# How do you solve |4x - 2| = | - x + 5|?

Apr 2, 2018

The solutions are $S = \left\{- 1 , \frac{7}{5}\right\}$

#### Explanation:

Let's build a sign chart

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a a a}$$\frac{1}{2}$$\textcolor{w h i t e}{a a a a a a a a a a}$$5$$\textcolor{w h i t e}{a a a a a a a a a}$$+ \infty$

$\textcolor{w h i t e}{a a a a}$$4 x - 2$$\textcolor{w h i t e}{a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a}$$0$$\textcolor{w h i t e}{a a a a}$$+$$\textcolor{w h i t e}{a a a a a a a a}$$+$

$\textcolor{w h i t e}{a a a a}$$- x + 5$$\textcolor{w h i t e}{a a a a a a}$$+$$\textcolor{w h i t e}{a a}$color(white)(aaaaaaaaa)+$\textcolor{w h i t e}{a a a a a}$$0$$\textcolor{w h i t e}{a a a}$$-$

$\textcolor{w h i t e}{a a a a}$$| 4 x - 2 |$$\textcolor{w h i t e}{a a a a a a}$$- 4 x + 2$$\textcolor{w h i t e}{a a}$color(white)(aaaa)4x-2$\textcolor{w h i t e}{a a a a a}$$4 x - 2$

$\textcolor{w h i t e}{a a a a}$$| - x + 5 |$$\textcolor{w h i t e}{a a a a a a a}$$x - 5$$\textcolor{w h i t e}{a a}$color(white)(aaaa)x-5$\textcolor{w h i t e}{a a a a a}$$- x + 5$

Therefore,

In the interval $x \in \left(- \infty , \frac{1}{2}\right)$

$- 4 x + 2 = x - 5$

$5 x = 7$

$x = \frac{7}{5}$

In the interval $x \in \left(\frac{1}{2} , 5\right)$

$4 x - 2 = x - 5$

$3 x = - 3$

$x = - 1$

In the interval $x \in \left(5 , + \infty\right)$

$4 x - 2 = - x + 5$

$5 x = 7$

$x = \frac{7}{5}$

Therefore,

the solutions are $S = \left\{- 1 , \frac{7}{5}\right\}$

graph{(y-|4x-2|)(y-|-x+5|)=0 [-7.11, 8.69, -0.55, 7.35]}