Answer
$\text{the interval }
\left( \dfrac{3}{2},5 \right]
$
Work Step by Step
Using the properties of inequalities, the solution to the given inequality, $
-3 \lt 2(x-3) \le 4
,$ is
\begin{array}{l}\require{cancel}
-3 \lt 2x-6 \le 4
\\\\
-3+6 \lt 2x-6+6 \le 4+6
\\\\
3 \lt 2x \le 10
\\\\
\dfrac{3}{2} \lt \dfrac{2}{2}x \le \dfrac{10}{2}
\\\\
\dfrac{3}{2} \lt x \le 5
.\end{array}
In interval notation, the solution is $
\text{the interval }
\left( \dfrac{3}{2},5 \right]
.$