#### Answer

$x=\left\{ -\dfrac{5}{3},3 \right\}$

#### Work Step by Step

Using the properties of equality, the given equation, $
|3x-2|+1=8
,$ is equivalent to
\begin{array}{l}\require{cancel}
|3x-2|=8-1
\\\\
|3x-2|=7
.\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=7 \text{ OR } 3x-2=-7
.\end{array}
Solving each equation results to
\begin{array}{l}\require{cancel}
3x-2=7
\\\\
3x=7+2
\\\\
3x=9
\\\\
x=\dfrac{9}{3}
\\\\
x=3
\\\\\text{ OR }\\\\
3x-2=-7
\\\\
3x=-7+2
\\\\
3x=-5
\\\\
x=-\dfrac{5}{3}
.\end{array}
Hence, the solutions are $
x=\left\{ -\dfrac{5}{3},3 \right\}
.$