78 Chapter 1 Linear Equations and Inequalities in One Variable Multiplying both sides of an equation by the same nonzero quantity results in an equivalent equation. However, the same is not always true for an inequality. If you multiply or divide an inequality by a negative quantity, the direction of the inequality symbol must be reversed. 4 6 5 For example, consider multiplying or dividing the inequality by 1. Multiply/divide by 1: 4 5 6 5 4 3 2 1 0 1 2 3 4 5 6 4 5 4 5 ac 6 bc and ac 7 bc and a 6 b, a 6 b, Solving a Linear Inequality Solve the inequality. Solution: Add 5 to both sides. a c 6 a c 7 b c b c Divide by 2 (reverse the inequality sign). 2x 5 6 2 2x 5 5 6 2 5 2x 6 7 Set-builder notation: Interval notation: 1 72 , 2 ) 7 22 5x 0 x 7 72 6 x 7 7 2 or x 7 3.5 2x 2 7 7 2 2x 5 6 2 Example 2 4 5 The number 4 lies to the left of 5 on the number line. However,4 lies to the right of 5. Changing the signs of two numbers changes their relative position on the number line.This is stated formally in the multiplication and division properties of inequality. Multiplication and Division Properties of Inequality Let a, b, and c represent real numbers. *If c is positive and then *If c is negative and then 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. Avoiding Mistakes Do not forget to reverse the direction of the inequality sign when multiplying or dividing by a negative number.
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