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