Answer
$\left( \dfrac{1}{6},\dfrac{7}{2} \right)$
Work Step by Step
Using the properties of inequality, the solution set of the given inequality, $
\dfrac{2}{3}\lt x+\dfrac{1}{2}\lt4
,$ is
\begin{array}{l}\require{cancel}
\dfrac{2}{3}-\dfrac{1}{2}\lt x+\dfrac{1}{2}-\dfrac{1}{2}\lt4-\dfrac{1}{2}
\\\\
\dfrac{4}{6}-\dfrac{3}{6}\lt x\lt\dfrac{8}{2}-\dfrac{1}{2}
\\\\
\dfrac{1}{6}\lt x\lt\dfrac{7}{2}
.\end{array}
Hence, the solution set is $
\left( \dfrac{1}{6},\dfrac{7}{2} \right)
.$
