Simple Inequalities Involving Addition And Subtraction - NJCTL

Slide 61 / 182 Slide 62 / 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. 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 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. -10 Slide 65 / 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? -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 10 Slide 66 / 182 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? A B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011 9 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Slide 67 / 182 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? Slide 68 / 182 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 6 rides B 7 rides A C 8 rides B D 9 rides C D Slide 69 / 182 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 Slide 70 / 182 Solving Compound Inequalities B r 8.6 C r 8.975 Return to Table of Contents D r 8.6 Slide 71 / 182 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. Slide 72 / 182

Slide 73 / 182 Slide 74 / 182 31 Which inequality is represented in the graph below? -5 -4 -3 -2 -1 0 1 2 3 4 5 A B C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Slide 75 / 182 Slide 76 / 182 32 Which inequality is represented in the graph below? -5 -4 -3 -2 -1 0 1 2 3 4 5 Solving Compound Inequalities that contain an AND statement is the same as writing A AND B You will need to solve both of these inequalities and graph their intersection. C D From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; 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: A A -5 -4 -3 -2 -1 0 1 2 3 4 5 B 2 1/2 -5 -4 -3 -2 -1 0 1 2 3 4 5 B -5 -4 -3 -2 -1 0 1 2 3 4 5 C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 C -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 Slide 81 / 182 35 Which result below is correct for this inequality: 2 1/2 -4 -3 -2 -1 0 1 2 3 4 5 Slide 82 / 182 36 Which result below is correct for this inequality: A A -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 B B C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 C -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 83 / 182 37 Which result below is correct for this inequality: A B C Slide 84 / 182

Slide 85 / 182 Slide 86 / 182 Writing a Compound Inequality From a Graph -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 How would you write this? Slide 87 / 182 Slide 88 / 182 Compound Inequalities Writing a Compound Inequality From a Graph Solve and graph the solution set. 1. -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 How would you write this? 2. or -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 Slide 89 / 182 Compound Inequalities Solve and graph the solution set. 3. or Slide 90 / 182 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? A -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 4. B 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 C 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 D 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Slide 91 / 182 Slide 92 / 182 40 Which graph represents the solution set for and ? A B C D 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Slide 93 / 182 Slide 94 / 182 Slide 95 / 182 Slide 96 / 182 41 Solve A B C D #

Slide 97 / 182 Slide 98 / 182 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? Slide 99 / 182 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"? Slide 100 / 182 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. What inequality symbol will we use? Why? What is the inequality? Solve and graph the inequality. Slide 101 / 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? A B C D Slide 102 / 182

Slide 103 / 182 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? 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? A 115 x 225 A B x 115 or x 225 B C 215 x 325 C D x 215 or x 325 D 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 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 Slide 106 / 182 Special Cases of Compound Inequalities C Return to Table of Contents D Slide 107 / 182 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. Slide 108 / 182

Slide 109 / 182 Slide 110 / 182 Special Cases Solve each set of compound inequalities. 1. 2. Slide 111 / 182 and or Slide 112 / 182 Special Cases Solve each set of compound inequalities. 3. 4. and Graphing Linear Inequalities in Slope-Intercept Form and Return to Table of Contents Slide 113 / 182 Slide 114 / 182 Graphing The following are graphs of linear inequalities. Shading is above the dotted line.This means the solutions areabove the line but NOT on it. Shading is below the dotted line.This means the solutions arebelow the line but NOT on it.

Slide 115 / 182 Slide 116 / 182 Graphing How to Graph a Linear Inequality The following are graphs of linear inequalities. 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). Shading is above a solid line.This means the solutions are above the line AND on it. Shading is below a solid line. This means the solutions arebelow the line AND on it. 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 Slide 117 / 182 Slide 118 / 182 Graphing Graph Step 1: Solve for y: (Think ), m -2 and b 1 Step 2: The line should be dashed because the inequality is Step 3: Graph boundary Graphing Graph Step 1: Solve for y Step 2: The line should be solid because the inequality is Step 4: Shade below the boundary line because y Step 3: Graph boundary Step 4: Shade above the boundary line because y Slide 119 / 182 Graphing Graph 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 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 120 / 182 51 Why are there dashed boundaries on some graphs of inequalities? A B C D 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.

