Question

How do you solve `(-x-3)/(x+2)<=0`?

Answer 1

The answer is `x in] -oo,-3 ] uu] -2, oo[`

Explanation:

Let `f(x)=(-x-3)/(x+2)`

As we cannot divide by `0` , the domain of `f(x)` is `D_f(x)=RR-{-2}`

To solve the inequalty, we do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaaa)` `x+2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

So,

`f(x<=0)` when

`x in] -oo,-3 ] uu] -2, oo[`