane that does not contain (0, 0) Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is not less than 1, shade the half-plane that does not contain (0, 0) 25 Graph y > 2x + 5 Then determine which of the ordered pairs in {( 2, 0), ( 1, 5), (2, 3), (7, 3)} are in the solution set y > 2x + 5 Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point 24 2x + 3y 12 Since 0 is not greater than 5, shade the half-plane that does not contain (0, 0) Plot the points on the graph Solve for y in terms of x Page 7

132 Practice Test - Chapter 5 25 Graph y > 2x + 5 Then determine which of the ordered pairs in {( 2, 0), ( 1, 5), (2, 3), (7, 3)} are in the solution set y > 2x + 5 Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point The ordered pairs that are in the solution set are (2, 3) and (7, 3) 26 PRESCHOOL Mrs Jones is buying new books and puzzles for her preschool classroom Each book costs $6, and each puzzle costs $4 Write and graph an inequality to determine how many books and puzzles she can buy for $96 Let x = the number of books and y = the number of puzzles 6x + 4y 96 Solve for y in terms of x Since 0 is not greater than 5, shade the half-plane that does not contain (0, 0) Plot the points on the graph The ordered pairs that are in the solution set are (2, 3) and (7, 3) 26 PRESCHOOL Mrs Jones is buying new books and puzzles for her preschool classroom Each book costs $6, and each puzzle costs $4 Write and graph an inequality to determine how many books and puzzles she can buy for $96 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is less than or equal to 24, shade the halfplane that contains (0, 0) Let x = the number of books and y = the number of puzzles 6x + 4y 96 Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Page 8

133 Preparing for Standardized Tests - Chapter 5 It will take her more than 8 years to spend $400 Read each problem Identify what you need to know Then use the information in the problem to solve 1 Craig has $20 to order a pizza The pizza costs $1250 plus $095 per topping If there is also a $3 delivery fee, how many toppings can Craig order? Let t = the number of toppings Because it is not possible to buy a fraction of a topping, Craig can buy up to 4 toppings 2 To join an archery club, Nina had to pay an initiation fee of $75, plus $40 per year in membership dues a Write an equation to model the total cost, y, of belonging to the club for x years b How many years will it take her to spend more than $400 to belong to the club? 3 The area of the triangle shown is no more than 84 square millimeters What is the height of the triangle? The height of the triangle is no more than 12 mm 4 Rosa earns $200 a month delivering newspapers, plus an average of $11 per hour babysitting If her goal is to earn at least $295 this month, how many hours will she have to babysit? Let h = the number of hours Rosa needs to babysit a y = 40x + 75 b Rosa will need to babysit for at least 9 hours to earn at least $295 this month It will take her more than 8 years to spend $400 3 The area of the triangle shown is no more than 84 square millimeters What is the height of the triangle? 5 To earn money for a new bike, Ethan is selling some of his baseball cards He has saved $245 If the bike costs $1400, and he can sell 154 cards, for how much money will he need to sell each card to reach his goal? Let p = the price he will need to sell each card for Page 1

134 p b Rosa will to babysit for at least 9 hours 5to earn at Preparing forneed Standardized Tests - Chapter least $295 this month 5 To earn money for a new bike, Ethan is selling some of his baseball cards He has saved $245 If the bike costs $1400, and he can sell 154 cards, for how much money will he need to sell each card to reach his goal? Let p = the price he will need to sell each card for Ethan will need to sell his baseball cards for at least $750 each There can be at most 1320 players in this league 7 Sarah has $120 to shop for herself and to buy some gifts for 6 of her friends She has purchased a shirt for herself for $32 Assuming she spends an equal amount on each friend, what is the maximum that she can spend per person? Let f = the amount Sarah can spend on each friend The maximum that Sarah can spend on each friend is $ In a certain lacrosse league, there can be no more than 22 players on each team, and no more than 10 teams per age group There are 6 age groups a Write an inequality to represent this situation b What is the greatest number of players that can play lacrosse in this league? a Let p = the number of players in a league p b There can be at most 1320 players in this league 7 Sarah has $120 to shop for herself and to buy some gifts for 6 of her friends She has purchased a shirt for herself for $32 Assuming she spends an equal amount on each friend, what is the maximum that she can spend per person? Let f = the amount Sarah can spend on each friend The maximum that Sarah can spend esolutions Manual - Powered by Cognero $1466 on each friend is Page 2