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? A A B B C C D D Slide 123 / 182 54 Slide 124 / 182 Which inequality is graphed? A B C D 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. PARCC - EOY - Question #2 Non-Calculator Section - SMART Response Format

Slide 127 / 182 Slide 128 / 182 Modeling with Inequalities 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. 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 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. Slide 129 / 182 Slide 130 / 182 Modeling with Inequalities Modeling with Inequalities 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. 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. 10 5 0 5 10 15 20 x Slide 131 / 182 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 Slide 132 / 182 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. y 20 15 10 5 D 0 5 10 15 20 x

Slide 133 / 182 Slide 134 / 182 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) 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 (2, 3) B C (4, 6) D (3, 3) C E (1, 4) D Slide 135 / 182 Slide 136 / 182 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. 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) 10 B (10, 5) C (8, 4) 5 D (5, 5) 0 5 10 15 20 Slide 137 / 182 x E (3, 7) Slide 138 / 182 Vocabulary Solving Systems of Inequalities Return to Table of 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.

Slide 139 / 182 Slide 140 / 182 Graphing a System of Linear Inequalities Step 1: Graph the boundary lines of each inequality. Remember: - dashed line for and - solid line for and Example Solve the following system of linear inequalities. Step 1: y 10 5 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. 0 -5 -10 5 10 x -5 -10 Slide 141 / 182 Slide 142 / 182 Example Continued Step 2 : Example Continued Step 3 : y -10 10 10 5 5 0 -5 y 5 10 x -10 -5 0 -5 -5 -10 -10 Slide 143 / 182 Example y y 10 10 5 5 -10 -5 -5 -10 x Example Continued Step 2: 0 10 Slide 144 / 182 Solve the following system of linear inequalities. Step 1: 5 5 10 x -10 -5 0 -5 -10 5 10 x

Slide 145 / 182 Slide 146 / 182 Example Continued Example Solve the following system of linear inequalities. Step 3: y Step 1: 10 y 10 5 -10 -5 5 0 5 10 x -10 -5 -10 -10 Step 2: 10 5 5 10 x -10 -5 0 -5 -5 -10 -10 Slide 149 / 182 x y 10 5 10 Example Continued Step 3: y 0 5 Slide 148 / 182 Example Continued -5 0 -5 Slide 147 / 182 -10 -5 5 10 Slide 150 / 182 x

Slide 151 / 182 Slide 152 / 182 63 Choose the graph below that displays the solution to the following system of linear inequalities: A B Slide 153 / 182 C Slide 154 / 182 65 Choose the graph below that displays the solution to the following system of linear inequalities: B A Slide 155 / 182 66 Choose the graph below that displays the solution to the following system of linear inequalities: A B C Slide 156 / 182 67 Choose all of the linear inequalities that correspond to the following graph: C A C B D

Slide 157 / 182 68 Which point is in the solution set of the system of inequalities shown in the accompanying graph? A B (0, 4) (2, 4) C D (-4, 1) (4, -1) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Slide 158 / 182 69 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph? A (0, 0) C B (1, 5) (0, 1) D (3, 2) From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011. Slide 159 / 182 70 Which ordered pair is in the solution set of the following system of linear inequalities? A (0,

Solving & Graphing Inequalities 2016-01-11 www.njctl.org Slide 3 / 182 Table of Contents Simple Inequalities Addition/Subtraction Simple Inequalities Multiplication/Division Solving Compound Inequalities Special Cases of Compound Inequalities Graphing Linear Inequalities in Slope-Intercept Form click on the topic to go to that section Glossary .