# How do you solve  |x + 3| = abs(x-2)?

Apr 28, 2016

$x = - \frac{1}{2}$

#### Explanation:

$| x + 3 | = | x - 2 |$

$\implies x + 3 = | x - 2 |$
or
$- \left(x + 3\right) = | x - 2 |$

$\implies x + 3 = x - 2$
or
$x + 3 = - \left(x - 2\right)$
or
$- \left(x + 3\right) = x - 2$
or
$- \left(x + 3\right) = - \left(x - 2\right)$

The first and fourth equations have no solution as you can cancel $x$ (or $- x$) from both sides to obtain $2 = - 3$, which is a contradiction.

The second and third equations have the same solution, as one can be obtained from the other by multiplying both sides by $- 1$. Then, we may simply solve

$x + 3 = - \left(x - 2\right)$

$\implies x + 3 = - x + 2$

$\implies 2 x = - 1$

$\therefore x = - \frac{1}{2}$