Question

How do solve `(3-x)/(x+5)<=0` and write the answer as a inequality and interval notation?

Answer 1

The solution is `x in (-oo,-5) uu [3, +oo)` or
`x<-5` and `x>=3`

Explanation:

Let `f(x)=(3-x)/(x+5)`

We build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-5` `color(white)(aaaaaaa)` `3` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `x+5` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

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

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

Therefore,

`f(x)<=0` when `x in (-oo,-5) uu [3, +oo)` in interval notation or

`x<-5` and `x>=3` as inequality