135 Standardized Test Practice - Cumulative, Chapter 1-5 Read each question Then fill in the correct answer on the answer document provided by your teacher or on a sheet of paper 1 Miguel received a $100 gift certificate for a graduation gift He wants to buy a CD player that costs $38 and CDs that cost $12 each Which of the following inequalities represents how many CDs Miguel can buy? A B C D CD player + (cost of one CD number of CDs) gift certificate Let n = number of CDs purchased Round to 5 so This means that the correct choice is D 2 Craig is paid time-and-a-half for any additional hours over 40 that he works Craig s goal is to earn at least $600 next week, what is the minimum number of hours he needs to work? F 43 hours G 44 hours H 45 hours J 46 hours If Wilson works 40 hours, he will earn 40(1280) = $512 So he needs to work more than 40 hours Let h represent the number of hours more than 40 hours he needs to work Round to 5 so This means that the correct choice is D 2 Craig is paid time-and-a-half for any additional hours over 40 that he works So he needs to work at least more hours, for a total of hours Since 44 hours would not quite be enough to work to earn $600, 45 hours is the first choice given that would be over $600 Therefore, the correct choice is H 3 Which equation has a slope of and a y-intercept of 6? Craig s goal is to earn at least $600 next week, what is the minimum number of hours he needs to work? F 43 hours G 44 hours H 45 hours J 46 hours If Wilson works 40 hours, he will earn 40(1280) = $512 So he needs to work more than 40 hours Let h represent the number of hours more than 40 hours he needs to work A B C D Slope intercept form is where m is the slope and b is the y-intercept Substituting in the given slope and y-intercept into the slope-intercept form, Therefore, the correct choice is C Page 1

136 So he needs to work at least more hours, for a total of hours Since 44 hours would not quite be enough to work to earn $600, 45 hours is the Standardized Practice - Cumulative, Chapter 1-5 first choicetest given that would be over $600 Therefore, the correct choice is H 3 Which equation has a slope of and a y-intercept of 6? A with < must be used The equal to must also be used to include both the lowest and highest score since they are also part of the range Therefore is the correct inequality and J is the correct answer 5 The current temperature is 82 If the temperature rises more than 4 degrees, there will be a new record high for the date Which number line represents the temperatures that would set a new record high? B C D Slope intercept form is where m is the slope and b is the y-intercept Substituting in the given slope and y-intercept into the slope-intercept form, Therefore, the correct choice is C 4 The highest score that is on record on a video game is 10,219 points The lowest score on record is 257 points Which of the following inequalities best shows the range of scores recorded on the game? F G H J The range of scores for the video game has both a lowest and highest score so a compound inequality with < must be used The equal to must also be used to include both the lowest and highest score since they are also part of the range Therefore is the correct inequality and J is the correct answer If the current temperature rises more than 4 degrees, a new record high h will be set The current temperature is 82 degrees Therefore the inequality that can be used to represent the temperatures at which a new record high will be set is h > , or h > 86 The number line that corresponds to this inequality has an open circle at 86 and an arrow that includes all of the values greater than 86 This is shown in choice B 6 The girls volleyball team is selling T-shirts and pennants to raise money for new uniforms The team hopes to raise more than $250 Which of the following combinations of items sold would meet this goal? F 16 T-shirts and 20 pennants G 20 T-shirts and 12 pennants H 18 T-shirts and 18 pennants J 15 T-shirts and 20 pennants 5 The current temperature is 82 If the temperature rises more than 4 degrees, there will be a new record high for the date Which number line represents the temperatures that would set a new record high? Let x = the number of T-shirts and y = the number of pennants Write an inequality $10 times the number of T-shirts plus $4 times the number of pennants is at least $250 Find the slope by rearranging the inequality similar to slopeintercept form Page 2

