#### Answer

$x=-12$

#### Work Step by Step

$\bf{\text{Solution Outline:}}$
To solve the given equation, $
\sqrt[3]{2x-3}-2=-5
,$ isolate first the radical expression. Then raise both sides to the exponent equal to the index of the radical. Use the properties of equality to isolate the variable. Finally, do checking of the solution/s with the original equation.
$\bf{\text{Solution Details:}}$
Using the properties of equality, the given equation is equivalent to
\begin{array}{l}\require{cancel}
\sqrt[3]{2x-3}=-5+2
\\\\
\sqrt[3]{2x-3}=-3
.\end{array}
Get rid of the radical symbol by raising both sides of the equation above to the exponent equal to $
3
$ (the same index as the radical). This results to
\begin{array}{l}\require{cancel}
(\sqrt[3]{2x-3})^3=(-3)^3
\\\\
2x-3=-27
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
2x=-27+3
\\\\
2x=-24
\\\\
x=-\dfrac{24}{2}
\\\\
x=-12
.\end{array}
Upon checking, $
x=-12
$ satisfies the original equation.