#### Answer

$x=\left\{ -3,\dfrac{11}{2} \right\}$

#### Work Step by Step

Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
|4x-5|=17
,$ is equivalent to
\begin{array}{l}\require{cancel}
4x-5=17 \text{ OR } 4x-5=-17
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
4x-5=17
\\\\
4x=17+5
\\\\
4x=22
\\\\
x=\dfrac{22}{4}
\\\\
x=\dfrac{11}{2}
\\\\\text{ OR }\\\\
4x-5=-17
\\\\
4x=-17+5
\\\\
4x=-12
\\\\
x=-\dfrac{12}{4}
\\\\
x=-3
.\end{array}
Hence, the solutions are $
x=\left\{ -3,\dfrac{11}{2} \right\}
.$