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

Apr 7, 2015

We have two situations
(1) $x - 7 < 0$
and
(2) $x - 7 \ge 0$

Case (1)
If $x - 7 < 0$ which implies $x < 7$
then
$\left\mid x - 7 \right\mid < 10$
is equivalent to
$7 - x < 10$
$- x < 3$
$x > - 3$
so the inequality holds if $- 3 < x < 7$

Case (2)
If $x - 7 \ge 0$ which implies $x \ge 7$
$\left\mid x - 7 \right\mid < 10$
is equivalent to
$x - 7 < 10$
$x < 17$
so the inequality holds if $7 \le x < 17$

Solution
The solution to the given inequality includes all values in the 2 ranges
$- 3 < x < 7$
and
$7 \le x < 17$
which can be combined into a single range
$- 3 < x < 17$