137 of pennants Write an inequality $10 times the number of T-shirts plus $4 times the number of pennants is at least $250 Find the slope by rearranging the inequality similar to slope- 1-5 Standardized Test Practice - Cumulative, Chapter intercept form Horizontal lines have a zero slope Parallel lines have the same slope Perpendicular lines have opposite reciprocal slopes Vertical lines are the only lines with an undefined slope The correct choice is D 8 Which expression below illustrates the Associative Property? F G H J Next, test each possibility in the inequality 16 T-shirts and 20 pennants 20 T-shirts and 12 pennants The Associative Property states that the way you group three or numbers when adding or multiplying does not change their sum or product Grouping is accomplished by ( )s Choice H is the only answer that changes the order of the numbers added but does not change the sum In choice F, the order of the factors changes (Commutative Property) Choice G distributes the factor of 2 (Distributive Property) Choice J is the Additive Inverse The correct choice is H Record your answers on the answer sheet provided by your teacher or on a sheet of paper 9 Solve 18 T-shirts and 18 pennants First, express solve each inequality using and Then 15 T-shirts and 20 pennants Choice C is the only choice that produces a correct statement and 7 What type of line does not have a defined slope? A horizontal B parallel C perpendicular D vertical Horizontal lines have a zero slope Parallel lines have the same slope Perpendicular lines have opposite reciprocal slopes Vertical lines are the only lines with an undefined slope The correct choice is D 8 Which expression below illustrates the Associative Property? F G H The solution is 10 GRIDDED RESPONSE Tien is saving money for a new television She needs to save at least $720 to pay for her expenses Each week Tien saves $50 toward her new television How many weeks will it take so she can pay for the television? Page 3 $50 per week number of weeks at least $720

138 that the inequality will also be equal to The line is shaded below, which means that the inequality is less than or equal to Therefore, the inequality is Standardized Test The solution is Practice - Cumulative, Chapter GRIDDED RESPONSE Tien is saving money for a new television She needs to save at least $720 to pay for her expenses Each week Tien saves $50 toward her new television How many weeks will it take so she can pay for the television? $50 per week number of weeks at least $720 Let w = number of weeks of saving 12 Solve Rewrite for Case 1 and Case 2 Case 1 is positive Case 2 is negative Round 144 to the next whole week of 15 to have enough money to pay for the television It will take her 15 weeks 11 Write an inequality that best represents the graph The answer is Use the slope- intercept form,, where m = the slope and b = the y-intercept From the graph, the y-intercept is 1 A second point can be located by moving down 2 units and left 3 units, so the slope is Substituting these values into 13 GRIDDED RESPONSE Daniel wants to ship a set of golf clubs and several boxes of golf balls in box that can hold up to 20 pounds If the set of clubs weighs 9 pounds and each box of golf balls weighs 12 ounces, how many boxes of golf balls can Daniel ship? Golf clubs + (cost of one box of golf balls number of boxes of golf balls) 20 pounds Let b = number of boxes of golf balls Before writing an inequality, convert all weights to the same type of units, The line is a solid line, which means that the inequality will also be equal to The line is shaded below, which means that the inequality is less than or equal to Therefore, the inequality is 12 Solve Rewrite for Case 1 and Case 2 Case 1 is positive Page 4

139 Standardized Test Practice - Cumulative, Chapter 1-5 The answer is 13 GRIDDED RESPONSE Daniel wants to ship a set of golf clubs and several boxes of golf balls in box that can hold up to 20 pounds If the set of clubs weighs 9 pounds and each box of golf balls weighs 12 ounces, how many boxes of golf balls can Daniel ship? The largest whole number (since partial boxes cannot be shipped) is 14 He can ship 14 boxes of golf balls 14 Graph the solution set for the inequality First, express Then solve each inequality using and Golf clubs + (cost of one box of golf balls number of boxes of golf balls) 20 pounds Let b = number of boxes of golf balls Before writing an inequality, convert all weights to the same type of units and The solution is The graph will be the numbers in-between and 5 with the circles filled in to include both and 5 15 Write an equation that represents the data in the table The largest whole number (since partial boxes cannot be shipped) is 14 He can ship 14 boxes of golf balls 14 Graph the solution set for the inequality First, express Then solve each inequality using and First determine if there is a constant rate of change by finding the change in the x values and y values Each x value increases by 1 and each value increases by 1 and each y value increases by 35 The slope of a line is determined by and Use the slope-intercept form, y = mx + b, and Page 5 substitute in the rate of change for m and the point (3, 125) for (x, y)

