Slide 1 / 182 Slide 2 / 182 lgebra I Solving & Graphing Inequalities 2016-011 www.njctl.org Slide 3 / 182 Slide 4 / 182 Table of ontents Simple Inequalities ddition/subtraction click on the topic to go to that section Simple Inequalities Multiplication/ivision Two-Step and Multiple-Step Inequalities Solving ompound Inequalities Special ases of ompound Inequalities Graphing Linear Inequalities in Slope-Intercept Form Solving Systems of Inequalitites Glossary & Standards Simple Inequalities Involving ddition and Subtraction Return to Table of ontents Slide 5 / 182 Inequality Slide 6 / 182 What do these symbols mean? (when read from LEFT to RIGHT) n Inequality is a mathematical sentence that uses symbols, such as <,, > or to compare to quantities. Less Than click Less Than or Equal To Greater Than click Greater Than or Equal To
Slide 7 / 182 Slide 8 / 182 Inequality Write an inequality for the sentence below: Three times a number, n, is less than 210. lick The sum of a number, n, and fifteen is greater than or equal to nine. lick Slide 9 / 182 Graphing Inequalities Remember! Open circle means that number is not included in the solution set and is used to represent < or >. Slide 10 / 182 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. losed circle means the solution set includes that number and is used to represent or. Slide 11 / 182 Slide 12 / 182 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.
Slide 13 / 182 Slide 14 / 182 Solving Inequalities Step 2: ecide whether or not the circle on your boundary should be open or closed based on the symbol used. You can check the computation by substituting the end point of 6 for x. In this case, the end point is not included (open circle) since x < 6. -9-8 -7-6 Slide 15 / 182 Review of Solving Inequalities Using ddition and Subtraction The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at: Slide 16 / 182 1 Which graph is the solution to the inequality: a number, n, minus is greater than one third? 5 2 6 0 1 2 3 4 5 2 5 6 0 1 2 3 4 5 http://www.njctl.org/courses/math/7th-grade/ equations-inequalities-7th-grade/ 2 5 6 0 1 2 3 4 5 2 5 6 0 1 2 3 4 5 Slide 17 / 182 Slide 18 / 182 2 Which graph is the solution to the inequality? 3 Which graph is the solution to the inequality? -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 -9-8 -7-6
Slide 19 / 182 4 Which graph is the solution to the inequality? Slide 20 / 182 5 Which graph is the solution to the inequality? -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 1.5-9 -8-7 -6 1.5-9 -8-7 -6 1.5-9 -8-7 -6 1.5-9 -8-7 -6 Slide 21 / 182 Slide 22 / 182 Inequalities Involving Multiplication and ivision Simple Inequalities Involving Multiplication and ivision gain, 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. Return to Table of ontents Slide 23 / 182 Slide 24 / 182 Multiplying or ividing by a Positive Number Since x is multiplied by 3, divide both sides by 3 to isolate the variable. -9-8 -7-6
Slide 25 / 182 Review of Solving Inequalities Using Multiplication and ivision The following formative assessment questions are review from 7th grade. If further instruction is need, see the presentation at: Slide 26 / 182 6 Which graph is the solution to the inequality, the product of 4 and a number, x, is greater than 24? -9-8 -7-6 -9-8 -7-6 http://www.njctl.org/courses/math/7th-grade/ equations-inequalities-7th-grade/ -9-8 -7-6 -9-8 -7-6 Slide 27 / 182 Slide 28 / 182 Slide 29 / 182 Slide 30 / 182 9 Find the solution to the inequality. 10 Find the solution to the inequality.
