# How do you solve |x+5| = 2x+3 and find any extraneous solutions?

Jun 22, 2016

$x = 2$

#### Explanation:

$| x + 5 | = 2 x + 3 = x + 5 + x - 2$

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

but for $x \ne - 5 \to \frac{x + 5}{\left\mid x + 5 \right\mid} = \pm 1$

$1 \pm 1 = \frac{x - 2}{\left\mid x + 5 \right\mid}$ we have two possibilities

1) $0 = \frac{x - 2}{\left\mid x + 5 \right\mid} \to x = 2$
2) $2 = \frac{x - 2}{\left\mid x + 5 \right\mid} \to x = \emptyset$
because for $x > 2 \to \frac{x - 2}{\left\mid x + 5 \right\mid} < 1$