# How do you solve absolute value inequality abs(2x-4)-1>0?

##### 1 Answer
Apr 7, 2015

Split the absolute value sub-expression into its two cases
$\left(2 x - 4\right) < 0 \rightarrow x < 2$
and
$\left(2 x - 4\right) \ge 0 \rightarrow x \ge 2$

If $\left(2 x - 4\right) < 0$
then
$\left\mid 2 x - 4 \right\mid - 1 > 0$
is equivalent to
$- 2 x + 4 - 1 > 0$
$- 2 x > - 3$
$x < \frac{2}{3}$ remember multiplying by a negative reverses the inequality.
(Note that this is consistent with the requirement $x < 2$)

If $\left(2 x - 4\right) \ge 0$
then
$\left\mid 2 x - 4 \right\mid - 1 > 0$
is equivalent to
$2 x - 4 - 1 \ge 0$
$2 x > 5$
$x > \frac{5}{2}$
(Again, note that this is consistent with the requirement $x \ge 2$)

So the solution to the given inequality is all values of $x$ such that
$x < \frac{2}{3}$ or $x > 2 \frac{1}{2}$