Slide 31 / 182 Multiplying or ividing by a Negative Number Slide 32 / 182 Solve and Graph 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. *Note: ividing each side by changes the to. The direction of the inequality changes only if the number you are using to multiply or divide by is negative. -9-8 -7-6 00 11 22 334 45 56 67 78 89 91010 click for answer Slide 33 / 182 Slide 34 / 182 11 Solve the inequality and graph the solution. 12 Solve the inequality and graph the solution. -9-8 -7-6 -9-8 -7-6 Slide 35 / 182 Slide 36 / 182 13 Solve the inequality and graph the solution. 14 Solve the inequality and graph the solution. -9-8 -7-6 -9-8 -7-6
Slide 37 / 182 Slide 38 / 182 Summary In review, an inequality symbol stays the same direction when you: dd, subtract, multiply or divide by the same positive number on both sides. dd or subtract the same negative number on both sides. n inequality symbol changes direction when you: Multiply or divide by the same negative number on both sides. Solving Two-Step and Multiple-Step Inequalities Return to Table of ontents Slide 39 / 182 Slide 40 / 182 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. Slide 41 / 182 Multiplying or ividing by a Negative Number Slide 42 / 182 nother reminder! If you multiply or divide by a negative number, reverse the direction of the inequality symbol!
Slide 43 / 182 Two Step Inequalities Example: Solve the inequality and graph the solution. Slide 44 / 182 Solve and Graph Try these. Solve each inequality and graph each solution. 1. dd 9 to both sides ivide both sides by 4 (sign stays the same) -9-8 -7-6 2. -9-8 -7-6 0 11 22 3344 556 67 78 89 91010-9 -8-7 -6 click for answer Slide 45 / 182 Slide 46 / 182 Solve and Graph 15 Solve and graph the solution. Try these. Solve each inequality and graph the solution. 2.5 3. 0 1 2 3 4 5 2.5-9 -8-7 -6 4. 0 1 2 3 4 5 2.5 0 1 2 3 4 5-9 -8-7 -6 2.5 0 1 2 3 4 5 Slide 47 / 182 Slide 48 / 182
Slide 49 / 182 Slide 50 / 182 19 Solve and graph the solution. -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 Slide 51 / 182 20 Which graph represents the solution set for: Question from P lgebra I End-of-ourse Practice Test 21 Slide 52 / 182 Find all negative odd integers that satisfy the following inequality. Select all that apply. 0 1 2 0 1 2 E F G 0 1 2 H 0 1 2 From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 53 / 182 22 Which value of x is in the solution set of? Slide 54 / 182 23 What is the solution of? 8 9 12 16 From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011.
Slide 55 / 182 Slide 56 / 182 24 In the set of positive integers, what is the solution set of the inequality? From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 57 / 182 26 Given: etermine all elements of set that are in the solution of the inequality. 18 6 2 Slide 58 / 182 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. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 59 / 182 Slide 60 / 182 Write an Inequality and Solve Example #2: You have $65.00 in birthday money and want to buy some s and a V. Suppose a V cost $15.00 and a cost $12.00. Write an inequality and solve to find out how many s you can buy along with one V.
Slide 61 / 182 Write an Inequality and Solve 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. Slide 62 / 182 Write an Inequality and Solve 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? How much could one pair of pants cost, if they were all the same price? Write an inequality and solve. Slide 63 / 182 Write an Inequality and Solve 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? Slide 64 / 182 Write an Inequality and Solve Example #6: To earn an 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? efine a variable, write an inequality and graph the solutions. -9-8 -7-6 Slide 65 / 182 Slide 66 / 182 Write an Inequality and Solve Example #7: Thelma and Laura start a lawn-mowing business and buy a lawnmower for $225. They plan to charge $15 to mow one lawn. What is the minimum number of lawns they need to mow if they wish to earn a profit of at least $750? 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 inequality could be used to determine how many packages of hot dogs Roger will need to buy? From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011 From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011.
Slide 67 / 182 28 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? Slide 68 / 182 29 dmission 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? 6 rides 7 rides 8 rides 9 rides Slide 69 / 182 30 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? r 8.975 r 8.6 Slide 70 / 182 Solving ompound Inequalities r 8.975 r 8.6 Return to Table of ontents Slide 71 / 182 Slide 72 / 182 ompound Inequalities When two inequalities are combined into one statement by the words N/OR, the result is called a compound inequality. solution of a compound inequality joined by and is any number that makes both inequalities true. solution of a compound inequality joined by or is any number that makes either inequality true.