140 The solution is The graph will be the numbers in-between and 5 with the circles filled in to include both and 5 Standardized Test Practice - Cumulative, Chapter 1-5 Substitute 2 for b and 35 for m to get: y = 35x Write an equation that represents the data in the table 16 A sporting goods company near the beach rents bicycles for $10 plus $5 per hour Write an equation in slope-intercept form that shows the total cost, y, of renting a bicycle for x hours How much would it cost Emily to rent a bicycle for 6 hours? Total cost equals (cost per hour number of hours) + initial bike cost Let y = total cost and x = renting a bicycle for x hours To calculate Emily s cost for 6 hours, let x = 6 First determine if there is a constant rate of change by finding the change in the x values and y values Each x value increases by 1 and each value increases by 1 and each y value increases by 35 The slope of a line is determined by Use the slope-intercept form, y = mx + b, and substitute in the rate of change for m and the point (3, 125) for (x, y) Substitute 2 for b and 35 for m to get: y = 35x A sporting goods company near the beach rents bicycles for $10 plus $5 per hour Write an equation in slope-intercept form that shows the total cost, y, of renting a bicycle for x hours How much would it cost Emily to rent a bicycle for 6 hours? It will cost Emily $40 to rent the bike for 6 hours Record your answers on a sheet of paper Show your work 17 Theresa is saving money for a vacation She needs to save at least $640 to pay for her expenses Each week, she puts $35 towards her vacation savings Let w represent the number of weeks Theresa saves money a Write an inequality to model the situation b Solve the inequality from part a What is the minimum number of weeks Theresa must save money in order to reach her goal? c If Theresa were to save $45 each week instead, by how many weeks would the minimum savings time be decreased? a $35 per week number of weeks is at least $640 Let w = number of weeks of saving Total cost equals (cost per hour number of hours) + initial bike cost Let y = total cost and x = renting a bicycle for x hours b To calculate Emily s cost for 6 hours, let x = 6 Round to the next whole week of 19 to have enough money to pay for the television It will take her 19 weeks It will cost Emily $40 to rent the bike for 6 hours c $45 per week number of weeks is at leastpage 6 $640 Let w = number of weeks of saving

141 Standardized Test Practice - Cumulative, Chapter 1-5 It will cost Emily $40 to rent the bike for 6 hours Record your answers on a sheet of paper Show your work 17 Theresa is saving money for a vacation She needs to save at least $640 to pay for her expenses Each week, she puts $35 towards her vacation savings Let w represent the number of weeks Theresa saves money a Write an inequality to model the situation b Solve the inequality from part a What is the minimum number of weeks Theresa must save money in order to reach her goal? c If Theresa were to save $45 each week instead, by how many weeks would the minimum savings time be decreased? a $35 per week number of weeks is at least $640 Let w = number of weeks of saving b Round to the next whole week of 19 to have enough money to pay for the television It will take her 19 weeks c $45 per week number of weeks is at least $640 Let w = number of weeks of saving Round to the next whole week of 15 to have enough money to pay for the television It will take her 15 weeks Calculate the decrease The minimum savings time would decrease by 4 weeks Page 7

