Answer
$x=\dfrac{3}{2}$
Work Step by Step
Factoring the given expression, $
\dfrac{1}{x-2}-\dfrac{3x}{x^2-4}=\dfrac{2}{x+2}
,$ results to
\begin{array}{l}\require{cancel}
\dfrac{1}{x-2}-\dfrac{3x}{(x+2)(x-2)}=\dfrac{2}{x+2}
.\end{array}
Multiplying both sides by the $LCD=
(x+2)(x-2)
$, the value of the variable that satisfies the equation above, $
\dfrac{1}{x-2}-\dfrac{3x}{(x+2)(x-2)}=\dfrac{2}{x+2}
,$ is
\begin{array}{l}\require{cancel}
(x+2)(1)-1(3x)=(x-2)(2)
\\\\
x+2-3x=2x-4
\\\\
x-3x-2x=-4-2
\\\\
-4x=-6
\\\\
x=\dfrac{-6}{-4}
\\\\
x=\dfrac{3}{2}
.\end{array}
