Answer
$x=\left\{ -\dfrac{9}{2},\dfrac{27}{2} \right\}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ OR $x=-a$, then the given equation, $
|2x-9|=18
,$ is equivalent to
\begin{array}{l}\require{cancel}
2x-9=18 \text{ OR } 2x-9=-18
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
2x-9=18
\\\\
2x=18+9
\\\\
2x=27
\\\\
x=\dfrac{27}{2}
\\\\\text{ OR }\\\\
2x-9=-18
\\\\
2x=-18+9
\\\\
2x=-9
\\\\
x=-\dfrac{9}{2}
.\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{9}{2},\dfrac{27}{2} \right\}
.$