Question

How do you solve `\frac { x + 4} { x - 3} \leq 0`?

Answer 1

The solution is `x in ]-oo,-4] uu ]3,+oo[`

Explanation:

We solve this inequality with a sign chart

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

The domain of `f(x)` is `D_f(x)=RR-{3}`

The sign chart is

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

`color(white)(aaaa)` `x+4` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-3` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)>=0`, when `x in ]-oo,-4] uu ]3,+oo[`