Answer
$x=\{ -8,18 \}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
|x-5|=13
,$ is equivalent to
\begin{array}{l}\require{cancel}
x-5=13 \text{ OR } x-5=-13
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-5=13
\\\\
x=13+5
\\\\
x=18
\\\\\text{ OR }\\\\
x-5=-13
\\\\
x=-13+5
\\\\
x=-8
.\end{array}
Hence, the solutions are $
x=\{ -8,18 \}
.$