Slide 73 / 182 Slide 74 / 182 31 Which inequality is represented in the graph below? 0 1 2 3 4 5 From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 75 / 182 32 Which inequality is represented in the graph below? Slide 76 / 182 Solving ompound Inequalities 0 1 2 3 4 5 that contain an N statement N is the same as writing You will need to solve both of these inequalities and graph their intersection. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 77 / 182 Slide 78 / 182
Slide 79 / 182 Slide 80 / 182 33 Which result below is correct for this inequality: 34 Which result below is correct for this inequality: 2 1 / 2 0 1 2 3 4 5 0 1 2 3 4 5 0 1 2 3 4 5-9 -8-7 -6 2 1 / 2 0 1 2 3 4 5 0 1 2 3 4 5 Slide 81 / 182 Slide 82 / 182 35 Which result below is correct for this inequality: 36 Which result below is correct for this inequality: -9-8 -7-6 -9-8 -7-6 -9-8 -7-6 Slide 83 / 182 Slide 84 / 182 37 Which result below is correct for this inequality:
Slide 85 / 182 Slide 86 / 182 Writing a ompound Inequality From a Graph -9-8 -7-6 How would you write this? Slide 87 / 182 Writing a ompound Inequality From a Graph Slide 88 / 182 ompound Inequalities Solve and graph the solution set. 1. -9-8 -7-6 -9-8 -7-6 How would you write this? 2. or -9-8 -7-6 Slide 89 / 182 Slide 90 / 182 3. or ompound Inequalities Solve and graph the solution set. 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 tall. Which graph represents these conditions? 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50-9 -8-7 -6 4. 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50-9 -8-7 -6 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011.
Slide 91 / 182 Slide 92 / 182 40 Which graph represents the solution set for and? 1112 13 1415 16171819 20 1112 13 1415 16171819 20 1112 13 1415 16171819 20 1112 13 1415 16171819 20 From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 93 / 182 Slide 94 / 182 41 Solve Slide 95 / 182 Slide 96 / 182
Slide 97 / 182 Slide 98 / 182 pplication of ompound 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? Slide 99 / 182 pplication of ompound Inequalities The sum of 3 times a number and two lies between 8 and 11. How will we translate "lies between 8 and 11"? Slide 100 / 182 pplication of ompound Inequalities cell phone plan offers free minutes for no more than 250 minutes per month. efine a variable and write an inequality for the possible number of free minutes. Graph the solution. What inequality symbol will we use? Why? What is the inequality? Solve and graph the inequality. Slide 101 / 182 Slide 102 / 182 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?
Slide 103 / 182 48 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? $115 x $225 x < $115 or x > $225 $215 x $325 x < $215 or x > $325 Slide 104 / 182 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? Slide 105 / 182 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 in the course. The final exam counts as two test grades, and a is received if the final course average is from 85 to 92. Slide 106 / 182 Special ases of ompound Inequalities Return to Table of ontents Slide 107 / 182 Slide 108 / 182 Special ases 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 ll Reals.
Slide 109 / 182 Slide 110 / 182 Special ases Solve each set of compound inequalities. 1. and 2. or Slide 111 / 182 Slide 112 / 182 Special ases Solve each set of compound inequalities. 3. and Graphing Linear Inequalities in Slope-Intercept Form 4. and Return to Table of ontents Slide 113 / 182 Slide 114 / 182 Graphing The following are graphs of linear inequalities. Shading is above the dotted line.this means the solutions are above the line but NOT on it. Shading is below the dotted line.this means the solutions are below the line but NOT on it.
