Algebra I Solving & Graphing Inequalities

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Slide 2 / 182 Algebra I Solving & Graphing Inequalities 2016-01-11 www.njctl.org

Slide 3 / 182 Table of Contents Simple Inequalities Addition/Subtraction click on the topic to go to that section 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

Slide 4 / 182 Simple Inequalities Involving Addition and Subtraction Return to Table of Contents

Slide 5 / 182 Inequality An Inequality is a mathematical sentence that uses symbols, such as <,, > or to compare to quantities.

Slide 6 / 182 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

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Slide 8 / 182 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

Slide 9 / 182 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.

Slide 9 (Answer) / 182 Graphing Inequalities The hand Remember! gestures/visuals below can assist students in remembering when to use an open or closed circle. Open circle < means & > that number make is not included in the solution set and is used to represent < or >. Teacher Notes & make Closed circle means the solution set includes that The number finger that and represents is used to the represent or. [This object is a pull tab] "equal to" fills in the circle.

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.

Slide 11 / 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.

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Slide 13 / 182 Solving Inequalities Step 2: Decide 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. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

Slide 13 (Answer) / 182 Step 2: Decide When whether graphing or each not the example, circle on make your boundary should be open sure or that closed the students based on understand the symbol when used. You can check the computation the circle is open by substituting and closed. the end point of 6 for x. In this case, the end point is not included (open circle) since Additional x < 6. Questions to address MPs: Is the circle on _ going to be open or closed? Why? (MP.6) Math Practice Solving Inequalities MP.6: Attend to precision [This object is a pull tab] -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

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Slide 15 / 182 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/

Slide 16 / 182 1 Which graph is the solution to the inequality: a number, n, minus is greater than one third? A -5-4 -3-2 -1 2 0 1 2 3 4 5 5 6 B -5-4 -3-2 -1 2 5 6 0 1 2 3 4 5 C 2 5 6-5 -4-3 -2-1 0 1 2 3 4 5 D -5-4 -3-2 -1 2 5 6 0 1 2 3 4 5

Slide 16 (Answer) / 182 1 Which graph is the solution to the inequality: a number, n, minus is greater than one third? A B -5-5 -4-4 -3-3 Answer -2-2 -1-1 2 0 1 2 3 4 5 B 2 5 6 0 1 2 3 4 5 5 6 C -5-4 -3-2 -1 2 5 6 [This object is a pull tab] 0 1 2 3 4 5 D -5-4 -3-2 -1 2 5 6 0 1 2 3 4 5

Slide 17 / 182 2 Which graph is the solution to the inequality? A -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 B -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 D -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10

Slide 17 (Answer) / 182 2 Which graph is the solution to the inequality? A B -10-9 -8-7 -6-10-9-8 -7-6 -5-4 -3-2 -1 Answer -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 D 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 [This object is a pull tab] D -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10

Slide 18 / 182 3 Which graph is the solution to the inequality? A -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 B -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 D -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10

Slide 18 (Answer) / 182 3 Which graph is the solution to the inequality? A B -10-9 -8-7 -6-10-9-8 -7-6 -5-4 -3-2 -1 Answer -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 C 0 1 2 3 4 5 6 7 8 9 10 C D -10-9 -8-7 -6-10-9-8 -7-6 -5-4 -3-2 -1-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 [This object is a pull tab] 0 1 2 3 4 5 6 7 8 9 10

Slide 19 / 182 4 Which graph is the solution to the inequality? A -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 B -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 D -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10

Slide 19 (Answer) / 182 4 Which graph is the solution to the inequality? A -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 B -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 Answer -5-4 -3-2 -1 A 0 1 2 3 4 5 6 7 8 9 10 D -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 [This object is a pull tab]

Slide 20 / 182 5 Which graph is the solution to the inequality? A B C D -10-9 -8-7 -6-5 -4-3 -2-1 -10-9 -8-7 -6-5 -4-3 -2-1 -10-9 -8-7 -6-5 -4-3 -2-1 -10-9 -8-7 -6-5 -4-3 -2-1 1.5 0 1 2 3 4 5 6 7 8 9 10 1.5 0 1 2 3 4 5 6 7 8 9 10 1.5 0 1 2 3 4 5 6 7 8 9 10 1.5 0 1 2 3 4 5 6 7 8 9 10

