Answer
$x=\left\{ -\dfrac{17}{2},\dfrac{19}{2} \right\}$
Work Step by Step
Since $|x|=a$ implies $x=a \text{ OR } x=-a$, then the statement $
\left|\dfrac{2x-1}{3} \right|=6
$ evaluates to
\begin{array}{l}
\dfrac{2x-1}{3}=6\\\\
2x-1=18\\
2x=19\\\\
x=\dfrac{19}{2}
,\\\\\text{OR}\\\\
\dfrac{2x-1}{3}=-6\\\\
2x-1=-18\\
2x=-17\\\\
x=-\dfrac{17}{2}
.\end{array}
Hence, the solution set is $
x=\left\{ -\dfrac{17}{2},\dfrac{19}{2} \right\}
$.