Table of Contents Simple Inequalities Addition/Subtraction Simple Inequalities Multiplication/Division Two-Step and Multiple-Step Inequalities Solving Compound Inequalities Special Cases of Compound Inequalities Graphing Linear Inequalities in Slope-Intercept Form Solving Systems of Inequalitites Glossary & Standards Simple Inequalities Involving Addition and Subtraction 1
Inequality An Inequality is a mathematical sentence that uses symbols, such as <,, > or to compare to quantities. What do these symbols mean? (when read from LEFT to RIGHT) Less Than click Less Than or Equal To Greater Than click Greater Than or Equal To Reading Inequalities Remember: Inequalities can be read either direction, as long as you use the correct vocabulary for the symbol. "15 is less than x" OR "x is greater than 15" "7 is greater than or equal to x" OR "x is less than or equal to 7" 2
Inequality Write an inequality for the sentence below: Three times a number, n, is less than 210. Click The sum of a number, n, and fifteen is greater than or equal to nine. Click Graphing Inequalities Remember! Open circle means that number is not included in the solution set and is used to represent < or >. Closed circle means the solution set includes that number and is used to represent or. Solving Inequalities Solving one-step inequalities is much like solving one-step equations. To solve an inequality, you need to isolate the variable using the properties of inequalities and inverse operations. 3
Isolate the Variable To find the solution, isolate the variable x. Remember, it is isolated when it appears by itself on one side of the equation. Solving Inequalities Step 1: Since 6 is added to x and subtraction is the inverse of addition, subtract 6 from both sides to undo the addition. Solving Inequalities of 6 for x. In this case, the end point is not included (open circle) since x < 6. 4
Solving Inequalities Step 3: Shade your number line to represent the solution set. To confirm, choose a number from your line (such as 4) and check that it Review of Solving Inequalities Using Addition and Subtraction The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at: http://www.njctl.org/courses/math/7th-grade/ equations-inequalities-7th-grade/ 1 5
2? 3? 4? 6
5? 1.5 1.5 1.5 1.5 Simple Inequalities Involving Multiplication and Division Inequalities Involving Multiplication and Division Again, similarly to solving equations, we can use the properties of multiplication and division to solve and graph inequalities - with one minor difference, which we will encounter in the upcoming slides. 7
Multiplying or Dividing by a Positive Number Since x is multiplied by 3, divide both sides by 3 to isolate the variable. Solve and Graph Since r is multiplied by 2 / 3, we can multiply both sides by the reciprocal of 2 / 3, which is 3 / 2. click for answer Review of Solving Inequalities Using Multiplication and Division The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at: http://www.njctl.org/courses/math/7th-grade/ equations-inequalities-7th-grade/ 8
6 Which graph is the solution to the inequality, the product of 4 and a number, x, is greater than 24? 7 Which inequality is the solution to: 8 9
9 10 Find the solution to the inequality. Multiplying or Dividing by a Negative Number So far, all the operations we have used worked the same as solving equations. The difference between solving equations versus inequalities is revealed when multiplying or dividing by a negative number. The direction of the inequality changes only if the number you are using to multiply or divide by is negative. 10
Solve and Graph *Note: Dividing each side by -3 changes the to. click for answer 11 Solve the inequality and graph the solution. 12 Solve the inequality and graph the solution. 11
13 Solve the inequality and graph the solution. 14 Solve the inequality and graph the solution. Summary In review, an inequality symbol stays the same direction when you: Add, subtract, multiply or divide by the same positive number on both sides. Add or subtract the same negative number on both sides. An inequality symbol changes direction when you: Multiply or divide by the same negative number on both sides. 12
Solving Two-Step and Multiple-Step Inequalities Inequalities Now we'll solve more complicated inequalities that have multi-step solutions because they involve more than one operation. Solving inequalities is like solving a puzzle. Keep working through the steps until you get the variable you're looking for alone on one side of the inequality using the same strategies as solving an equation. Two Step Inequalities You can solve two step inequalities in the same way you solve equations. is solved in the same way as You can add any positive or negative number to both sides of the inequality. You can multiply or divide both sides of an equality by any positive number. 13
