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Chapter 3 Introduction to Graphing and Equations of Lines Section 7 Linear Inequalities in Two Variables

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3.7 - 2 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Section 3.7 Objectives 1Determine Whether an Ordered Pair Is a Solution to a Linear Inequality 2Graph Linear Inequalities 3Solve Problems Involving Linear Inequalities

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3.7 - 3 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Linear Inequalities Linear inequalities in two variables are inequalities in one of the forms Ax + By C Ax + By C Ax + By C where A and B are not both zero. A linear inequality in two variables x and y is satisfied by an ordered pair (a, b) if a true statement results when x is replaced by a and y is replaced by b.

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3.7 - 4 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Solutions to Linear Inequalities Example: Determine whether the ordered pair (5, – 1) is a solution to the inequality 4x – 5y 12. Let x = 5 and y = – 1 in the inequality. 4x – 5y 12 True. (5, – 1) is a solution. 4(5) – 5(– 1) 12 20 + 5 12 25 12

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3.7 - 5 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Graphing Linear Inequalities A graph of a linear inequality in two variables x and y consists of all points (x, y) whose coordinates satisfy the inequality. A dashed line indicates that the line is NOT part of the solution. y x 4 4 y < – 3x + 9 A solid line indicates that the line IS part of the solution. y x 4 4 y – 3x + 9

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3.7 - 6 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Graphing Linear Inequalities in Two Variables Graphing Linear Inequalities Step 1: Replace the inequality symbol with an equal sign and graph the resulting equation. If the inequality is strict ( ), use dashes to graph the line; if the inequality is nonstrict ( or ), use a solid line. The graph separates the xy-plane into two half-planes. Step 2: Select a test point P that is not on the line (that is, select a test point in one of the half-planes). (a) If the coordinates of P satisfy the inequality, then shade the half-plane containing P. (b) If the coordinates of P do not satisfy the inequality, then shade the half-plane that does not contain P.

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3.7 - 7 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Graphing Linear Inequalities Example: Graph the inequality 2x + 5y 10. Graph the line 2x + 5y = 10. The solid line indicates that the line is part of the solution. Look for a test point. Is (0, 0) a solution? 2x + 5y 10 2(0) + 5(0) 10 0 9 False (0, 0) is not included in the solution. y x 1 4 1 4

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3.7 - 8 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Graphing Linear Inequalities Example: Graph the inequality 2x – 3y < 0 Graph the line 2x – 3y = 0 The dashed line indicates that the line is NOT part of the solution. Look for a test point. Is (1, 2) a solution? 2x – 3y < 0 2(1) – 3(2) < 0 – 4 < 0 True (1, 2) is included in the solution. x y 11 22 33 44 123 4 22 33 44 1 2 3 4 2 – 6 < 0

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3.7 - 9 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Solving Problems Involving Inequalities Example: Sharon gets $15 for allowance. She went to the store and bought gum that costs $1.50 a pack and candy bars that cost $2.00 each. (a) Write a linear inequality that describes Sharon’s options for buying gum and candy bars with the $15. (b) Can she buy 5 packs of gum and 6 candy bars? (c) Can she buy 4 packs of gum and 4 candy bars? Continued.

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3.7 - 10 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Solving Problems Involving Inequalities Example continued: (a) Step 1: Identify We want to determine the number of packs of gum and the number of candy bars that Sharon can buy for $15. Step 2: Name g = the number of packs of gum c = the number of candy bars Step 3: Translate If Sharon buys g packs of gum, she will spend 1.50g. If she buys c candy bars, she will spend 2c. Continued. 1.5g + 2c 15 Use the information to write the linear inequality.

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3.7 - 11 Copyright © 2010 Pearson Education, Inc. Sullivan, III & Struve, Elementary and Intermediate Algebra Solving Problems Involving Inequalities Example continued: (c) Let g = 4 and c = 4. 1.5(5) + 2(6) 15 7.5 + 12 15 19.5 15 False 1.5(4) + 2(4) 15 6 + 8 15 14 15 True (b) Let g = 5 and c = 6. 1.5g + 2c 15 Sharon cannot buy 5 packs of gum and 6 candy bars. Sharon can buy 4 packs of gum and 4 candy bars.

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