Answer
$\left( -\infty,-3 \right] \cup \left[ \dfrac{5}{3},\infty \right)$
Work Step by Step
Since for any $c\gt0$, $|x|\gt c$ implies $x\gt c$ or $x\lt-c$, the given inequality, $
|3x+2|\ge7
,$ is equivalent to
\begin{array}{l}\require{cancel}
3x+2\ge7 \text{ OR } 3x+2\le-7
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
3x+2\ge7
\\\\
3x\ge7-2
\\\\
3x\ge5
\\\\
x\ge\dfrac{5}{3}
\\\\\text{ OR }\\\\
3x+2\le-7
\\\\
3x\le-7-2
\\\\
3x\le-9
\\\\
x\le-\dfrac{9}{3}
\\\\
x\le-3
.\end{array}
Hence, the solution to the given inequality is the interval $
\left( -\infty,-3 \right] \cup \left[ \dfrac{5}{3},\infty \right)
.$
Note that $"\gt"$ may be replaced with $"\ge"$.