Multiplying or Dividing by a Negative Number Another reminder! If you multiply or divide by a negative number, reverse the direction of the inequality symbol! Example: Solve the following inequality: Step 1: Use additive inverse Step 2: Use multiplicative inverse Example: Solve the inequality and graph the solution. Add 9 to both sides Divide both sides by 4 (sign stays the same) click for answer 14
Try these. Solve each inequality and graph each solution. 1. Solve and Graph 2. Solve and Graph Try these. Solve each inequality and graph the solution. 15 2.5 2.5 2.5 2.5 15
16 17 18 16
19 Solve and graph the solution. 20 Which graph represents the solution set for: 21 Find all negative odd integers that satisfy the following inequality. Select all that apply. A B C D E F G H From the New York State Education Department. Office of Assessment Policy, Development and 17
22 From the New York State Education Department. Office of Assessment Policy, Development and 23 From the New York State Education Department. Office of Assessment Policy, Development and 24 From the New York State Education Department. Office of Assessment Policy, Development and 18
25 From the New York State Education Department. Office of Assessment Policy, Development and 26 From the New York State Education Department. Office of Assessment Policy, Development and Inequalities in the Real World Inequalities are helpful when applied to real life scenarios. These inequalities can be used for budgeting purposes, speed limits, cell phone data usage, and building materials management, just to name a few. Translating between the languages of English words to numbers/ symbols is imperative in being able to solve the correct inequality. The next slides will provide ample practice in setting up and solving these inequality applications. 19
Write an Inequality and Solve Example #1: Your town is having a fall carnival. Admission into the carnival is $3.00 and each game inside costs $0.25. Write an inequality that represents the possible number of games that can be played if you have $10.00. What is the maximum number of games that can be played? Maximum number of games that can be played is 28. Example #2: You have $65.00 in birthday money and want to buy some CDs and a DVD. Suppose a DVD cost $15.00 and a CD cost $12.00. Write an inequality and solve to find out how many CDs you can buy along with one DVD. Example #3: Matt was getting ready to go back to school. He had $150 to buy school supplies. Matt bought 3 pairs of pants and spent $30 on snacks and other items. How much could one pair of pants cost, if they were all the same price? Write an inequality and solve. 20
Example #4: You have $60 to spend on a concert. Tickets cost $18 each and parking is $8. Write an inequality to model the situation. How many tickets can you buy? Example #5: If you borrow the $60 from your mom and pay her back at a rate of $7 per week, when will your debt be under $15? Example #6: To earn an A in math class, you must earn a total of at least 180 points on three tests. On the first two tests, your scores were 58 and 59. What is the minimum score you must get on the third test in order to earn an A? Define a variable, write an inequality and graph the solutions. 21
From the New York State Education Department. Office of Assessment Policy, Development and 27 Roger is having a picnic for 78 guests. He plans to serve each guest at least one hot dog. If each package, p, contains eight hot dogs, which From the New York State Education Department. Office of Assessment Policy, Development and 28 A school group needs a banner to carry in a parade. The narrowest street the parade is marching down measures 36 ft across, but some space is taken up by parked cars. The students have decided the banner should be 18 ft long. There is 45 ft of trim available to sew around the border of the banner. What is the greatest possible width for the banner? A B C D 22
29 Admission to a town fair is $7.00. You plan to spend $6.00 for lunch and $4.50 for snacks. Each ride costs $2.25. If you have $35 to spend, what is the number of rides you can go on? A B C D 6 rides 7 rides 8 rides 9 rides 30 A female gymnast is participating in a 4-event competition. Each event is scored on a ten-point scale. She scored a 9.1 in uneven bars, an 8.5 on the balance beam, and a 9.4 on the vault. Which inequality represents the remaining score required in the floor exercise for the gymnast to receive at least an 8.9 average? A r 8.975 B r 8.6 C r 8.975 D r 8.6 Solving Compound Inequalities 23
Compound Inequalities When two inequalities are combined into one statement by the words AND/OR, the result is called a compound inequality. A solution of a compound inequality joined by and is any number that makes both inequalities true. A solution of a compound inequality joined by or is any number that makes either inequality true. Here are some examples. AND Compound Inequalities AND NOTE: "and" means intersection, so you graph the intersection of the two inequalities Compound Inequalities Here are some additional examples. OR OR NOTE: "or" means union, so you graph the union of the two inequalities 24
