Question

How do you solve `(x-4)/(x+3)<=0` using a sign chart?

Answer 1

The answer is `x in ]-3,4]`

Explanation:

As you canot divide by `0`, therefore `x!=-3`

Let `f(x)=(x-4)/(x+3)`

let's do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaaa)` `4` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+3` `color(white)(aaa)` `-` `color(white)(aaaa)` `∣∣` `color(white)(aa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaa)` `-` `color(white)(aaaa)` `∣∣` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaa)` `+` `color(white)(aaaa)` `∣∣` `color(white)(aa)` `-` `color(white)(aaaa)` `+`

Therefore, `f(x)<=0` `=>` `x in ]-3,4]`
graph{(x-4)/(x+3) [-32.46, 32.46, -16.24, 16.25]}