Answer
$\dfrac{2(2x-3)}{3(6+x)}$
Work Step by Step
The given expression, $
\left( \dfrac{2}{3}-\dfrac{1}{x} \right)\div\left( \dfrac{3}{x}+\dfrac{1}{2} \right)
,$ simplifies to
\begin{array}{l}\require{cancel}
\dfrac{x(2)-3(1)}{3x}\div\dfrac{2(3)+x(1)}{2x}
\\\\
\dfrac{2x-3}{3x}\div\dfrac{6+x}{2x}
\\\\
\dfrac{2x-3}{3x}\cdot\dfrac{2x}{6+x}
\\\\
\dfrac{2x-3}{3\cancel{x}}\cdot\dfrac{2\cancel{x}}{6+x}
\\\\
\dfrac{2(2x-3)}{3(6+x)}
.\end{array}
