Answer
$x=\dfrac{-1\pm\sqrt{33}}{4}$
Work Step by Step
Multiplying both sides by the $LCD=x(x-2)$, the expression $
\dfrac{3x}{x-2}-\dfrac{x+1}{x}=\dfrac{6}{x(x-2)}
$ simplifies to
\begin{array}{l}
x(x-2)\left( \dfrac{3x}{x-2}-\dfrac{x+1}{x}\right)=\left(\dfrac{6}{x(x-2)}\right)x(x-2)
\\\\
x(3x)-(x-2)(x+1)=1(6)
\\\\
3x^2-(x^2+x-2x-2)=6
\\\\
3x^2-x^2-x+2x+2=6
\\\\
2x^2+x-4=0
.\end{array}
Using the Quadratic Formula, $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$, the values of $x$ are
\begin{array}{l}
x=\dfrac{-1\pm\sqrt{(1)^2-4(2)(-4)}}{2(2)}
\\\\
x=\dfrac{-1\pm\sqrt{33}}{4}
.\end{array}
