Answer
$\varnothing$.
The graph is shown below.
Work Step by Step
The given compound inequality is
$5(x-2)\gt15$ and $\frac{x-6}{4}\leq -2$.
Solve each inequality separately.
$\Rightarrow 5(x-2)\gt15$ and $\frac{x-6}{4}\leq -2$.
$\Rightarrow 5x-10\gt15$ and $x-6\leq -8$.
$\Rightarrow 5x\gt15+10$ and $x\leq -8+6$.
$\Rightarrow 5x\gt25$ and $x\leq -2$.
$\Rightarrow x\gt\frac{25}{5}$ and $x\leq -2$.
$\Rightarrow x\gt5$ and $x\leq -2$.
First graph then take the intersection of the solution sets of the two inequalities..
The graph is shown in the image file.
We can write the compound inequality.
$x\gt 5$ as $(5,\infty)$ and $x\leq-2$ as $(-\infty,-2]$
The intersection is
$(-\infty,-2]\cap(5,\infty)=\varnothing$.
