Answer
$x=\left\{ -\dfrac{1}{2},\dfrac{1}{3} \right\}$
Work Step by Step
Multiplying both sides by the $LCD=
x(x-2)
,$ the solution to the given equation, $
\dfrac{5x}{x-2}-\dfrac{x+1}{x}=\dfrac{3}{x(x-2)}
,$ is
\begin{array}{l}\require{cancel}
x(5x)-(x-2)(x+1)=1(3)
\\\\
5x^2-(x^2+x-2x-2)=3
\\\\
5x^2-(x^2-x-2)=3
\\\\
5x^2+x^2+x+2-3=0
\\\\
6x^2+x-1=0
\\\\
(2x+1)(3x-1)=0
\\\\
x=\left\{ -\dfrac{1}{2},\dfrac{1}{3} \right\}
.\end{array}
