Section 2.8 Linear Inequalities 171 TIP: The solution to an inequality gives a set of values that make the original inequality true. Therefore, you can test your final answer by using test points. That is, pick a value in the proposed solution set and verify that it makes the original inequality true. Furthermore, any test point picked outside the solution set should make the original inequality false. For example, ( 6 5 4 3 2 1 0 1 2 3 4 5 6 p 4 p 3 Pick as an arbitrary test point Pick as an arbitrary test point outside within the proposed solution set. the proposed solution set. 2p 5 6 3p 6 2p 5 6 3p 6 ? ? 2142 5 6 3142 6 2132 5 6 3132 6 ? ? 8 5 6 12 6 6 5 6 9 6 ? 13 6 18 ✔ True 1 6 3 False 4. Multiplication and Division Properties of Inequality Multiplying both sides of an equation by the same 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. For example, consider multiplying or dividing the inequality, Multiply/Divide by 1 4 7 5 Figure 2-12 4 6 5 4 6 5 by 1. 5 4 3 2 1 0 1 2 3 4 5 6 4 > 5 4 < 5 6 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 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.
miller_beginning_intermediate_algebra_4e_ch1_3
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