Answer
$x=\left\{ -4,-\dfrac{2}{3} \right\}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ or $x=-a$, then the given inequality, $
\left| 3x+7 \right|-5=0
,$ is equivalent to
\begin{array}{l}\require{cancel}
\left| 3x+7 \right|=5
\\\\
3x+7=5
\\\text{ OR }\\
3x+7=-5
.\end{array}
Solving each equation separately, then,
\begin{array}{l}\require{cancel}
3x+7=5
\\\\
3x=5-7
\\\\
3x=-2
\\\\
x=-\dfrac{2}{3}
\\\text{ OR }\\
3x+7=-5
\\\\
3x=-5-7
\\\\
3x=-12
\\\\
x=-\dfrac{12}{3}
\\\\
x=-4
.\end{array}
Hence, $
x=\left\{ -4,-\dfrac{2}{3} \right\}
.$