Slide 20 (Answer) / 182 5 Which graph is the solution to the inequality? A B C D -10-9 -8-7 -6-5 -4-3 -2-1 -10-9 -8-7 -6-5 -4-3 -2-1 -10-9 -8-7 -6-5 -4-3 -2-1 Answer -10-9 -8-7 -6-5 -4-3 -2-1 1.5 0 1 2 3 4 5 6 7 8 9 10 1.5 0 1 2 3 4 5 6 7 8 9 10 1.5 C 0 1 2 3 4 5 6 7 8 9 10 1.5 0 1 2[This 3object 4 is 5a pull 6 tab] 7 8 9 10

Slide 21 / 182 Simple Inequalities Involving Multiplication and Division Return to Table of Contents

Slide 22 / 182 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.

Slide 23 / 182 Multiplying or Dividing by a Positive Number Since x is multiplied by 3, divide both sides by 3 to isolate the variable. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

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Slide 25 / 182 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/

Slide 26 / 182 6 Which graph is the solution to the inequality, the product of 4 and a number, x, is greater than 24? A -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10 B -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 D -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10

Slide 26 (Answer) / 182 6 Which graph is the solution to the inequality, the product of 4 and a number, x, is greater than 24? A B -10-9 -8-7 -6-10-9-8 -7-6 -5-4 -3-2 -1 Answer -5-4 -3-2 -1 0 1 2 3 4 5 6 7 8 9 10 B 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 D -10-9 -8-7 -6-5 -4-3 -2-1 [This object is a pull tab] 0 1 2 3 4 5 6 7 8 9 10

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Slide 29 / 182 9 Find the solution to the inequality. A B C D

Slide 29 (Answer) / 182 9 Find the solution to the inequality. A B C D Answer A [This object is a pull tab]

Slide 30 / 182 10 Find the solution to the inequality. A B C D

Slide 30 (Answer) / 182 10 Find the solution to the inequality. A B C D Answer D [This object is a pull tab]

Slide 31 / 182 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.

Slide 31 (Answer) / 182 Multiplying or Dividing by a Negative Number If the students need to see the So far, all derivation the operations as to we why have the used inequality worked sign the same as solving gets equations. flipped when The difference multiplying between or dividing solving equations by a negative versus inequalities number, there is revealed is a lab when multiplying located or in dividing the 7th grade by a negative unit and number. can accessed using the URL Teacher Notes http://www.njctl.org/courses/math/7thgrade/equations-inequalities-7th-grade/ The direction of the inequality changes only if the number you are using to multiply or divide by is negative. [This object is a pull tab]

Slide 32 / 182 Solve and Graph *Note: Dividing each side by -3 changes the to. -10-9 -8-7 -6-5 -4-4-3-3-2-2-1-1 00 11 22 334 45 56 67 78 89 91010 click for answer

Slide 33 / 182 11 Solve the inequality and graph the solution. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

Slide 33 (Answer) / 182 11 Solve the inequality and graph the solution. Answer -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10-10 -9-8 -7-6 -5-4 -3-2 -1 1 0 2 3 4 5 6 7 8 9 10 [This object is a pull tab]

Slide 34 / 182 12 Solve the inequality and graph the solution. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

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Slide 35 / 182 13 Solve the inequality and graph the solution. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

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Slide 36 / 182 14 Solve the inequality and graph the solution. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

Slide 36 (Answer) / 182 14 Solve the inequality and graph the solution. Answer -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 [This object is a pull tab] 0 1 2 3 4 5 6 7 8 9 10

Slide 37 / 182 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.

Slide 38 / 182 Solving Two-Step and Multiple-Step Inequalities Return to Table of Contents

Slide 39 / 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.

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Slide 41 / 182 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!

