Question

How do you solve `(-x+2)/(x-4)>=0`?

Answer 1

The answer is `x in [2, 4 [`

Explanation:

Let `f(x)=(-x+2)/(x-4)`

The domain of `f(x)` is `D_f(x)=RR-{4}`

We have to make a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `2` `color(white)(aaaaaa)` `4` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `2-x` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aa)` `∥` `color(white)(aa)` `-`

`color(white)(aaaa)` `x-4` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aa)` `∥` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aa)` `∥` `color(white)(aa)` `-`

Therefore,

`f(x)>=0` when `x in [2, 4 [`