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Chapter 9 Equations, Inequalities and Problem Solving Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 9.7 Linear Inequalities and Problem Solving Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Linear Inequalities An inequality is a statement that contains of the symbols: < , >, ≤ or ≥. Equations x=3 Inequalities x>3 12 = 7 – 3y 12 ≤ 7 – 3y Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 3 Graphing Solutions Graphing solutions to linear inequalities in one variable • Use a number line • Use a closed circle at the endpoint of a interval if you want to include the point • Use an open circle at the endpoint if you DO NOT want to include the point Represents the set {xx 7} 7 Represents the set {xx > – 4} -4 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 4 Example Graph: 2 x 5 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 5 Addition Property of Inequality If a, b, and c are real numbers, then a < b and a + c < b + c are equivalent inequalities. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 6 Multiplication Property of Inequality 1. If a, b, and c are real numbers, and c is positive, then a < b and ac < bc are equivalent inequalities. 2. If a, b, and c are real numbers, and c is negative, then a < b and ac > bc are equivalent inequalities. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 7 Solving Linear Inequalities To Solve Linear Inequalities in One Variable Step 1: If an inequality contains fractions, multiply both sides by the LCD to clear the inequality of fractions. Step 2: Use distributive property to remove parentheses if they appear. Step 3: Simplify each side of inequality by combining like terms. Step 4: Get all variable terms on one side and all numbers on the other side by using the addition property of inequality. Step 5: Get the variable alone by using the multiplication property of inequality. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 8 Example Solve: 3x + 8 ≥ 5. Graph the solution set. 3x 8 5 3x 8 8 5 8 3 x 3 3x 3 3 3 x 1 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 9 Example Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set. 3x + 9 ≥ 5(x – 1) 3x + 9 ≥ 5x – 5 Apply the distributive property. 3x – 3x + 9 ≥ 5x – 3x – 5 9 ≥ 2x – 5 9 + 5 ≥ 2x – 5 + 5 14 ≥ 2x 7≥x The graph of solution set is{x|x ≤ 7}. Subtract 3x from both sides. Simplify. Add 5 to both sides. Simplify. Divide both sides by 2. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 10 Example Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set. 7(x – 2) + x > –4(5 – x) – 12 7x – 14 + x > –20 + 4x – 12 Apply the distributive property. 8x – 14 > 4x – 32 Combine like terms. 8x – 4x – 14 > 4x – 4x – 32 Subtract 4x from both sides. 4x – 14 > –32 Simplify. 4x – 14 + 14 > –32 + 14 Add 14 to both sides. 4x > –18 Simplify. 9 x Divide both sides by 4. 2 The graph of solution set is {x|x > –9/2}. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 11 Example You are having a catered event. You can spend at most $1200. The set up fee is $250 plus $15 per person, find the greatest number of people that can be invited and still stay within the budget. Let x represent the number of people Set up fee + cost per person × number of people ≤ 1200 250 + 15x ≤ 1200 continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 12 continued You are having a catered event. You can spend at most $1200. The set up fee is $250 plus $15 per person, find the greatest number of people that can be invited and still stay within the budget. 250 15x 1200 15 x 950 15 x 950 15 15 x 63.3 The number of people who can be invited must be 63 or less to stay within the budget. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Martin-Gay, Developmental Mathematics, 2e 13