Answer
$(-\infty,-\frac{1}{2})\cup(\frac{3}{2},\infty)$.
The graph is shown below.
Work Step by Step
The given expression is
$\Rightarrow 3\left | 2 x-1\right |+2\gt8$
Subtract $2$ from both sides.
$\Rightarrow 3\left | 2 x-1\right |+2-2\gt8-2$
Simplify.
$\Rightarrow 3\left | 2 x-1\right |\gt6$
Divide both sides by $3$.
$\Rightarrow \frac{3\left | 2 x-1\right |}{3}\gt\frac{6}{3}$
Simplify.
$\Rightarrow \left | 2 x-1\right |\gt2$
Rewrite the inequality without absolute value bars.
$\Rightarrow 2x-1\lt-2$ or $2x-1\gt2$
Solve each inequality separately.
Add $1$ to all parts.
$\Rightarrow 2x-1+1\lt-2+1$ or $2x-1+1\gt2+1$
Simplify.
$\Rightarrow 2x\lt-1$ or $2x\gt3$
Divide all parts by $2$.
$\Rightarrow \frac{2x}{2}\lt\frac{-1}{2}$ or $\frac{2x}{2}\gt\frac{3}{2}$
Simplify.
$\Rightarrow x\lt-\frac{1}{2}$ or $x\gt\frac{3}{2}$
The solution set is less than $-\frac{1}{2}$ or greater than $\frac{3}{2}$.
The interval notation is
$(-\infty,-\frac{1}{2})\cup(\frac{3}{2},\infty)$.
