Question

How do you solve and write the following in interval notation: `(x-2)/(x+1) <=0`?

Answer 1

The solution is `x in (-1,2]`

Explanation:

Let `f(x)=(x-2)/(x+1)`

We build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `-1` `color(white)(aaaaaa)` `2` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x+1` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `+` `color(white)(aa)` `0` `color(white)(aaa)` `+`

`color(white)(aaaa)` `x-2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+`

Therefore,

`f(x)<=0` when `x in (-1,2]`

graph{(x-2)/(x+1) [-14.24, 14.24, -7.12, 7.12]}