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Slide 43 / 182 Two Step Inequalities Example: Solve the inequality and graph the solution. Add 9 to both sides Divide both sides by 4 (sign stays the same) -10-9 -8-7 -6-5 -4-3 -2-2-1-1 0 11 22 3344 556 67 78 89 91010 click for answer

Slide 44 / 182 Try these. Solve each inequality and graph each solution. 1. Solve and Graph -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 2. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

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Slide 45 / 182 Try these. Solve each inequality and graph the solution. 3. Solve and Graph -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 4. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

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Slide 46 / 182 15 Solve A and graph the solution. 2.5-5 -4-3 -2-1 0 1 2 3 4 5 B 2.5-5 -4-3 -2-1 0 1 2 3 4 5 C 2.5-5 -4-3 -2-1 0 1 2 3 4 5 D 2.5-5 -4-3 -2-1 0 1 2 3 4 5

Slide 46 (Answer) / 182 15 Solve A and graph the solution. 2.5-5 -4-3 -2-1 0 1 2 3 4 5 B Answer 2.5 C -5-4 -3-2 -1 0 1 2 3 4 5 C 2.5-5 -4-3 -2-1 [This object is a pull tab] 0 1 2 3 4 5 D 2.5-5 -4-3 -2-1 0 1 2 3 4 5

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Slide 50 / 182 19 Solve and graph the solution. A B C D -10-9 -8-7 -6-10-9-8 -7-6 -10-9 -8-7 -6-5 -4-3 -2-1 -5-4 -3-2 -1-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 1 0 2 3 4 5 6 7 8 9 10 1 0 2 3 4 5 6 7 8 9 10-10-9-8 -7-6 -5-4 -3-2 -1 1 0 2 3 4 5 6 7 8 9 10

Slide 50 (Answer) / 182 19 Solve and graph the solution. A B C D -10-9 -8-7 -6-10-9-8 -7-6 -10-9 -8-7 -6-5 -4-3 -2-1 Answer -5-4 -3-2 -1-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 B 1 0 2 3 4 5 6 7 8 9 10 [This object is a pull tab] 0 1 2 3 4 5 6 7 8 9 10-10-9-8 -7-6 -5-4 -3-2 -1 1 0 2 3 4 5 6 7 8 9 10

Slide 51 / 182 20 Which graph represents the solution set for: Question from ADP Algebra I End-of-Course Practice Test A -2-1 0 1 2 B -2-1 0 1 2 C -2-1 0 1 2 D -2-1 0 1 2

Slide 51 (Answer) / 182 20 Which graph represents the solution set for: Question from ADP Algebra I End-of-Course Practice Test A -2-1 0 1 2 B -2-1 Answer A 0 1 2 C -2-1 0 1 2 [This object is a pull tab] D -2-1 0 1 2

Slide 52 / 182 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 Administration. Internet. Available from www.nysedregents.org/integratedalgebra; accessed 17, June, 2011.

Slide 52 (Answer) / 182 21 Find all negative odd integers that satisfy the following inequality. Select all that apply. A B C D Answer & Math Practice -3x + 1 17-3x 16 E x -5 1 / 3 x = -5, F-3, and -1 G D, E, G This example addresses MP.2 H [This object is a pull tab] 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 53 / 182 22 Which value of x is in the solution set of? A 8 B 9 C 12 D 16 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 53 (Answer) / 182 22 Which value of x is in the solution set of? A 8 B 9 C 12 D 16 Answer A [This object is a pull tab] 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 54 / 182 23 What is the solution of? 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 54 (Answer) / 182 23 What is the solution of? A B C D Answer A [This object is a pull tab] 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 55 / 182 24 In the set of positive integers, what is the solution set of the inequality? 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 55 (Answer) / 182 24 In the set of positive integers, what is the solution set of the inequality? A B C D Answer B [This object is a pull tab] 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.

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Slide 57 / 182 26 Given: Determine all elements of set A that are in the solution of the inequality. A 18 B 6 C -3 D -12 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 57 (Answer) / 182 26 Given: Determine all elements of set A that are in the solution of the inequality. A 18 B 6 C -3 D -12 Answer D [This object is a pull tab] 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 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.

