# How do you solve 1/2abs(4x – 8) - 7 = 3?

Jul 5, 2015

Rearrange and split into cases to find: $x = 7$ or $x = - 3$

#### Explanation:

First add $7$ to both sides to get:

$\frac{1}{2} \left\mid 4 x - 8 \right\mid = 10$

Divide both sides by $2$ to get:

$5 = \frac{1}{4} \left\mid 4 x - 8 \right\mid = \left\mid x - 2 \right\mid$

There are now two cases:

Case 1 : (x - 2) >= 0

If $\left(x - 2\right) \ge 0$ then $\left\mid x - 2 \right\mid = \left(x - 2\right)$ and we have:

$5 = x - 2$, hence $x = 7$

This is a valid solution since $x - 2 = 7 - 2 > 0$ thus satisfying the condition of the case.

Case 2 : (x - 2) < 0

If $\left(x - 2\right) < 0$ then $\left\mid x - 2 \right\mid = - \left(x - 2\right) = 2 - x$ and we have:

$5 = 2 - x$, hence $x = - 3$

This is a valid solution since $x - 2 = - 3 - 2 < 0$ thus satisfying the condition of the case.