Answer
$\left( \dfrac{23}{8},\dfrac{17}{8} \right)$
Work Step by Step
Using the properties of equality, the given expression, $
\left| 2x+\dfrac{3}{4} \right|-7\le-2
,$ is equivalent to
\begin{array}{l}\require{cancel}
\left| 2x+\dfrac{3}{4} \right|\le-2+7
\\\\
\left| 2x+\dfrac{3}{4} \right|\le5
.\end{array}
Since for any $a\gt0$, $|x|\le a$ implies $-a\le x\le a$, then the expression, $
\left| 2x+\dfrac{3}{4} \right|\le5
,$ is equivalent to
\begin{array}{l}\require{cancel}
-5\le 2x+\dfrac{3}{4} \le5
\\\\
-5-\dfrac{3}{4}\le 2x+\dfrac{3}{4}-\dfrac{3}{4} \le5-\dfrac{3}{4}
\\\\
-\dfrac{23}{4}\le 2x \le\dfrac{17}{4}
\\\\
-\dfrac{\dfrac{23}{4}}{2}\le \dfrac {2}{2}x \le\dfrac{\dfrac{17}{4}}{2}
\\\\
-\dfrac{23}{8}\lt x \le\dfrac{17}{8}
.\end{array}
Hence, the solution set is $
\left( \dfrac{23}{8},\dfrac{17}{8} \right)
.$
See the graph below.