## Intermediate Algebra (12th Edition)

Published by Pearson

# Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises: 49

#### Answer

$\left[ -\dfrac{12}{5},\dfrac{8}{5} \right]$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|5x+2| \le 10 ,$ use the definition of absolute value inequalities. Use the properties of inequalities to isolate the variable. For the interval notation, use a parenthesis for the symbols $\lt$ or $\gt.$ Use a bracket for the symbols $\le$ or $\ge.$ For graphing inequalities, use a hollowed dot for the symbols $\lt$ or $\gt.$ Use a solid dot for the symbols $\le$ or $\ge.$ $\bf{\text{Solution Details:}}$ Since for any $c\gt0$, $|x|\lt c$ implies $-c\lt x\lt c$ (or $|x|\le c$ implies $-c\le x\le c$), the inequality above is equivalent to \begin{array}{l}\require{cancel} -10 \le 5x+2 \le 10 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -10-2 \le 5x+2-2 \le 10-2 \\\\ -12 \le 5x \le 8 \\\\ -\dfrac{12}{5} \le \dfrac{5x}{5} \le \dfrac{8}{5} \\\\ -\dfrac{12}{5} \le x \le \dfrac{8}{5} .\end{array} In interval notation, the solution set is $\left[ -\dfrac{12}{5},\dfrac{8}{5} \right] .$ The colored graph is the graph of the solution set.

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