Question

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

Answer 1

The solutions are `S={-1, 7/5}`

Explanation:

Let's build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaaa)` `-oo` `color(white)(aaaaaaaaa)` `1/2` `color(white)(aaaaaaaaaa)` `5` `color(white)(aaaaaaaaa)` `+oo`

`color(white)(aaaa)` `4x-2` `color(white)(aaaaaaa)` `-` `color(white)(aaaaaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaaaaa)` `+`

`color(white)(aaaa)` `-x+5` `color(white)(aaaaaa)` `+` `color(white)(aa)` ###color(white)(aaaaaaaaa)#`+` `color(white)(aaaaa)` `0` `color(white)(aaa)` `-`

`color(white)(aaaa)` `|4x-2|` `color(white)(aaaaaa)` `-4x+2` `color(white)(aa)` ###color(white)(aaaa)#`4x-2` `color(white)(aaaaa)` `4x-2`

`color(white)(aaaa)` `|-x+5|` `color(white)(aaaaaaa)` `x-5` `color(white)(aa)` ###color(white)(aaaa)#`x-5` `color(white)(aaaaa)` `-x+5`

Therefore,

In the interval `x in (-oo, 1/2)`

`-4x+2=x-5`

`5x=7`

`x=7/5`

In the interval `x in (1/2, 5)`

`4x-2=x-5`

`3x=-3`

`x=-1`

In the interval `x in (5, +oo)`

`4x-2=-x+5`

`5x=7`

`x=7/5`

Therefore,

the solutions are `S={-1, 7/5}`

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