Answer
$\left( -\infty,-\dfrac{7}{6} \right)\cup\left( \dfrac{3}{2},\infty \right)$
Work Step by Step
Using the properties of inequality, the given statement, $
|6x-1|-2\gt6
,$ is equivalent to
\begin{array}{l}\require{cancel}
|6x-1|\gt6+2
\\\\
|6x-1|\gt8
.\end{array}
Since for any $a\gt0$, $|x|\gt a$ implies $x\gt a$ OR $x\lt-a$, then the equation above is equivalent to
\begin{array}{l}\require{cancel}
6x-1\gt8 \text{ OR } 6x-1\lt-8
.\end{array}
Solving each inequality results to
\begin{array}{l}\require{cancel}
6x-1\gt8
\\\\
6x\gt8+1
\\\\
6x\gt9
\\\\
x\gt\dfrac{9}{6}
\\\\
x\gt\dfrac{3}{2}
\\\\\text{ OR }\\\\
6x-1\lt-8
\\\\
6x\lt-8+1
\\\\
6x\lt-7
\\\\
x\lt-\dfrac{7}{6}
.\end{array}
Hence, the solution set is the interval $
\left( -\infty,-\dfrac{7}{6} \right)\cup\left( \dfrac{3}{2},\infty \right)
.$
