Answer
$\left( -\dfrac{3}{2},\dfrac{13}{10} \right)$
Work Step by Step
Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a,$ then the solution to the given equation, $
\left| 5x+\dfrac{1}{2} \right|-2\lt5
,$ is
\begin{array}{l}\require{cancel}
\left| 5x+\dfrac{1}{2} \right|\lt5+2
\\\\
\left| 5x+\dfrac{1}{2} \right|\lt7
\\\\
-7\lt 5x+\dfrac{1}{2} \lt7
\\\\
-7-\dfrac{1}{2}\lt 5x+\dfrac{1}{2}-\dfrac{1}{2} \lt7-\dfrac{1}{2}
\\\\
-\dfrac{15}{2}\lt 5x \lt \dfrac{13}{2}
\\\\
-\dfrac{\dfrac{15}{2}}{5}\lt \dfrac{5x}{5} \lt \dfrac{\dfrac{13}{2}}{5}
\\\\
-\dfrac{15}{10} \lt x \lt \dfrac{13}{10}
\\\\
-\dfrac{3}{2} \lt x \lt \dfrac{13}{10}
.\end{array}
Hence, the solution set is the interval $
\left( -\dfrac{3}{2},\dfrac{13}{10} \right)
.$