# How do you solve abs(x+10)=4x-8?

Aug 2, 2015

$x = 6$

#### Explanation:

Your absolute value equation looks like this

$| x + 10 | = 4 x - 8$

Right from the start, you know that the solutions to this equation must satisfy the condition

$4 x - 8 > 0 \iff x > 2$

That happens because the absolute value of a number, regardlesss if that number is positive or negative, is always positive.

$\textcolor{b l u e}{| a | = \left\{\begin{matrix}a \text{ & " & "if " a>=0 \\ -a" & " & "if } a < 0\end{matrix}\right.}$

So, with this in mind, determine the equation's two possible solutions

• If $\left(x + 10\right) > 0$, you have

$| x + 10 | = x + 10$

and the equation becomes

$x + 10 = 4 x - 8 \implies x = \frac{18}{3} = \textcolor{g r e e n}{6}$

• If $\left(x + 10\right) < 0$, you have

$| x + 10 | = - \left(x + 10\right) = - x - 10$

This will get you

$- x - 10 = 4 x + 8 \implies x = \textcolor{red}{- \frac{18}{5}}$

This solution, $x = - \frac{18}{5}$, will be an extraneous solution because it does not satisfy the contion $x > 2$.

Therefore, $x = 6$ is the only solution to this equation.