Question

How do you solve `(w-6)/w<=0` using a sign chart?

Answer 1

The answer is `x in ]0,6]`

Explanation:

Let `f(w)=(w-6)/w`

The domain of `f(w)` is `D_f(w)=RR-{0}`

Let's build the sign chart

`color(white)(aaaa)` `w` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `0` `color(white)(aaaaa)` `6` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `w` `color(white)(aaaaaaaa)` `-` `color(white)(aa)` `||` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `w-6` `color(white)(aaaaa)` `-` `color(white)(aa)` `||` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(w)` `color(white)(aaaaaa)` `+` `color(white)(aa)` `||` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(w)<=0` when `x in ]0,6]`