Question

How do you solve `(x+2)/x<=0` using a sign chart?

Answer 1

The answer is `x in [-2,0[`

Explanation:

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

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

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaaaaa)` `0` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+2` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aa)` `+`

Therefore

`f(x)<=0` when `x in [-2,0[`