Slide 58 (Answer) / 182 Inequalities in the Real World The problems on the next 7 slides Inequalities are helpful address when MP.1, applied MP.2, to MP.3, real life MP.4, scenarios. MP.5, These and MP.7. inequalities can be used for budgeting purposes, speed limits, cell phone data usage, and Additional building Questions materials to address management, MPs: just to name a few. What information do you have? (MP.1) What is the problem asking? (MP.1) Create an inequality to represent the problem? (MP.2) Translating between the languages of English words to numbers/ What do you already know about solving symbols is imperative this in problem? being able (MP.4) to solve the correct inequality. The next slides will Why provide did you ample decide practice to use in this setting model? up and solving these inequality applications. (MP.3 & MP.5) What do you know about inequalities that [This object is a pull tab] would apply to this situation? (MP.7) Math Practice

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Slide 60 / 182 Write an Inequality and Solve 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.

Slide 60 (Answer) / 182 Write an Inequality and Solve 15 + 12x 65 Example #2: You have $65.00-15 in birthday -15 money and want to buy some CDs and a DVD. Suppose a DVD cost $15.00 and a CD cost $12.00. 12x 50 12 12 Write an inequality and solve to find out how many CDs you can buy along with one DVD. x 4.16 Answer Can you buy 0.16 of a CD? You can buy 4 CDs and 1 DVD. [This object is a pull tab]

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. How much could one pair of pants cost, if they were all the same price? Write an inequality and solve.

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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?

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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?

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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. -10-9 -8-7 -6-5 -4-3 -2-1 0 1 2 3 4 5 6 7 8 9 10

Slide 64 (Answer) / 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 mum scores score 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. ore at least 63 points to is object is a pull tab] -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

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? 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 65 (Answer) / 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? Answer They must mow at least 65 lawns. [This object is a pull tab] 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 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.

Slide 66 (Answer) / 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 Answer B [This object is a pull tab] 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? A B C D

Slide 67 (Answer) / 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? A B Answer D C [This object is a pull tab] D

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 B C D 6 rides 7 rides 8 rides 9 rides

Slide 68 (Answer) / 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 B 6 rides 7 rides Answer B C 8 rides D 9 rides [This object is a pull tab]

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 B r 8.6 C r 8.975 D r 8.6

Slide 69 (Answer) / 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 B r 8.6 C r 8.975 Answer B D r 8.6 [This object is a pull tab]

Slide 70 / 182 Solving Compound Inequalities Return to Table of Contents

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.

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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 74 (Answer) / 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 Answer B [This object is a pull tab] 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 32 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 (Answer) / 182 32 Which inequality is represented in the graph below? -5-4 -3-2 -1 0 1 2 3 4 5 A B C D Answer D [This object is a pull tab] 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 76 / 182 Solving Compound Inequalities that contain an AND statement AND is the same as writing You will need to solve both of these inequalities and graph their intersection.

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Slide 79 / 182 33 Which result below is correct for this inequality: A -5-4 -3-2 -1 0 1 2 3 4 5 B -5-4 -3-2 -1 0 1 2 3 4 5 C -5-4 -3-2 -1 0 1 2 3 4 5

Slide 79 (Answer) / 182 33 Which result below is correct for this inequality: A B -5-4 -3 Answer -2-1 0 1 2 3 4 5 B -5-4 -3-2 -1 0 1 2 3 4 5 C [This object is a pull tab] -5-4 -3-2 -1 0 1 2 3 4 5

Slide 80 / 182 34 Which result below is correct for this inequality: A 2 1 / 2-5 -4-3 -2-1 0 1 2 3 4 5 B -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 C 2 1 / 2-5 -4-3 -2-1 0 1 2 3 4 5

