#### Answer

$x=\left\{ -6,9 \right\}$

#### Work Step by Step

Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
\left|\dfrac{2}{3}x-1\right|=5
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{2}{3}x-1=5 \text{ OR } \dfrac{2}{3}x-1=-5
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
\dfrac{2}{3}x-1=5
\\\\
3\left(\dfrac{2}{3}x-1\right)=(5)3
\\\\
2x-3=15
\\\\
2x=15+3
\\\\
2x=18
\\\\
x=\dfrac{18}{2}
\\\\
x=9
\\\\\text{ OR }\\\\
3\left(\dfrac{2}{3}x-1\right)=(-5)3
\\\\
2x-3=-15
\\\\
2x=-15+3
\\\\
2x=-12
\\\\
x=-\dfrac{12}{2}
\\\\
x=-6
.\end{array}
Hence, the solutions are $
x=\left\{ -6,9 \right\}
.$