# How do you solve abs(x - 7)<10?

Apr 1, 2015

Absolute value equations or inequalities must be split in two.

For $x \ge 7 , \left(x - 7\right)$ is positive, so the inequality is

$x - 7 < 10 \to x < 17$ (in total this means $7 \le x < 17$)

For $x \le 7 , \left(x - 7\right)$ is negative, but the absolute turns it around:

$7 - x < 10 \to x > - 3$ (this means -3 < x <=7)

Our total "solution space" is thus: $- 3 < x < 17$
graph{|x-7| [-16.38, 24.15, -4.57, 15.7]}