Slide 80 (Answer) / 182 34 Which result below is correct for this inequality: A -5-4 -3-2 Answer -1 2 A 1 / 2 0 1 2 3 4 5 B -10-9 -8-7 -6-5 -4-3 -2-1 [This object is a pull tab] 0 1 2 3 4 5 6 7 8 9 10 C 2 1 / 2-5 -4-3 -2-1 0 1 2 3 4 5

Slide 81 / 182 35 Which result below is correct for this inequality: A B C

Slide 81 (Answer) / 182 35 Which result below is correct for this inequality: A B C Answer B [This object is a pull tab]

Slide 82 / 182 36 Which result below is correct for this inequality: A B -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 C -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10-10-9-8 -7-6 -5-4 -3-2 -1 1 0 2 3 4 5 6 7 8 9 10

Slide 82 (Answer) / 182 36 Which result below is correct for this inequality: A B -10-9 -8-7 -6 Answer -5-4 -3-2 -1 B 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 [This object is a pull tab] -10-9 -8-7 -6-5 -4-3 -2-1 1 0 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 83 (Answer) / 182 37 Which result below is correct for this inequality: A B C Answer C [This object is a pull tab]

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Slide 86 / 182 Writing a Compound Inequality From a Graph -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 How would you write this?

Slide 86 (Answer) / 182 Writing a Compound Inequality From a Graph -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 Answer & Math Practice x -6 OR x 0 How would you write this? The question on this slide addresses MP.2. [This object is a pull tab]

Slide 87 / 182 Writing a Compound Inequality From a Graph -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 How would you write this?

Slide 87 (Answer) / 182 Writing a Compound Inequality From a Graph -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 Answer -5 < x 2 How would you write this? [This object is a pull tab]

Slide 88 / 182 Compound Inequalities Solve and graph the solution set. 1. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 2. or -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

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Slide 89 / 182 Compound Inequalities Solve and graph the solution set. 3. or -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10 4. -10-9 -8-7 -6-5 -4-3 -2-1 1 0 2 3 4 5 6 7 8 9 10

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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 B C 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 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 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 90 (Answer) / 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 B C 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 Answer A 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 [This object is a pull tab] 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.

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Slide 92 / 182 40 Which graph represents the solution set for and? A 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 B 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 C 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 D 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 92 (Answer) / 182 40 Which graph represents the solution set for and? A B 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 Answer 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 B C D 0 1 2 3 4 5 6 7 8 9 10 1112 13 1415 16171819 20 [This object is a pull tab] 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 41 Solve A B C D

Slide 93 (Answer) / 182 41 Solve A B C D Answer B [This object is a pull tab]

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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 98 (Answer) / 182 Application of Compound Inequalities Let's start off by translating the words of an applied problem into math. Answer & Math Practice 3x + 2 The sum of 3 times a number and two lies The question between on 8 this and slide 11. addresses MP.2 "The sum of 3 times a number and two" Additional translates Questions into what? to address MPs: How can you represent the problem with symbols and numbers? (MP.2) [This object is a pull tab]

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"? What inequality symbol will we use? Why? What is the inequality? Solve and graph the inequality.

Slide 99 (Answer) / 182 Application of Compound Inequalities The sum of 3 times a number and two lies between 8 and 11. Answer & Math Practice How will we translate "lies between 8 and 11"? What inequality symbol will we use? Why? 0 1 2 3 4 5 What is the inequality? The questions Solve and on this graph slide the address inequality. MP.2, MP.5, MP.6 & MP.7 [This object is a pull tab]

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.

Slide 100 (Answer) / 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. Let m Graph = number the solution. of minutes 0 m 250 Answer -100-50 0 50 100 150 200 250 300 350 400 [This object is a pull tab]

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 101 (Answer) / 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 Answer D [This object is a pull tab]

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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? A $115 x $225 B x < $115 or x > $225 C $215 x $325 D x < $215 or x > $325

Slide 103 (Answer) / 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? A $115 x $225 B x < $115 or x > $225 C $215 x $325 Answer A D x < $215 or x > $325 [This object is a pull tab]

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 B C D

Slide 104 (Answer) / 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 B C D Answer D [This object is a pull tab]

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 C D

Slide 105 (Answer) / 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 C D Answer B [This object is a pull tab]

Slide 106 / 182 Special Cases of Compound Inequalities Return to Table of Contents

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.

