Answer
$(-\infty,-10]\cup[2,\infty)$.
Work Step by Step
The given expression is
$\Rightarrow 7-\left | \frac{x}{2}+2\right |\leq4$
Subtract $7$ from both sides.
$\Rightarrow 7-\left | \frac{x}{2}+2\right |-7\leq4-7$
Simplify.
$\Rightarrow -\left | \frac{x}{2}+2\right |\leq-3$
Multiply all parts by −1 and change the sense of the inequality.
$\Rightarrow (-1)(-\left | \frac{x}{2}+2\right |)\geq(-1)(-3)$
Simplify.
$\Rightarrow \left | \frac{x}{2}+2\right |\geq3$
Rewrite the inequality without absolute value bars.
$\Rightarrow \frac{x}{2}+2\leq-3$ or $\frac{x}{2}+2\geq3$
Solve each inequality separately.
Subtract $2$ from all parts.
$\Rightarrow \frac{x}{2}+2-2\leq-3-2$ or $\frac{x}{2}+2-2\geq3-2$
Simplify.
$\Rightarrow \frac{x}{2}\leq-5$ or $\frac{x}{2}\geq1$
Multiply all parts by $2$.
$\Rightarrow (2)\cdot \frac{x}{2}\leq(2)\cdot(-5)$ or $(2)\cdot\frac{x}{2}\geq(2)\cdot1$
Simplify.
$\Rightarrow x\leq-10$ or $x\geq2$
The solution set is less than or equal to $-10$ or greater than or equal to $2$.
The interval notation is
$(-\infty,-10]\cup[2,\infty)$.
