# How do you solve abs(3x-8)=x?

Mar 24, 2018

The solutions are $S = \left\{2 , 4\right\}$

#### Explanation:

Let's rewrite the equation

$| 3 x - 8 | - x = 0$

$| 3 x - 8 | = 3 x - 8$, if $3 x - 8 \ge 0$

and

$| 3 x - 8 | = - \left(3 x - 8\right) = - 3 x + 8$, if $3 x - 8 < 0$

Let $f \left(x\right) = | 3 x - 8 | - x = 0$

Let's build a variation table

$\textcolor{w h i t e}{a a a a}$$x$$\textcolor{w h i t e}{a a a a a a a a}$$- \infty$$\textcolor{w h i t e}{a a a a a a a a}$$0$$\textcolor{w h i t e}{a a a a a a a a a a}$$\frac{8}{3}$$\textcolor{w h i t e}{a a a a a a a}$$+ \infty$

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

$\textcolor{w h i t e}{a a a a}$$| 3 x - 8 |$$\textcolor{w h i t e}{a a a a a a}$$- 3 x + 8$$\textcolor{w h i t e}{a}$color(white)(aaa)-3x+8$\textcolor{w h i t e}{a a a a a}$$3 x - 8$

$\textcolor{w h i t e}{a a a a}$$f \left(x\right)$$\textcolor{w h i t e}{a a a a a a a a a a a}$$x = 2$$\textcolor{w h i t e}{a}$color(white)(aaaaa)x=2$\textcolor{w h i t e}{a a a a a a}$$x = 4$

The solutions are $S = \left\{2 , 4\right\}$

graph{|3x-8|-x [-7.9, 7.904, -3.95, 3.95]}