Answer
$x= \left\{ -\dfrac{4}{3},\dfrac{2}{9} \right\}$
Work Step by Step
Since for any $a\gt0$, $|x|=a$ implies $x=a$ or $x=-a$, then the solution/s to the given equation, $
\left| \dfrac{6x+1}{x-1} \right|=3
,$ is/are
\begin{array}{l}\require{cancel}
\dfrac{6x+1}{x-1}=3
\\\\
6x+1=3(x-1)
\\\\
6x+1=3x-3
\\\\
6x-3x=-3-1
\\\\
3x=-4
\\\\
x=-\dfrac{4}{3}
\\\\\text{OR}\\\\
\dfrac{6x+1}{x-1}=-3
\\\\
6x+1=-3(x-1)
\\\\
6x+1=-3x+3
\\\\
6x+3x=3-1
\\\\
9x=2
\\\\
x=\dfrac{2}{9}
.\end{array}
Hence, the solutions are $
x= \left\{ -\dfrac{4}{3},\dfrac{2}{9} \right\}
.$