# What is the solution set for abs(3x-1) = x+5?

Aug 5, 2015

x = {-1; 3}

#### Explanation:

The first thing that you need to notice here is that the expression on the right side of the equation must be positive because it represents the absolute value of th expression $3 x - 1$.

So, any solution that does not satisfy the condition

$x + 5 \ge 0 \implies x \ge - 5$

will be an extraneous solution.

You need to take into account two possibilities for this equation

• $\left(3 x - 1\right) > 0$, which means that

$| 3 x - 1 | = 3 x - 1$

and the equation becomes

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

• $\left(3 x - 1\right) < 0$, which means that

$| 3 x - 1 | = - \left(3 x - 1\right) = - 3 x + 1$

and the equation becomes

$- 3 x + 1 = x + 5$

$- 4 x = 4 \implies x = \frac{4}{- 4} = \textcolor{g r e e n}{- 1}$

Since both values satisfy the condition $x \ge - 5$, they are both valid solutions to the equation.