# How do you solve abs(3x+5)-2x=3x+4?

Aug 2, 2015

$x = \frac{1}{2}$

#### Explanation:

Start by isolating the modulus on one side of the equation

$| 3 x + 5 | - \textcolor{red}{\cancel{\textcolor{b l a c k}{2 x}}} + \textcolor{red}{\cancel{\textcolor{b l a c k}{2 x}}} = 3 x + 4 + 2 x$

$| 3 x + 5 | = 5 x + 4$

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

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

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

$5 x + 4 > 0 \iff x > - \frac{4}{5}$

Now, this equation can produce two solutions, depending on which condition is true

• If $\left(3 x + 5\right) > 0$, you have

$| 3 x + 5 | = 3 x + 5$

and the equation becomes

$3 x + 5 = 5 x + 4 \implies x = \textcolor{g r e e n}{\frac{1}{2}}$

• If $\left(3 x + 5\right) < 0$, you have

$| 3 x + 5 | = - \left(3 x + 5\right) = - 3 x - 5$

This will get you

$- 3 x - 5 = 5 x + 4 \implies x = \textcolor{red}{- \frac{9}{8}}$

Since $x = - \frac{9}{8}$ does not satisfy the condition $x > - \frac{4}{5}$, this solution will be extraneous.

As a result, th only solution to this equation is $x = \frac{1}{2}$.