# Chapter 9 - Inequalities and Problem Solving - 9.3 Absolute-Value Equations and Inequalities - 9.3 Exercise Set - Page 599: 43

$x=\left\{ -8.7,8.7 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$ Given that $f(x)=|x|-3 ,$ to find $x$ for which $f(x)=5.7 ,$ use substitution. Then isolate the absolute value expression and use the definition of an absolute value equality. $\bf{\text{Solution Details:}}$ Replacing $f(x)$ with $5.7 ,$ then \begin{array}{l}\require{cancel} f(x)=|x|-3 \\\\ 5.7=|x|-3 \\\\ |x|-3=5.7 .\end{array} Using the properties of equality to isolate the variable results to \begin{array}{l}\require{cancel} |x|-3=5.7 \\\\ |x|=5.7+3 \\\\ |x|=8.7 .\end{array} Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the given equation is equivalent to \begin{array}{l}\require{cancel} x=8.7 \\\\\text{OR}\\\\ x=-8.7 .\end{array} Hence, $x=\left\{ -8.7,8.7 \right\} .$

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