Question

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

Answer 1

The solutions are `S={2, 4}`

Explanation:

Let's rewrite the equation

`|3x-8|-x=0`

`|3x-8|=3x-8`, if `3x-8>=0`

and

`|3x-8|=-(3x-8)=-3x+8`, if `3x-8<0`

Let `f(x)=|3x-8|-x=0`

Let's build a variation table

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-oo` `color(white)(aaaaaaaa)` `0` `color(white)(aaaaaaaaaa)` `8/3` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaaaaaaa)` `-` `color(white)(aaaa)` ###color(white)(aaaaa)#`+` `color(white)(aaaaaaaa)` `+`

`color(white)(aaaa)` `|3x-8|` `color(white)(aaaaaa)` `-3x+8` `color(white)(a)` ###color(white)(aaa)#`-3x+8` `color(white)(aaaaa)` `3x-8`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaaaaa)` `x=2` `color(white)(a)` ###color(white)(aaaaa)#`x=2` `color(white)(aaaaaa)` `x=4`

The solutions are `S={2, 4}`

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