# Chapter 1 - Section 1.7 - Absolute Value Equations and Inequalities - 1.7 Exercises: 62

$\left[ 0,6 \right]$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ To solve the given inequality, $|2x-6| \le 6 ,$ 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} -6 \le 2x-6 \le 6 .\end{array} Using the properties of inequality, the inequality above is equivalent to \begin{array}{l}\require{cancel} -6+6 \le 2x-6+6 \le 6+6 \\\\ 0 \le 2x \le 12 \\\\ \dfrac{0}{2} \le \dfrac{2x}{2} \le \dfrac{12}{2} \\\\ 0 \le x \le 6 .\end{array} In interval notation, the solution set is $\left[ 0,6 \right] .$ The colored graph is the graph of the solution set.

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