Question

How do solve `(x+4)/(1-x)<=0` and write the answer as a inequality and interval notation?

Answer 1

The answer is `-oo < x <= -4` or `1 < x <+oo`
`x in ]-oo,-4]uu]1,+oo[` in interval notation

Explanation:

We solve this equation with a sign chart

Let `f(x)=(x+4)/(1-x)`

The domain of `f(x)` is `D_f(x)=RR-{1}`

Now, we build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaa)` `-4` `color(white)(aaaaaa)` `1` `color(white)(aaaaaaa)` `+oo`

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

`color(white)(aaaa)` `1-x` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaa)` `||` `color(white)(aaaa)` `-`

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

Therefore,

`f(x)<=0` when `x in ]-oo,-4]uu]1,+oo[` in interval notation

As an inequality `-oo < x <= -4` or `1 < x <+oo`