142 Study Guide and Review - Chapter 5 State whether each sentence is true or false If false, replace the underlined term to make a true sentence 1 Set-builder notation is a less concise way of writing a solution set The statement is false Set-builder notation is a more concise way of writing a solution set because it fully describes it in fewer words This is the form of a linear inequality The graph of a linear inequality involves shading one side of a boundary, which divides the coordinate plane into two half-planes So, the statement is true 6 A point defines the boundary of an open half-plane The statement is false A line defines the boundary between the two halfplanes 2 There are two types of compound inequalities There are two types of compound inequalities: those that include and (graphs that include intersections) and those that include or (graphs that include unions) So, the statement is true A point can be used as an open (circle) or closed (dot) endpoint in the solution set of an inequality 3 The graph of a compound inequality containing and shows the union of the individual graphs The statement is false The graph of a compound inequality containing and shows the intersection of two inequalities, which is only where the two overlap The graph of a union of two inequalities shows both inequalities, whether they overlap or not 4 A compound inequality containing or is true if one or both of the inequalities is true Its graph is the union of the graphs of the two inequalities A compound inequality containing or is true if at least one of the inequalities is true This union of the two inequalities has a graph in which both inequalities are shown So, the statement is true 5 The graph of an inequality of the form y < ax + b is a region on the coordinate plane called a half-plane This is the form of a linear inequality The graph of a linear inequality involves shading one side of a boundary, which divides the coordinate plane into two half-planes So, the statement is true 6 A point defines the boundary of an open half-plane The statement is false A line defines the boundary between the two halfplanes 7 The boundary is the graph of the equation of the line that defines the edge of each half-plane The boundary is the graph of the equation of the line that defines the edge of each half-plane If it is included in the solution, the solution is a closed halfplane If it is not included in the solution, then the solution is an open half-plane So, the statement is true 8 The solution set to the inequality y x includes the boundary The solution set to the inequality y x includes the boundary because y is greater than or equal to x Its graph would include a boundary with a solid line So, the statement is true 9 When solving an inequality, multiplying each side by a negative number reverses the inequality symbol When solving an inequality, multiplying or dividing each side by a negative number reverses the inequality symbol The direction of the inequality sign is reversed to make the resulting inequality also true So, the statement is true 10 The graph of a compound inequality that contains and is the intersection of the graphs of the two Page 1 inequalities

143 When solving an inequality, multiplying or dividing each side by a negative number reverses the inequality symbol The direction of the inequality sign Study Guide and Review Chapterinequality 5 is reversed to make the -resulting also true So, the statement is true 10 The graph of a compound inequality that contains and is the intersection of the graphs of the two inequalities The solution set is {h h < 5} 14 5 < a + 2 The graph of a compound inequality that contains and is the intersection of the graphs of the two inequalities It is where the two inequalities overlap So, the statement is true Solve each inequality Then graph it on a number line 11 w 4 > 9 Since 7 < a is the same as a > 7, the solution set is {a a > 7} p 15 The solution set is {w w > 13} The solution set is {p p 2} 12 x y The solution set is {x x 5} h < 1 The solution set is {y y 7} 17 FIELD TRIP A bus can hold 44 people If there are 35 students in Samantha s class, how many more people can ride on the bus? The solution set is {h h < 5} The bus holds a maximum of 44 people Let s equal how many more students can ride the bus Write an inequality 14 5 < a + 2 Since 7 < a is the same as a > 7, the solution set is {a a > 7} The solution set is {s s 9} So, no more than 9 more students can ride the bus Solve each inequality Graph the solution onpage a 2 number line p 15 18

144 The solution set is {p p < 8} Study Guide andset Review 5 more than 9 The solution is {s s- Chapter 9} So, no more students can ride the bus Solve each inequality Graph the solution on a number line w 18 The solution set is {w w 11} The solution set is {x x > 18} 22 2m > The solution set is {m m < 50} 23 The solution set is {g g 20} 20 4p < 32 The solution set is {t t < 72} The solution set is {p p < 8} w The solution set is {w w 11} 24 MOVIE RENTAL Jack has no more than $24 to spend on DVDs for a party Each DVD rents for $4 Find the maximum number of DVDs Jack can rent for his party The phrase no more than means the same as less than or equal to Let d be the number of DVDs Jack rents Write an inequality Page 3

145 The solution set is {t t < 72} The solution set is {b b > 6} Study Guide and Review - Chapter 5 24 MOVIE RENTAL Jack has no more than $24 to spend on DVDs for a party Each DVD rents for $4 Find the maximum number of DVDs Jack can rent for his party x + 8 The phrase no more than means the same as less than or equal to Let d be the number of DVDs Jack rents Write an inequality The solution set is {x x 5} The solution set is {d d 6} So, the maximum number of DVDs Jack can rent is 6 Solve each inequality Graph the solution on a number line 25 3h 7 < The solution set is {h h < 7} b > 34 The solution set is {t t > 6} 29 Four times a number decreased by 6 is less than 2 Define a variable, write an inequality, and solve for the number Let x = the number The solution set is {b b > 6} x + 8 The solution set is {x x < 1} Page 4 30 TICKET SALES The drama club collected $160 from ticket sales for the spring play They need to