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Slide 110 / 182 Special Cases Solve each set of compound inequalities. 1. and 2. or

Slide 110 (Answer) / 182 Special Cases Solve each set of compound inequalities. 1. and Answer 1. No solution 2. All reals 2. or [This object is a pull tab]

Slide 111 / 182 Special Cases Solve each set of compound inequalities. 3. and 4. and

Slide 111 (Answer) / 182 Special Cases Solve each set of compound inequalities. 3. and 3. No solution Answer 4. and, which simplifies to 4. and [This object is a pull tab]

Slide 112 / 182 Graphing Linear Inequalities in Slope-Intercept Form Return to Table of Contents

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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. 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 are below the line AND on it.

Slide 116 / 182 How to Graph a Linear Inequality 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

Slide 117 / 182 Graph Graphing 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 Step 4: Shade below the boundary line because y <

Slide 118 / 182 Graphing Graph Step 1: Solve for 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

Slide 119 / 182 Graph 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 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 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.

Slide 120 (Answer) / 182 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. Answer C The slope of the line depends on the line type. D The y-intercept depends on the line type. A [This object is a pull tab]

Slide 121 / 182 52 For which of these inequalities would the graph have a solid boundary and be shaded above? A B C D

Slide 121 (Answer) / 182 52 For which of these inequalities would the graph have a solid boundary and be shaded above? A B C D Answer D [This object is a pull tab]

Slide 122 / 182 53 For which of these inequalities would the graph have a dashed boundary and be shaded above? A B C D

Slide 122 (Answer) / 182 53 For which of these inequalities would the graph have a dashed boundary and be shaded above? A B C D Answer C [This object is a pull tab]

Slide 123 / 182 54 Which inequality is graphed? A B C D

Slide 123 (Answer) / 182 54 Which inequality is graphed? A B Answer B C [This object is a pull tab] D

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Slide 125 / 182 56 Graph the solution set of. When you finish, type the number "1" into your responder. PARCC - EOY - Question #2 Non-Calculator Section - SMART Response Format

Slide 125 (Answer) / 182 56 Graph the solution set of. When you finish, type the number "1" into your responder. Answer [This object is a pull tab] PARCC - EOY - Question #2 Non-Calculator Section - SMART Response Format

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.

Slide 126 (Answer) / 182 Modeling with Inequalities Math Practice The example on the next 4 slides address MP.1, Throughout MP.2, MP.4, this unit, MP.5, you MP.6, have and learned MP.7 how to solve and graph inequalities, both on a number line and in Additional the coordinate Questions plane. to address MPs: How could you start this problem? (MP.1) How can you represent this problem with We can apply these skills to solve realistic word symbols and numbers? (MP.2) Which problems, tool/manipulative such as purchasing would be items best for at a store within a this budget problem? and earning (MP.5) money through various jobs. How do you know that your answer is accurate? Let's get (MP.6) started. What do you know about writing inequalities that would apply to this situation? (MP.7) [This object is a pull tab]

Slide 127 / 182 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.

Slide 128 / 182 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.

Slide 128 (Answer) / 182 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. Answer Part A Write an inequality that would be used to model the situation. [This object is a pull tab]

Slide 129 / 182 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. 10 5 0 5 10 15 20 x

5 Slide 129 (Answer) / 182 Modeling with Inequalities At a department store, dress shirts ycost $12.50 each and each pair of dress pants cost $25 y 20 each. You have $125 to spend. Let x represents the dress shirts and y 20 15 represents the number of pairs of dress pants. 15 10 Part B 5 Graph the inequality in a coordinate plane. 10 x Answer 0 5 10 [This object is a pull tab] 15 20 0 5 10 15 20 x

Slide 130 / 182 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.

