Question

How do you solve `x/(x+3)>=2`?

Answer 1

The answer is `x in [-6, -3[`

Explanation:

We cannot do crossing over.

We rewrite the inequality as `x/(x+3)-2>=0`

`(x-2(x+3))/(x+3)>=0`

`(x-2x-6)/(x+)=(-x-6)/(x+3)>=0`

Let `f(x)=-(x+6)/(x+3)`

We can make a time chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaa)` `-6` `color(white)(aaaaa)` `-3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `-x-6` `color(white)(aaaa)` `+` `color(white)(aaaaaa)` `-` `color(white)(aa)` `∣∣` `color(white)(aa)` `-`

`color(white)(aaaa)` `x+3` `color(white)(aaaaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aa)` `∣∣` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `-` `color(white)(aaaaaa)` `+` `color(white)(aa)` `∣∣` `color(white)(aa)` `-`

Therefore,

`f(x)>=0`, when `x in [-6, -3[`