# How do you solve and check for extraneous solutions in abs(x-1) = 5x + 10?

Aug 2, 2015

Solution: $x = - \frac{3}{2}$
Extraneous solution: $x = - \frac{11}{4}$

#### Explanation:

If you take into account the fact that the absolute value of a number is always positive regardless if said number is positive or negative

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

then you can say that the solutions to this equation must satisfy the condition

$5 x + 10 > 0 \iff x > - 2$

Now, your absolute value equation will produce two solutions, depending on which condition is true

• If $\left(x - 1\right) > 0$, then

$| x - 1 | = x - 1$

This will get you

$x - 1 = 5 x + 10 \implies x = \textcolor{red}{- \frac{11}{4}}$

• If $\left(x - 1\right) < 0$, then

$| x - 1 | = - \left(x - 1\right) = - x + 1$

The solution to the equation will be

$- x + 1 = 5 x + 10 \implies x = - \frac{9}{6} = \textcolor{g r e e n}{- \frac{3}{2}}$

As you can see, $x = - \frac{11}{4}$ does not satisfy the condition $x > - 2$, which means that this solution will be extraneous.

The only solution to this equation will thus be $x = - \frac{3}{2}$.