Slide 130 (Answer) / 182 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. Answers will vary. Part C Sample answers include: List 3 combinations of dress shirts (0, 5), and (4, pairs 3), (6, of dress 1), (2, pants 4), (7, that 1) could be purchased within your budget. Answer [This object is a pull tab]

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 D

Slide 131 (Answer) / 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 D Answer B [This object is a pull tab]

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. 20 15 10 y 5 0 5 10 15 20 x

Slide 132 (Answer) / 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. Answer 20 15 10 5 y 20 15 10 5 y 0 5 10 15 20 [This object is a pull tab] x 0 5 10 15 20 x

Slide 133 / 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) B (2, 3) C (4, 6) D (3, 3) E (1, 4)

Slide 133 (Answer) / 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) B (2, 3) C (4, 6) D (3, 3) E (1, 4) Answer B and E [This object is a pull tab]

Slide 134 / 182 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

Slide 134 (Answer) / 182 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 Answer C C D [This object is a pull tab]

Slide 135 / 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. 10 5 0 5 10 15 20 x

Slide 135 (Answer) / 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, y which costs $4.50 per bucket. Let x represent the 20 y number of sodas purchased and y 15 20 represent the buckets of popcorn 10 purchased. 15 Part B Graph your solution in the coordinate plane below. When you are finished, type the number "1" into your responder. Answer 10 5 5 0 5 10 15 20 [This object is a pull tab] x 0 5 10 15 20 x

Slide 136 / 182 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)

Slide 136 (Answer) / 182 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) Answer A, C, D C (8, 4) [This object is a pull tab] D (5, 5) E (3, 7)

Slide 137 / 182 Solving Systems of Inequalities Return to Table of Contents

Slide 138 / 182 Vocabulary 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 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.

Slide 140 / 182 Example Solve the following system of linear inequalities. Step 1: y 10 5-10 -5 0 5 10 x -5-10

Slide 141 / 182 Example Continued Step 2: y 10 5-10 -5 0 5 10 x -5-10

Slide 142 / 182 Example Continued Step 3: y 10 5-10 -5 0 5 10 x -5-10

Slide 143 / 182 Example Solve the following system of linear inequalities. Step 1: y 10 5-10 -5 0 5 10 x -5-10

Slide 144 / 182 Example Continued Step 2: y 10 5-10 -5 0 5 10 x -5-10

Slide 145 / 182 Example Continued Step 3: y 10 5-10 -5 0 5 10 x -5-10

Slide 146 / 182 Example Solve the following system of linear inequalities. Step 1: y 10 5-10 -5 0 5 10 x -5-10

Slide 146 (Answer) / 182 Example Solve the following system of linear inequalities. y 10 Step 1: Answer y 10-10 5-5 5 0 5 10 x -10-5 0-5 5 10-10 x -5 [This object is a pull tab] -10

Slide 147 / 182 Example Continued Step 2: y 10 5-10 -5 0 5 10 x -5-10

Slide 147 (Answer) / 182 Example Continued y 10 Step 2: Answer -10 y 10 5-5 5 0 5 10 x -10-5 0-5 5 10 x -5-10 [This object is a pull tab] -10

Slide 148 / 182 Example Continued Step 3: y 10 5-10 -5 0 5 10 x -5-10

Slide 148 (Answer) / 182 Example Continued y 10 Step 3: Answer -10 y 10 5-5 5 0 5 10 x -10-5 0-5 5 10 x -5-10 [This object is a pull tab] -10

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Slide 152 / 182 63 Choose the graph below that displays the solution to the following system of linear inequalities: A B C

Slide 152 (Answer) / 182 63 Choose the graph below that displays the solution to the following system of linear inequalities: Answer A B C B [This object is a pull tab]

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Slide 154 / 182 65 Choose the graph below that displays the solution to the following system of linear inequalities: A B C

Slide 154 (Answer) / 182 65 Choose the graph below that displays the solution to the following system of linear inequalities: Answer A A B C [This object is a pull tab]

Slide 155 / 182 66 Choose the graph below that displays the solution to the following system of linear inequalities: A B C

