#### Answer

$x=\left\{ -4,\dfrac{22}{5} \right\}$

#### Work Step by Step

Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
|5x-1|=21
,$ is equivalent to
\begin{array}{l}\require{cancel}
5x-1=21 \text{ OR } 5x-1=-21
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
5x-1=21
\\\\
5x=21+1
\\\\
5x=22
\\\\
x=\dfrac{22}{5}
\\\\\text{ OR }\\\\
5x-1=-21
\\\\
5x=-21+1
\\\\
5x=-20
\\\\
x=-\dfrac{20}{5}
\\\\
x=-4
.\end{array}
Hence, the solutions are $
x=\left\{ -4,\dfrac{22}{5} \right\}
.$