31 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from 32 From the New York State Education Department. Office of Assessment Policy, Development and Solving Compound Inequalities that contain an AND statement You will need to solve both of these inequalities and graph their intersection. 25
Solving Compound Inequalities that contain an AND statement AND AND Step 1 Rewrite as 2 separate inequalities. Step 2 Solve each inequality for x. AND Step 3 Graph your solution. click for answer Solving Compound Inequalities that contain an AND statement What do I do first? AND And now? AND What do I do with my solution? click to reveal 33 Which result below is correct for this inequality: 26
34 Which result below is correct for this inequality: 2 1 / 2 2 1 / 2 35 Which result below is correct for this inequality: 36 Which result below is correct for this inequality: 27
37 Which result below is correct for this inequality: Solving Compound Inequalities that contain an OR statement OR Just like before, solve each one separately. However, with OR statements, graph their union. OR OR Solving Compound Inequalities that contain an OR statement OR Solve each one separately, then graph their union. OR OR click for answer 28
Writing a Compound Inequality From a Graph How would you write this? Writing a Compound Inequality From a Graph How would you write this? Compound Inequalities Solve and graph the solution set. 1. 29
Compound Inequalities Solve and graph the solution set. 4. 38 In order to be admitted for a certain ride at an amusement park, a child must be greater than or equal to 36 inches tall and less than 48 inches From the New York State Education Department. Office of Assessment Policy, Development and 39 Which graph shows the solution to this compound inequality? or 30
40 From the New York State Education Department. Office of Assessment Policy, Development and 41 Solve 42 31
43 From the New York State Education Department. Office of Assessment Policy, Development and 44 Write the inequality shown by the graph. 45 Write the inequality shown by the graph. 32
Application of Compound Inequalities Let's start off by translating the words of an applied problem into math. The sum of 3 times a number and two lies between 8 and 11. "The sum of 3 times a number and two" translates into what? Application of Compound Inequalities The sum of 3 times a number and two lies between 8 and 11. How will we translate "lies between 8 and 11"? What inequality symbol will we use? Why? What is the inequality? Solve and graph the inequality. Application of Compound Inequalities A cell phone plan offers free minutes for no more than 250 minutes per month. Define a variable and write an inequality for the possible number of free minutes. Graph the solution. 33
46 Each type of marine mammal thrives in a specific range of temperatures. The optimal temperatures for dolphins range from 50 F to 90 F. Which inequality represents the temperatures where dolphins will not thrive? A B C D 47 About 20% of the time that you sleep is spent in rapid eye movement (REM) sleep, which is associated with dreaming. If a teen sleeps 8 to 10 hours, which inequality represents the time spent in REM sleep? A B C D 48 A store is offering a $50 mail in rebate on all color printers. Nathan is looking at different color printers that range in price from $165 to $275. How much can he expect to spend after the rebate? A $115 x $225 B x < $115 or x > $225 C $215 x $325 D x < $215 or x > $325 34
49 One quarter of a number decreased by 7 is at most 11 or greater than 15. Which compound inequality represents the possible values of the number? A B C D 50 Lyla has scores of 82, 92, 93, and 99 on her math tests. Use a compound inequality to find the range of scores she can make on her final exam to receive a B in the course. The final exam counts as two test grades, and a B is received if the final course average is from 85 to 92. A B C D Special Cases of Compound Inequalities 35
Special Cases A solution of a compound inequality joined by and is any number that makes both inequalities true. When there is no number that makes both inequalities true, we say there is no solution. When all numbers make both inequalities true, we say the solution is the set of Reals or All Reals. No Solution AND AND AND The solution set is No Solution since there are no numbers that are both greater than or equal to 9 and less than -4. We write this solution as { } or. All Real Numbers OR OR OR OR The solution set is All Reals since all numbers are either less than or equal to -7 or greater than -10. We write this solution set as R. -11-10 -9-8 -7-6 -5-4 -3-2 -1 36