Slide 115 / 182 Graphing The following are graphs of linear inequalities. Slide 116 / 182 How to Graph a Linear Inequality Shading is above a solid line.this means the solutions are above the line N on it. Shading is below a solid line. This means the solutions are below the line N on it. 1) ecide where the boundary goes: Solve inequality for y, for example y > 2x - 1 2) ecide 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) ecide 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 Slide 117 / 182 Graphing Step 1: Solve for y: (Think ), m = and b = 1 Step 2: The line should be dashed because the inequality is < Graph Step 1: Solve for y Slide 118 / 182 Graphing Step 3: Graph boundary Step 4: Shade below the boundary line because y < Step 2: The line should be solid because the inequality is Step 3: Graph boundary Step 4: Shade above the boundary line because y Graph Slide 119 / 182 Graphing Is the equation already solved for y? 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 Slide 120 / 182 51 Why are there dashed boundaries on some graphs of inequalities? Points on the line make the inequality false. Points on the line make the inequality true. The slope of the line depends on the line type. The y-intercept depends on the line type. 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
Slide 121 / 182 52 For which of these inequalities would the graph have a solid boundary and be shaded above? Slide 122 / 182 53 For which of these inequalities would the graph have a dashed boundary and be shaded above? Slide 123 / 182 Slide 124 / 182 54 Which inequality is graphed? Slide 125 / 182 56 Graph the solution set of. When you finish, type the number "1" into your responder. Slide 126 / 182 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. PR - EOY - Question #2 Non-alculator Section - SMRT Response Format
Slide 127 / 182 Modeling with Inequalities t 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 Write an inequality that would be used to model the situation. Slide 128 / 182 Modeling with Inequalities t 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 Write an inequality that would be used to model the situation. Part Graph the inequality in a coordinate plane. Part List 3 combinations of dress shirts and pairs of dress pants that could be purchased within your budget. Slide 129 / 182 Modeling with Inequalities t 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 Graph the inequality in a coordinate plane. 10 Slide 130 / 182 Modeling with Inequalities t 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 List 3 combinations of dress shirts and pairs of dress pants that could be purchased within your budget. 5 0 5 10 15 20 x Slide 131 / 182 57 t 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 Which inequality would be used to model this situation? Slide 132 / 182 58 t 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 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
Slide 133 / 182 59 t 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 Which pairs (x, y) can represent the amount of soccer balls and footballs purchased at the sports shop? Select all that apply. (7, 1) (2, 3) (4, 6) (3, 3) E (1, 4) Slide 134 / 182 60 group of friends went to the movies on Friday night. fter 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 Which inequality would be used to model this situation? Slide 135 / 182 61 group of friends went to the movies on Friday night. fter 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 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 Slide 136 / 182 62 group of friends went to the movies on Friday night. fter 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 Which pairs (x, y) can represent the amount spent on soda and buckets of popcorn at the theater? Select all that apply. (17, 1) (10, 5) (8, 4) (5, 5) E (3, 7) Slide 137 / 182 Slide 138 / 182 Vocabulary Solving Systems of Inequalities 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. Return to Table of ontents
Slide 139 / 182 Graphing a System of Linear Inequalities Slide 140 / 182 Example Solve the following 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 1: 10 5 y Step 3: Identify the intersection of the half-planes. This is the solution to the system of linear inequalities. 0 5 10 x Slide 141 / 182 Slide 142 / 182 Example ontinued Example ontinued Step 2: y Step 3: y 10 10 5 5 0 5 10 x 0 5 10 x Slide 143 / 182 Example Solve the following system of linear inequalities. Slide 144 / 182 Example ontinued Step 1: 10 y Step 2: 10 y 5 5 0 5 10 x 0 5 10 x
Slide 145 / 182 Example ontinued Slide 146 / 182 Example Solve the following system of linear inequalities. Step 3: 10 y Step 1: 10 y 5 5 0 5 10 x 0 5 10 x Slide 147 / 182 Slide 148 / 182 Example ontinued Example ontinued Step 2: y Step 3: y 10 10 5 5 0 5 10 x 0 5 10 x Slide 149 / 182 Slide 150 / 182
Slide 151 / 182 Slide 152 / 182 63 hoose the graph below that displays the solution to the following system of linear inequalities: Slide 153 / 182 Slide 154 / 182 65 hoose the graph below that displays the solution to the following system of linear inequalities: Slide 155 / 182 66 hoose the graph below that displays the solution to the following system of linear inequalities: Slide 156 / 182 67 hoose all of the linear inequalities that correspond to the following graph:
