188 Chapter 2 Linear Equations and Inequalities For example, consider multiplying or dividing the inequality, Multiply/Divide 4 6 5 by 1 4 7 5 5 4 3 2 1 0 1 2 3 4 5 6 4 > 5 4 < 5 The number 4 lies to the left of 5 on the number line. However, lies to the right of 5 (Figure 2-12). Changing the sign of two numbers changes their relative position on the number line. This is stated formally in the multiplication and division properties of inequality. 4 Instructor Note: Show that this works for other situations. Start with two negative numbers or one positive and one negative. Show that multiplying or dividing by 1 changes the direction of the inequality sign. Multiplication and Division Properties of Inequality Let a, b, and c represent real numbers, c 0. a 6 b, ac 6 bc *If c is positive and then and a 6 b, ac 7 bc a c 6 *If c is negative and then and a c 7 b c b c The second statement indicates that if both sides of an inequality are multiplied or divided by a negative quantity, the inequality sign must be reversed. *These properties may also be stated for a b, a 7 b, and a b. 4 6 5 by 1. Figure 2-12 6 Solving a Linear Inequality Solve the inequality and graph the solution set. Express the solution set in set-builder notation and in interval notation. Solution: Add 3 to both sides. 5x 3 12 5x 3 3 12 3 5x 15 5x 5 15 5 x 3 5x ƒ x 36 Set-builder notation: Interval notation: 33, 2 5x 3 12 Example 5 Classroom Example: p. 195, Exercise 74 5 Divide by . Reverse the direction of the inequality sign. 6 5 4 3 2 1 0 1 2 3 4 5 6 Skill Practice Solve the inequality and graph the solution set. Express the solution set in set-builder notation and in interval notation. 11. 5p 2 7 22 Answer 11. 4 5p ƒ p 6 46; (, 4)
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