146 The solution set is {t t > 6} Study Guide and Review - Chapter 5 29 Four times a number decreased by 6 is less than 2 Define a variable, write an inequality, and solve for the number Let x = the number The solution set is {t t 80} So, at least 80 more tickets need to be sold Solve each compound inequality Then graph the solution set 31 m 3 < 6 and m + 2 > 4 and The solution set is {m 2 < m < 9} To graph the solution set, graph m < 9 and graph m > 2 Then find the intersection The solution set is {x x < 1} 30 TICKET SALES The drama club collected $160 from ticket sales for the spring play They need to collect at least $400 to pay for new lighting for the stage If tickets sell for $3 each, how many more tickets need to be sold? 32 4 < 2t 6 < 8 First, express 4 < 2t 6 < 8 using and Then solve each inequality and The phrase at least means the same as greater than or equal to Let t be the number of tickets that still need to be sold Write an inequality The solution set is {t 1 < t < 7} To graph the solution set, graph t > 1 and graph t < 7 Then find the intersection The solution set is {t t 80} So, at least 80 more tickets need to be sold Solve each compound inequality Then graph the solution set 31 m 3 < 6 and m + 2 > x or 5x 8 > 22 or and The solution set is {m 2 < m < 9} To graph the solution set, graph m < 9 and graph m > 2 Then find the intersection 32 4 < 2t 6 < 8 The solution set is {x x 3 or x > 6} Notice that the graphs do not intersect To graph the solution set, graph x 3 and graph x > 6 Then find the union 34 KITES A large dragon kite can be flown in wind Page 5 speeds no less than 7 miles per hour and no more than 16 miles per hour Write an inequality for the

147 Notice that the graphs do not intersect To graph the solution set, graph x 3 and graph x > 6 Then find the union Study Guide and Review - Chapter 5 34 KITES A large dragon kite can be flown in wind speeds no less than 7 miles per hour and no more than 16 miles per hour Write an inequality for the wind speeds at which the kite can fly The solution set is {x 5 < x < 13} 36 Case 1 p + 2 is positive Let x be the wind speed The phrase no less than means the same as greater or equal to The phrase no more than means the same as less than or equal to The word and indicates that the problem represents an intersection and x 7 x 16 So, an inequality that represents the wind speeds for which the kite can be flown is 7 x 16 or Case 2 p + 2 is negative Solve each inequality Then graph the solution set 35 Case 1 x 4 is positive The solution set is {p p < 9 or p > 5} and Case 2 x 4 is negative 37 Case 1 2c + 3 is positive and Case 2 2c + 3 is negative The solution set is {x 5 < x < 13} 36 Case 1 p + 2 is positive The solution set is {c 7 c 4} 38 or Case 2 p + 2 is negative Case 1 f 9 is positive or Case 2 f 9 is negative The solution set is {p p < 9 or p > 5} Page 6 The solution set is {f f 7 or f 11}

148 The solution set is The solution set is {c 7 c 4} Study Guide and Review - Chapter Case 1 f 9 is positive or Case 2 f 9 is negative Case 1 is positive The solution set is {f f 7 or f 11} 39 Case 1 3d 1 is positive and Case 2 3d 1 is negative and Case 2 The solution set is is negative 40 Case 1 is positive The solution set is {b } 41 Case 1 Page 7 is positive

149 The solution set is {b } The solution set is {t t < 13 or t > 7} Study Guide and Review - Chapter Case 1 4y 3 is positive Case 1 is positive and Case 2 4y 3 is negative or Case 2 is negative The solution set is {y } 43 Case 1 m + 19 is positive The solution set is {t t < 13 or t > 7} and 42 Case 1 4y 3 is positive Case 2 m + 19 is negative Page 8

