# Solve |x-1|-|2+x|=3 ?

Jun 26, 2016

$x = - 5$

#### Explanation:

$| x - 1 | - | 2 + x | = 3$

Making $y = \left\mid 2 + x \right\mid + 3$ we have

$\left\mid x - 1 \right\mid = y \to \left\{\begin{matrix}x - 1 = y \\ x - 1 = - y\end{matrix}\right.$

1) From $x - 1 = y \to x = y + 1$ so
$y = \left\mid 2 + y + 1 \right\mid + 3 = \left\mid y + 3 \right\mid + 3$ but

$y + 3 = \left\mid y + 3 \right\mid + 6$ and
$\pm 1 = 1 + \frac{6}{y + 3}$ for $y \ne - 3$ so the only possibility is
$- 2 = \frac{6}{y + 3} \to y = - 6 \to x = - 5$

2)From $x - 1 = - y \to x = - \left(y + 1\right)$ so
$y = \left\mid y - 1 \right\mid + 3$ but
$y - 1 = \left\mid y - 1 \right\mid + 2$ then
$\pm 1 = 1 + \frac{2}{y - 1}$ for $y \ne 1$ so the only possibility is
$- 2 = \frac{2}{y - 1} \to y = 0 \to x = 1$

Substituting in the main equation we certify the solution: $x = - 5$