Special Cases Solve each set of compound inequalities. Special Cases Solve each set of compound inequalities. Graphing Linear Inequalities in Slope-Intercept Form 37
Graphing Graphs of inequalities are similar to linear equations because they both have points on a coordinate plane and a line connecting the points. However, a linear equation is ONLY the line but an inequality extends beyond that line. Linear Equation: Inequality: Graphing Graphing 38
1) Decide where the boundary goes: Solve inequality for y, for example y > 2x - 1 2) Decide whether boundary should be: - solid ( or : points on the boundary make the inequality true) or - dashed (< or >: points on the boundary make the inequality false) 3) Graph the boundary (the line). 4) Decide where to shade: y > or y : shade above (referring to y-axis) the boundary y < or y : shade below (referring to y-axis) the boundary Or, you can test a point Graph Graphing Step 1: Solve for y m = -2 and b = 1 Step 2: The line should be dashed because the inequality is < Step 3: Graph boundary Step 4: Shade below the boundary line because y < Graph Graphing Step 2: The line should be solid because the inequality is Step 3: Graph boundary Step 4: Shade above the boundary line because y 39
Graph Graphing Is the line solid or dashed? Explain why this is the case. The line is dashed because it is not included in the inequality. click to reveal Will we shade above or below the line? Explain why this is the case. You shade above the line because the inequality shows that y is greater than the expression on the right hand side. Or, if you test a point (0, 0), it satisfies the inequality, so click to reveal you shade in that direction. click to reveal the inequality graph 51 Why are there dashed boundaries on some graphs of inequalities? A Points on the line make the inequality false. B Points on the line make the inequality true. C The slope of the line depends on the line type. D The y-intercept depends on the line type. 52 solid boundary and be shaded above? 40
53 For which of these inequalities would the graph have a dashed boundary and be shaded above? 54 55 Which inequality matches the given graph? A B C D 41
56 When you finish, type the number "1" into your responder. PARCC - EOY - Question #2 Non-Calculator Section - SMART Response Format Modeling with Inequalities Throughout this unit, you have learned how to solve and graph inequalities, both on a number line and in the coordinate plane. We can apply these skills to solve realistic word problems, such as purchasing items at a store within a budget and earning money through various jobs. Let's get started. Modeling with Inequalities At a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants. Part A Write an inequality that would be used to model the situation. Part B Graph the inequality in a coordinate plane. Part C List 3 combinations of dress shirts and pairs of dress pants that could be purchased within your budget. 42
Modeling with Inequalities At a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants. Part A Write an inequality that would be used to model the situation. At a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 y each. You have $125 to spend. Let x represents 20 the dress shirts and y represents the number of pairs of dress pants. 15 Part B Graph the inequality in a coordinate plane. Modeling with Inequalities 10 5 0 5 10 15 20 x Modeling with Inequalities At a department store, dress shirts cost $12.50 each and each pair of dress pants cost $25 each. You have $125 to spend. Let x represents the dress shirts and y represents the number of pairs of dress pants. Part C List 3 combinations of dress shirts and pairs of dress pants that could be purchased within your budget. 43
57 At a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs. Part A Which inequality would be used to model this situation? A B C D 58 At a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs. Part B Graph your solution in the coordinate plane below. When you are finished, type the number "1" into your responder. 20 15 10 y 5 0 5 10 15 20 x 59 At a sports shop, soccer balls cost $18 each and footballs cost $15 each. You have $90 to spend. Let x represents the number of soccer balls and y represents the number of footballs. Part C Which pairs (x, y) can represent the amount of soccer balls and footballs purchased at the sports shop? Select all that apply. A (7, 1) B (2, 3) C (4, 6) D (3, 3) E (1, 4) 44
