Answer
$\left[ -3,-2 \right]$
Work Step by Step
Since for any $a\gt0$, $|x|\lt a$ implies $-a\lt x\lt a,$ then the given inequality, $
|2x+5|\le1
,$ is equivalent to
\begin{array}{l}\require{cancel}
-1\le 2x+5\le1
.\end{array}
Using the properties of inequality, then
\begin{array}{l}\require{cancel}
-1-5\le 2x+5-5\le1-5
\\\\
-6\le 2x\le-4
\\\\
-\dfrac{6}{2}\le \dfrac{2x}{2}\le-\dfrac{4}{2}
\\\\
-3\le x\le-2
.\end{array}
Hence, the solution is the interval $
\left[ -3,-2 \right]
.$
