#### Answer

$x=\left\{ 3,9 \right\}$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
3|x-6|=9
,$ use the properties of equality to isolate the absolute value expression. Then use the definition of absolute value equality. Use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Using the properties of equality to isolate the absolute value expression results to
\begin{array}{l}\require{cancel}
|x-6|=\dfrac{9}{3}
\\\\
|x-6|=3
.\end{array}
Since for any $c\gt0$, $|x|=c$ implies $x=c \text{ or } x=-c,$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
x-6=3
\\\\\text{OR}\\\\
x-6=-3
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
x-6=3
\\\\
x=3+6
\\\\
x=9
\\\\\text{OR}\\\\
x-6=-3
\\\\
x=-3+6
\\\\
x=3
.\end{array}
Hence, $
x=\left\{ 3,9 \right\}
.$