60 A group of friends went to the movies on Friday night. After purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket. Let x represent the number of sodas purchased and y represent the buckets of popcorn purchased. Part A Which inequality would be used to model this situation? A B C D 61 A group of friends went to the movies on Friday night. After purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket. Let x represent the y number of sodas purchased and y 20 represent the buckets of popcorn purchased. 15 Part B Graph your solution in the coordinate plane below. When you are finished, type the number "1" into your responder. 10 5 0 5 10 15 20 x 62 A group of friends went to the movies on Friday night. After purchasing the tickets, they had $30 left to spend on soda, which costs $1.50 per cup and popcorn, which costs $4.50 per bucket. Let x represent the number of sodas purchased and y represent the buckets of popcorn purchased. Part C Which pairs (x, y) can represent the amount spent on soda and buckets of popcorn at the theater? Select all that apply. A (17, 1) B (10, 5) C (8, 4) D (5, 5) E (3, 7) 45
Solving Systems of Inequalities Return to Contents A system of linear inequalities is two or more linear inequalities. The solution to a system of linear inequalities is the intersection of the half-planes formed by each linear inequality. The most direct way to find the solution to a system of linear inequalities is to graph the equations on the same coordinate plane and find the region of intersection. Graphing a System of Linear Inequalities Step 1: Graph the boundary lines of each inequality. Remember: - dashed line for < and > - solid line for and Step 2: Shade the half-plane for each inequality. Step 3: Identify the intersection of the half-planes. This is the solution to the system of linear inequalities. 46
Example Solve the following system of linear inequalities. Step 1: y 10 5-10 -5 0 5 10 x -5-10 Example Continued 10 y 5 x -10 Example Continued 10 y 5 x -10 47
Example Solve the following system of linear inequalities. Step 1: y 10 5-10 -5 0 5 10 x -5-10 Example Continued Example Continued 10 y 5-10 -5 0 5 10 x -5-10 48
Example Solve the following system of linear inequalities. Step 1: y 10 5-10 -5 0 5 10 x -5-10 Example Continued Step 2: y 10 5-10 -5 0 5 10 x -5-10 Example Continued Step 3: y 10 5-10 -5 0 5 10 x -5-10 49
Example Solve the following system of linear inequalities. Step 1: 10 y 5-10 -5 0 5 10 x -5-10 Example Continued Step 2: y 10 5-10 -5 0 5 10 x -5-10 Example Continued Step 3: 10 y 5-10 -5 0 5 10 x -5-10 50
63 Choose the graph below that displays the solution to the following system of linear inequalities: 64 Choose the graph below that displays the solution to the following system of linear inequalities: 65 Choose the graph below that displays the solution to the following system of linear inequalities: 51
66 Choose the graph below that displays the solution to the following system of linear inequalities: 67 Choose all of the linear inequalities that correspond to the following graph: A B C D 68 Which point is in the solution set of the system of inequalities shown in the accompanying graph? 52
69 70 71 53
72 A system of inequalities is given. Graph the solution set of the system of linear inequalities in the coordinate plane. When you finish, type the number "1" into your Responder. PARCC - PBA - Question #3 Non-Calculator Section - SMART Response Format Modeling with a System of Inequalities Similar to solving application problems by graphing a single inequality, we can also apply our skills with solving a system of inequalities to solve realistic word problems. Let's get started. Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli. Part A: Graph the solution set of the system of linear inequalities in a coordinate plane. Part B: Create 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions. Part C: Given the conditions in Part A, if Preston mows lawns for 9 hours this month, what is the minimum number of hours he would have to work at the deli to earn at least $150? Give your answer to the nearest whole hour. Part D: Given the conditions in Part A, Preston prefers mowing lawns over working at the deli. What is the maximum number of hours he can mow lawns to be able to earn at least $150? Give your answer to the nearest whole hour. 54
Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of y 15 hours per month. Let x represent the number 20 of hours Preston mows lawns and y represent the number of hours 15 Preston works at the deli. Part A: Graph the solution set of the system of linear inequalities in a coordinate plane. 10 5 0 5 10 15 20 x Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli. Part B: Create 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions. Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli. Part C: Given the conditions in Part A, if Preston mows lawns for 5 hours this month, what is the minimum number of hours he would have to work at the deli to earn at least $150? Give your answer to the nearest whole hour. 55
