Answer
$(-\infty,1)\cup(2,\infty)$.
The graph is shown below.
Work Step by Step
The given expression is
$\Rightarrow 2\left | 2 x-3\right |+10\gt12$
Subtract $10$ from both sides.
$\Rightarrow 2\left | 2 x-3\right |+10-10\gt12-10$
Simplify.
$\Rightarrow 2\left | 2 x-3\right |\gt2$
Divide both sides by $2$.
$\Rightarrow \frac{2\left | 2 x-3\right |}{2}\gt\frac{2}{2}$
Simplify.
$\Rightarrow \left | 2 x-3\right |\gt1$
Rewrite the inequality without absolute value bars.
$\Rightarrow 2x-3\lt-1$ or $2x-3\gt1$
Solve each inequality separately.
Add $3$ to all parts.
$\Rightarrow 2x-3+3\lt-1+3$ or $2x-3+3\gt1+3$
Simplify.
$\Rightarrow 2x\lt2$ or $2x\gt4$
Divide all parts by $2$.
$\Rightarrow \frac{2x}{2}\lt\frac{2}{2}$ or $\frac{2x}{2}\gt\frac{4}{2}$
Simplify.
$\Rightarrow x\lt1$ or $x\gt2$
The solution set is less than $1$ or greater than $2$.
The interval notation is
$(-\infty,1)\cup(2,\infty)$.
