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