Intermediate Algebra for College Students (7th Edition)

Published by Pearson
ISBN 10: 0-13417-894-7
ISBN 13: 978-0-13417-894-3

Chapter 4 - Section 4.3 - Equations and Inequalities Involving Absolute Value - Exercise Set - Page 284: 72

Answer

$(-\infty,2)\cup(8,\infty)$. The graph is shown below.

Work Step by Step

The given expression is $\Rightarrow -2\left | 5-x\right |\lt-6$ Divide both sides by $2$. $\Rightarrow \frac{-2\left | 5-x\right |}{2}\lt\frac{-6}{2}$ Simplify. $\Rightarrow -\left | 5-x\right |\lt-3$ Multiply all parts by −1 and change the sense of the inequality. $\Rightarrow -1(-\left | 5-x\right |)\gt-1(-3)$ Simplify. $\Rightarrow \left | 5-x\right |\gt3$ Rewrite the inequality without absolute value bars. $\Rightarrow 5-x\lt-3$ or $5-x\gt3$ Solve each inequality separately. Subtract $5$ from all sides. $\Rightarrow 5-x-5\lt-3-5$ or $5-x-5\gt3-5$ Simplify. $\Rightarrow -x\lt-8$ or $-x\gt-2$ Multiply all parts by −1 and change the sense of the inequality. $\Rightarrow -1(-x)\gt-1(-8)$ or $-1(-x)\lt-1(-2)$ Simplify. $\Rightarrow x\gt8$ or $x\lt2$ The solution set is less than $2$ or greater than $8$. The interval notation is $(-\infty,2)\cup(8,\infty)$.
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