Answer
$x=\left\{ 18,66 \right\}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
\left|14-\dfrac{1}{3}x\right|=8
,$ is equivalent to
\begin{array}{l}\require{cancel}
14-\dfrac{1}{3}x=8 \text{ OR } 14-\dfrac{1}{3}x=-8
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
14-\dfrac{1}{3}x=8
\\\\
-\dfrac{1}{3}x=8-14
\\\\
-\dfrac{1}{3}x=-6
\\\\
-3\left(-\dfrac{1}{3}x\right)=(-6)(-3)
\\\\
x=18
\\\\\text{ OR }\\\\
14-\dfrac{1}{3}x=-8
\\\\
-\dfrac{1}{3}x=-8-14
\\\\
-\dfrac{1}{3}x=-22
\\\\
-3\left(-\dfrac{1}{3}x\right)=(-22)(-3)
\\\\
x=66
.\end{array}
Hence, the solutions are $
x=\left\{ 18,66 \right\}
.$