Answer
$x=\left\{ -\dfrac{7}{3},1 \right\}$
Work Step by Step
Using the properties of equality, the given equation, $
|3x+2|+4=9
,$ is equivalent to
\begin{array}{l}\require{cancel}
|3x+2|=9-4
\\\\
|3x+2|=5
.\end{array}
Since for any $a\gt0$, $|x|=a$ implies $x=a$ or $x=-a$, then the equation above is equivalent to
\begin{array}{l}\require{cancel}
3x+2=5 \text{ OR } 3x+2=-5
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3x+2=5
\\\\
3x=5-2
\\\\
3x=3
\\\\
x=\dfrac{3}{3}
\\\\
x=1
\\\\\text{ OR }\\\\
3x+2=-5
\\\\
3x=-5-2
\\\\
3x=-7
\\\\
x=-\dfrac{7}{3}
.\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{7}{3},1 \right\}
.$