Answer
$(-\infty,-10]\cup[2,\infty)$
Work Step by Step
"$ y $ is at most $4$" $\qquad $ is written as $\quad y\leq 4.$
Substituing the expression for y,
$ 7-|\displaystyle \frac{x}{2}+2|\leq 4\qquad $ ... add $(-7)$
$-|\displaystyle \frac{x}{2}+2|\leq-3\qquad $ ... multiply with $(-2)$
$... \times (negative)$ changes direction
$|x+4|\geq 6$
... $|u| \geq c $ is equivalent to ($ u\leq -c $) or ($ u\geq c $)
$ \begin{array}{lllll}
x+4\leq-6 & /-4 & ...or... & x+4\geq 6 & /-4\\
x\leq-10 & & & x \geq 2 & \\
x\in(-\infty,-10] & & or & x\in[2,\infty) & \\
& & & &
\end{array} $
Solution set: $(-\infty,-10]\cup[2,\infty)$
