Question

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

Answer 1

The solution is `x in (-oo,-6] uu(-2,+oo)`

Explanation:

We cannot do crossing over, let's rewrite the inequality

`(x-2)/(x+2)<=2`

`(x-2)/(x+2)-2<=0`

`((x-2)-2(x+2))/(x+2)<=0`

`(x-2-2x-4)/(x+2)<=0`

`(-x-6)/(x+2)<=0`

`-(x+6)/(x+2)<=0`

Let `f(x)=-(x+6)/(x+2)`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaaaa)` `-oo` `color(white)(aaaa)` `-6` `color(white)(aaaaaaaa)` `-2` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `-(x+6)` `color(white)(aaaa)` `+` `color(white)(aaaa)` `0` `color(white)(aaaa)` `-` `color(white)(aaaaa)` `-`

`color(white)(aaaa)` `(x+2)` `color(white)(aaaaaa)` `-` `color(white)(aaaaaaaaa)` `-` `color(white)(aa)` `||` `color(white)(aa)` `+`

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

Therefore,

`f(x)<=0` when `x in (-oo,-6] uu(-2,+oo)`

graph{(x-2)/(x+2)-2 [-22.8, 22.81, -11.4, 11.42]}