Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Chapter 9 Equations, Inequalities and Problem Solving

Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. 9.7 Linear Inequalities and Problem Solving

Martin-Gay, Developmental Mathematics, 2e 33 Linear Inequalities An inequality is a statement that contains of the symbols:, ≤ or ≥. EquationsInequalities x = 3x > 3 12 = 7 – 3y12 ≤ 7 – 3y Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 44 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 {x  x  7} Represents the set {x  x > – 4} Graphing Solutions Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 55 Graph: Example Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 66 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 77 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 bc are equivalent inequalities. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 88 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. Solving Linear Inequalities Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 99 Solve: 3x + 8 ≥ 5. Graph the solution set. Example Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 10 Solve: 3x + 9 ≥ 5(x – 1). Graph the solution set. 3x + 9 ≥ 5x – 5 Apply the distributive property. 3x – 3x + 9 ≥ 5x – 3x – 5 Subtract 3x from both sides. 9 ≥ 2x – 5 Simplify. 14 ≥ 2x Simplify. 7 ≥ x Divide both sides by ≥ 2x – Add 5 to both sides. The graph of solution set is{x|x ≤ 7}. Example 3x + 9 ≥ 5(x – 1) Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 11 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 – > – Add 14 to both sides. 4x > –18 Simplify. Divide both sides by 4. Example Solve: 7(x – 2) + x > –4(5 – x) – 12. Graph the solution set. The graph of solution set is {x|x > –9/2}. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 12 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 ≤ x ≤ 1200 continued Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall.

Martin-Gay, Developmental Mathematics, 2e 13 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. 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.