Answer
$x=\left\{ -10,-2 \right\}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
\left|\dfrac{1}{2}x+3\right|=2
,$ is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{1}{2}x+3=2 \text{ OR } \dfrac{1}{2}x+3=-2
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
\dfrac{1}{2}x+3=2
\\\\
2\left(\dfrac{1}{2}x+3\right)=(2)2
\\\\
x+6=4
\\\\
x=4-6
\\\\
x=-2
\\\\\text{ OR }\\\\
\dfrac{1}{2}x+3=-2
\\\\
2\left(\dfrac{1}{2}x+3\right)=(-2)2
\\\\
x+6=-4
\\\\
x=-4-6
\\\\
x=-10
.\end{array}
Hence, the solutions are $
x=\left\{ -10,-2 \right\}
.$
