Answer
$x=\left\{ -\dfrac{1}{3},1 \right\}$
Work Step by Step
Since for any $c\gt0$, $|x|=c$ implies $x=c$ or $x=-c$, the given equation, $
|3x-1|=2
,$ is equivalent to
\begin{array}{l}\require{cancel}
3x-1=2 \text{ OR } 3x-1=-2
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3x-1=2
\\\\
3x=2+1
\\\\
3x=3
\\\\
x=\dfrac{3}{3}
\\\\
x=1
\\\\\text{ OR }\\\\
3x-1=-2
\\\\
3x=-2+1
\\\\
3x=-1
\\\\
x=-\dfrac{1}{3}
.\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{1}{3},1 \right\}
.$