# How do you solve and check for extraneous solutions in abs(x + 6) = 2x?

Aug 2, 2015

$x = 6$

#### Explanation:

Your absolute value equation looks like this

$| x + 6 | = 2 x$

Right from the start, you can say that any negative value of $x$ will be an extraneous solution because the absolute value of a number can only be positive.

So, you need to check two cases for your equation

• If $\left(x + 6\right) > 0$, you have

$| x + 6 | = x + 6$

The equation becomes

$x + 6 = 2 x \implies x = \textcolor{g r e e n}{6}$

• If $\left(x + 6\right) < 0$, you have

$| x + 6 | = - \left(x + 6\right) = - x - 6$

The equation will be

$- x - 6 = 2 x \implies 3 x = - 6 \implies x = \frac{- 6}{3} = \textcolor{red}{- 2}$

This solution will be extraneous because it implies that the absolute values of $4$ is negative, which is false.

$| \textcolor{red}{- 2} + 6 | = 2 \cdot \textcolor{red}{\left(- 2\right)}$

$| 4 | = - 4 \iff 4 \ne - 4$

The first solution is valid, since you have

$| \textcolor{g r e e n}{6} + 6 | = 2 \cdot \textcolor{g r e e n}{6}$

$| 12 | = 12 \iff 12 = 12$