150 The solution set is {y } The solution set is {m 20 m 18} Study Guide and Review - Chapter Case 1 m + 19 is positive Case 1 k 7 is positive and Case 2 m + 19 is negative or Case 2 k 7 is negative The solution set is {m 20 m 18} 44 Case 1 k 7 is positive The solution set is {k k 3 or k 11} Graph each inequality 45 y > x 3 y>x 3 Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point or Case 2 k 7 is negative Since 0 is greater than 3, shade the half-plane that contains (0, 0) esolutions - Powered by Cognero The Manual solution set is {k k 3 or k 11} Page 9

151 The solution set is {k k 3 or k 11} Study Guide and Review - Chapter 5 Graph each inequality 45 y > x 3 y>x 3 Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point 46 y < 2x + 1 y < 2x + 1 Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is less than 1, shade the half-plane that contains (0, 0) Since 0 is greater than 3, shade the half-plane that contains (0, 0) 47 3x y 4 46 y < 2x + 1 Solve for y in terms of x y < 2x + 1 Because the inequality involves <, graph the boundary using a dashed line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is less than 1, shade the half-plane that contains (0, 0) Since 0 is greater than or equal to -4, shade the halfplane that contains (0, 0) 47 3x y 4 Page 10

152 Study Guide and Review - Chapter x y 4 Solve for y in terms of x 48 y 2x + 6 y 2x + 6 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is not greater than or equal to 6, shade the half-plane that does not contain (0, 0) Since 0 is greater than or equal to -4, shade the halfplane that contains (0, 0) 49 5x 2y < 10 Solve for y in terms of x 48 y 2x + 6 y 2x + 6 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is not greater than or equal to 6, shade the half-plane that does not contain (0, 0) Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is greater than -5, shade the half-plane that contains (0, 0) Page 11

153 Study Guide and Review - Chapter x 2y < 10 Solve for y in terms of x 50 x + y 1 Solve for y in terms of x Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is not greater than 1, shade the half-plane that does not contain (0, 0) Since 0 is greater than -5, shade the half-plane that contains (0, 0) Graph each inequality Determine which of the ordered pairs are part of the solution set for each inequality 51 y 4 {(3, 6), (1, 2), ( 4, 8), (3, 2), (1, 7)} 50 x + y 1 Solve for y in terms of x y 4 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point 0 4 Since 0 is less than or equal to 4, shade the half-plane that contains (0, 0) Because the inequality involves >, graph the boundary using a dashed line Choose (0, 0) as a test point Since 0 is not greater than 1, shade the half-plane that does not contain (0, 0) To determine which of the ordered pairs are part of the solution set, plot them on the graph Page 12

154 52 2x + 3y 12 {( 2, 2), ( 1, 1), (0, 4), (2, 2)} Study Guide and Review - Chapter 5 Solve for y in terms of x Graph each inequality Determine which of the ordered pairs are part of the solution set for each inequality 51 y 4 {(3, 6), (1, 2), ( 4, 8), (3, 2), (1, 7)} y 4 Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point 0 4 Since 0 is less than or equal to 4, shade the half-plane that contains (0, 0) Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is not greater than or equal to 12, shade the half-plane that does not contain (0, 0) To determine which of the ordered pairs are part of the solution set, plot them on the graph Because (1, 2) and (3, 2) are in the shaded halfplane, they are part of the solution set To determine which of the ordered pairs are part of the solution set, plot them on the graph 52 2x + 3y 12 {( 2, 2), ( 1, 1), (0, 4), (2, 2)} Solve for y in terms of x Because (0, 4) lies on the boundary, it is part of the solution set Because the inequality involves, graph the 53 BAKERY Ben has $24 to spend on cookies and cupcakes Write and graph an inequality that Page 13 represents what Ben can buy

155 Study Guide(0, and - Chapter 5 it is part of the Because 4)Review lies on the boundary, solution set 53 BAKERY Ben has $24 to spend on cookies and cupcakes Write and graph an inequality that represents what Ben can buy Let x be the number of cookies and let y be the number of cupcakes The cookies are $2 each, so the cost of the cookies is 2x The cupcakes are $3 each, so the cost of the cupcakes is 3y Because Ben can spend no more than $24, the total cost must be less than or equal to $24 Write an inequality 2x + 3y 24 Solve for y in terms of x Because the inequality involves, graph the boundary using a solid line Choose (0, 0) as a test point Since 0 is less than or equal to 24, shade the halfplane that contains (0, 0) Page 14