Answer
$(−∞,-8]$ U $[-4,∞)$
Work Step by Step
$x/(x+4) \le 2$
$x/(x+4)-2 \le 2-2$
$\frac{x}{x+4}-2 \le 0$
$\frac{x}{x+4}-2*\frac{x+4}{x+4} \le 0$
$\frac{x}{x+4}-\frac{2(x+4)}{x+4} \le 0$
$\frac{x}{x+4}-\frac{2x+8)}{x+4} \le 0$
$\frac{x-2x-8}{x+4} \le 0$
$\frac{-x-8}{x+4} \le0$
The denominator is zero when $x=-4$, and the numerator is zero when $x=-8$.
We have three sections: $(−∞,-8)$, $(-8,-4)$, and $(-4,∞)$. We need to test one value for x in each section to determine if the section would be a solution set. Since we have the $\le$ sign, we include the end points and use brackets instead of parentheses.
Let $x=-10$, $x=-5$, and $x=0$
$x=-10$
$x/(x+4) \le 2$
$-10/(-10+4) \le 2$
$-10/-6 \le 2$
$5/3 \le2$ (true)
$x=-5$
$x/(x+4) \le 2$
$-5/(-5+4) \le 2$
$-5/-1 \le 2$
$5 \le 2$ (false)
$x=0$
$x/(x+4) \le 2$
$0/(0+4) \le 2$
$0/4 \le 2$
$0 \le 2$ (true)
