Answer
$x=\left\{ 12,36 \right\}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
\left|12-\dfrac{1}{2}x\right|=6
,$ is equivalent to
\begin{array}{l}\require{cancel}
12-\dfrac{1}{2}x=6 \text{ OR } 12-\dfrac{1}{2}x=-6
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
12-\dfrac{1}{2}x=6
\\\\
-\dfrac{1}{2}x=6-12
\\\\
-\dfrac{1}{2}x=-6
\\\\
-2\left(-\dfrac{1}{2}x\right)=(-6)(-2)
\\\\
x=12
\\\\\text{ OR }\\\\
12-\dfrac{1}{2}x=-6
\\\\
-\dfrac{1}{2}x=-6-12
\\\\
-\dfrac{1}{2}x=-18
\\\\
-2\left(-\dfrac{1}{2}x\right)=(-18)(-2)
\\\\
x=36
.\end{array}
Hence, the solutions are $
x=\left\{ 12,36 \right\}
.$