Slide 157 / 182 68 Which point is in the solution set of the system of inequalities shown in the accompanying graph? Slide 158 / 182 69 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph? (0, 4) (, 1) (0, 0) (1, 5) (2, 4) (4, ) (0, 1) (3, 2) From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 159 / 182 70 Which ordered pair is in the solution set of the following system of linear inequalities? Slide 160 / 182 71 Mr. raun has $75.00 to spend on pizzas and soda for a picnic. Pizzas cost $9.00 each and the drinks cost $0.75 each. Five times as many drinks as pizzas are needed. What is the maximum number of pizzas that Mr. raun can buy? (0, 3) (2, 0) ( 1, 0) ( 1, 4) From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. From the New York State Education epartment. Office of ssessment Policy, evelopment and dministration. Internet. vailable from www.nysedregents.org/integratedlgebra; accessed 17, June, 2011. Slide 161 / 182 72 system of inequalities is given. Slide 162 / 182 Modeling with a System of Inequalities Graph the solution set of the system of linear inequalities in the coordinate plane. When you finish, type the number "1" into your Responder. 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. PR - P - Question #3 Non-alculator Section - SMRT Response Format
Slide 163 / 182 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 : Graph the solution set of the system of linear inequalities in a coordinate plane. Part : reate 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions. Part : Given the conditions in Part, 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 : Given the conditions in Part, 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. Slide 165 / 182 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 : reate 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions. Slide 164 / 182 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 : Graph the solution set of the system of linear inequalities in a coordinate plane. 10 5 0 5 10 Slide 166 / 182 15 20 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 : Given the conditions in Part, 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. x Slide 167 / 182 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 : Given the conditions in Part, 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. Slide 168 / 182 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 : 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
Slide 169 / 182 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 Which pairs (x, y) represent the sales of comic books and packs of baseball cards to meet the given conditions? Select all that apply. (25, 25) (26, 8) (30, 10) (35, 25) E (18, 40) Slide 170 / 182 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 Given the conditions in Part, 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. Slide 171 / 182 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 Graph the solution set of the system of linear inequalities in the coordinate plane. When you finish, type the number "1" into your Responder. PR - EOY - Question #25 alculator Section - SMRT Response Format Slide 173 / 182 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 Given the conditions in Part, 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. Slide 172 / 182 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 Which pairs (x, y) represent hours that Leah could work to meet the given conditions? Select all that apply. (4, 15) (5, 12) (10, 9) (15, 5) E (19, 1) PR - EOY - Question #25 alculator Section Slide 174 / 182 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 Given the conditions in Part, Leah prefers babysitting over working at the ice cream store. What is the maximum number of hours she can babysit to be able to earn at least $120? Give your answer to the nearest whole hour. PR - EOY - Question #25 alculator Section PR - EOY - Question #25 alculator Section
Slide 175 / 182 Slide 176 / 182 Inequality Glossary & Standards Slide 177 / 182 Solution Set Return to Table of ontents ny number that, when substituted into an equation/inequality, will satisfy the equation/ inequality n Inequality is a mathematical sentence that uses symbols, such as <,, > or to compare to quantities. 2 < 18 x > 6 x Slide 178 / 182 r - 9 2 + 9 +9 ompound Inequality r 11 Two inequalities that are combined into one statement by the words N/OR ack to Instruction 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! x > N x < 3 < x < 3 "and" means intersection "or" means union x OR x 3 ack to Instruction ack to Instruction Slide 179 / 182 No Solution When there is no number that makes the equation/inequalities true Slide 180 / 182 Reals When all (any) numbers make the equation/inequalities true 2x + 8 = 2(x - 4) 2x + 8 = 2x - 8 8 = -8 2x 18 N x > 12 x 9 N x < { } or { } "no solution" 2x + 8 = 2(x + 4) 2x + 8 = 2x + 8 0 = 0 R x + 3 > 17 OR 5(x + 2) > 0 x -7 OR x > R "reals" "all real numbers" R ack to Instruction ack to Instruction
Slide 181 / 182 System of Linear Inequalities Slide 182 / 182 Throughout this unit, the Standards for Mathematical Practice are used. Two or more linear inequalities y > 2x - 3 y < -x + 4 y 10 5 0 5 10 x ack to Instruction MP1: Making sense of problems & persevere in solving them. MP2: Reason abstractly & quantitatively. MP3: onstruct viable arguments and critique the reasoning of others. MP4: Model with mathematics. MP5: Use appropriate tools strategically. MP6: ttend to precision. MP7: Look for & make use of structure. MP8: Look for and express regularity in repeated reasoning. dditional 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.