## Intermediate Algebra (12th Edition)

Published by Pearson

# Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises - Page 119: 63

#### Answer

$\left[ -\dfrac{7}{6},-\dfrac{5}{6} \right]$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|-6x-6| \le 1 ,$ 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} -1 \le -6x-6 \le 1 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -1+6 \le -6x-6+6 \le 1+6 \\\\ 5 \le -6x \le 7 .\end{array} Dividing by a negative number (and consequently reversing the sign), the inequality above is equivalent to \begin{array}{l}\require{cancel} \dfrac{5}{-6} \ge \dfrac{-6x}{-6} \ge \dfrac{7}{-6} \\\\ -\dfrac{5}{6} \ge x \ge -\dfrac{7}{6} \\\\ -\dfrac{7}{6} \le x \le -\dfrac{5}{6} .\end{array} In interval notation, the solution set is $\left[ -\dfrac{7}{6},-\dfrac{5}{6} \right] .$ The colored graph is the graph of the solution set.

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