Modeling with a System of Inequalities Preston would like to earn at least $150 per month. He mows lawns for $8 per hour and works at a deli for $12 per hour. Preston cannot work more than a total of 15 hours per month. Let x represent the number of hours Preston mows lawns and y represent the number of hours Preston works at the deli. Part D: Given the conditions in Part A, Preston prefers mowing lawns over working at the deli. What is the maximum number of hours he can mow lawns to be able to earn at least $150? Give your answer to the nearest whole hour. 73 Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at least $210. Gavin y knows that the will sell more than 20 40 comic books. Let x represent the number of comic books sold 30 and y represent the packs of baseball cards sold. Part A: Graph the solution set of the system of linear inequalities in a coordinate plane. When you finish, type the number "1" into your Responder. 20 10 0 10 20 30 40 x 74 Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at least $210. Gavin knows that the will sell more than 20 comic books. Let x represent the number of comic books sold and y represent the packs of baseball cards sold. Part B Which pairs (x, y) represent the sales of comic books and packs of baseball cards to meet the given conditions? Select all that apply. A (25, 25) B (26, 8) C (30, 10) D (35, 25) E (18, 40) 56
75 Gavin is selling comic books and baseball cards to make money for summer vacations. The comic books each cost $6 and baseball cards cost $5 for a single pack. He needs to make at least $210. Gavin knows that the will sell more than 20 comic books. Let x represent the number of comic books sold and y represent the packs of baseball cards sold. Part C Given the conditions in Part A, if Gavin sold 14 packs of baseball cards, what is the minimum number of comic books he would need to sell to earn at least $210? Give your answer to the nearest whole number. 76 Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop. Part A Graph the solution set of the system of linear inequalities in the coordinate plane. When you finish, type the number "1" into your Responder. PARCC - EOY - Question #25 Calculator Section - SMART Response Format 77 Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop. Part B Which pairs (x, y) represent hours that Leah could work to meet the given conditions? Select all that apply. A (4, 15) B (5, 12) C (10, 9) D (15, 5) E (19, 1) PARCC - EOY - Question #25 Calculator Section 57
78 Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop. Part C Given the conditions in Part A, if Leah babysits for 7 hours this month, what is the minimum number of hours she would have to work at the ice cream shop to earn at least $120? Give your answer to the nearest whole hour. PARCC - EOY - Question #25 Calculator Section 79 Leah would like to earn at least $120 per month. She babysits for $5 per hour and works at an ice cream shop for $8 per hour. Leah cannot work more than a total of 20 hours per month. Let x represent the number of hours Leah babysits and y represent the number of hours Leah works at the ice cream shop. Part D Given the conditions in Part A, Leah prefers babysitting over working at the ice cream store. What is the maximum number of answer to the nearest whole hour. PARCC - EOY - Question #25 Calculator Section Glossary & Standards 58
Inequality An Inequality is a mathematical sentence that uses symbols, such as <,, > or to compare to quantities. 2 < 18 x > 6 x -3 r - 9 2 + 9 +9 r 11 Solution Set Any number that, when substituted into an equation/inequality, will satisfy the equation/ inequality r - 9 = 2 + 9 +9 r = 11 {11} check: 11-9 = 2 2 = 2 r - 9 2 + 9 +9 r 11 Solution is not included! Solution is included! Compound Inequality Two inequalities that are combined into one statement by the words AND/OR x > -2 AND x < 3-2 < x < 3 "and" means intersection x -2 OR x 3 "or" means union 59
No Solution When there is no number that makes the equation/inequalities true 2x + 8 = 2(x - 4) 2x + 8 = 2x - 8 8 = -8 2x 18 AND -3x > 12 x 9 AND x < -4 { } or { } "no solution" Reals When all (any) numbers make the equation/inequalities true 2x + 8 = 2(x + 4) 2x + 8 = 2x + 8 0 = 0 R -2x + 3 > 17 OR 5(x + 2) > -40 x -7 OR x > -10 R "reals" "all real numbers" R System of Linear Inequalities Two or more linear inequalities 10 y 5-10 -5 0 5 10 x -5-10 60
Throughout this unit, the Standards for Mathematical Practice are used. MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: Construct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: Attend to precision. MP7: Look for & make use of structure. MP8: Look for and express regularity in repeated reasoning. Additional questions are included on the slides using the "Math Practice" Pull-tabs (e.g. a blank one is shown to the right on this slide) with a reference to the standards used. If questions already exist on a slide, then the specific MPs that the questions address are listed in the Pull-tab. 61