Answer
$x=\{ -6,12 \}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
|x-3|=9
,$ is equivalent to
\begin{array}{l}\require{cancel}
x-3=9 \text{ OR } x-3=-9
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-3=9
\\\\
x=9+3
\\\\
x=12
\\\\\text{ OR }\\\\
x-3=-9
\\\\
x=-9+3
\\\\
x=-6
.\end{array}
Hence, the solutions are $
x=\{ -6,12 \}
.$