How do you solve the inequality abs(4-x)<=8?

Feb 7, 2018

$- 4 \le x \le 12$

Explanation:

$\text{inequalities of the type "|x|<=a" always have solutions}$
$\text{of the form}$

$- a \le x \le a$

$\Rightarrow | 4 - x | \le 8 \text{ will have solutions}$

$- 8 \le 4 - x \le 8$

$\text{to isolate the x term in the middle subtract 4 from }$
$\text{each interval}$

$- 8 \textcolor{red}{- 4} \le 4 \textcolor{red}{- 4} - x \le 8 \textcolor{red}{- 4}$

$\Rightarrow - 12 \le - x \le 4$

$\text{multiply each interval by } - 1$

$\text{remembering to "color(red)"reverse the inequality signs}$

$\Rightarrow 12 \ge x \ge - 4$

$\Rightarrow - 4 \le x \le 12$