Slide 155 (Answer) / 182 66 Choose the graph below that displays the solution to the following system of linear inequalities: Answer A A B C [This object is a pull tab]

Slide 156 / 182 67 Choose all of the linear inequalities that correspond to the following graph: A B C D

Slide 156 (Answer) / 182 67 Choose all of the linear inequalities that correspond to the following graph: Answer A and D [This object is a pull tab] A B C D

Slide 157 / 182 68 Which point is in the solution set of the system of inequalities shown in the accompanying graph? A (0, 4) C (-4, 1) B (2, 4) D (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 157 (Answer) / 182 68 Which point is in the solution set of the system of inequalities shown in the accompanying graph? Answer C [This object is a pull tab] A (0, 4) C (-4, 1) B (2, 4) D (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) B (0, 1) C (1, 5) 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 158 (Answer) / 182 69 Which ordered pair is in the solution set of the system of inequalities shown in the accompanying graph? Answer D [This object is a pull tab] A (0, 0) B (0, 1) C (1, 5) 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, 3) B (2, 0) C ( 1, 0) D ( 1, 4) 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 (Answer) / 182 70 Which ordered pair is in the solution set of the following system of linear inequalities? Answer B A (0, 3) [This object is a pull tab] B (2, 0) C ( 1, 0) D ( 1, 4) 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 160 / 182 71 Mr. Braun 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. Braun can buy? 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 160 (Answer) / 182 71 Mr. Braun 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. Braun can buy? Answer 5 [This object is a pull tab] 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 161 / 182 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

Slide 161 (Answer) / 182 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. Answer [This object is a pull tab] PARCC - PBA - Question #3 Non-Calculator Section - SMART Response Format

Slide 162 / 182 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.

Slide 162 (Answer) / 182 Math Practice Modeling with a System of Inequalities The example on the next 5 slides address MP.1, MP.2, MP.4, MP.5, MP.6, and MP.7 Additional Similar to Questions solving application to address problems MPs: by graphing a How single could inequality, you start we this can problem? also apply (MP.1) our skills with How solving can a you system represent of inequalities this problem to with solve realistic word symbols problems. and numbers? (MP.2) Which tool/manipulative would be best for this Let's problem? get started. (MP.5) How do you know that your answer is accurate? (MP.6) What do you know about writing inequalities that would apply to this situation? (MP.7) [This object is a pull tab]

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 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.

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 A: Graph the solution set of the system of linear inequalities in a coordinate plane. 10 5 0 5 10 15 20 x

Slide 164 (Answer) / 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 y$12 per hour. Preston cannot work more than a total of y 20 15 hours per month. Let x represent the number 20 15 of hours Preston mows lawns and y represent 10 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. Answer 10 5 5 0 5 10 [This object is a pull tab] 15 20 x 0 5 10 15 20 x

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 B: Create 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions.

Slide 165 (Answer) / 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 B: Create 3 ordered pairs (x, y) that represent the hours that Preston could work to meet the given conditions. Answer Answers may vary. As long as they are in the shaded region, or points on either line, they are valid. Sample Answers: (0, 14), (1, 13), (2, 12), (3, 11), (5, 10) [This object is a pull tab]

Slide 166 / 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 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.

Slide 166 (Answer) / 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 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. Answer 10 hours, since 9 hours will not meet the minimum of $150. [This object is a pull tab]

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 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.

Slide 167 (Answer) / 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 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. Answer 7 hours [This object is a pull tab]

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 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

Slide 168 (Answer) / 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 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. Answer 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. 40 30 20 10 y 40 30 20 10 y 0 10 20 [This object is a pull tab] 0 10 20 30 40 x 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 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)

Slide 169 (Answer) / 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 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) Answer A, C, D [This object is a pull tab]

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 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.

Slide 170 (Answer) / 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 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. Answer Gavin must sell at least 27 comic books. [This object is a pull tab]

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 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

Slide 171 (Answer) / 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 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. Answer [This object is a pull tab] PARCC - EOY - Question #25 Calculator Section - SMART Response Format