Question

How do you solve `(2x+1) / (x-9) >= 0`?

Answer 1

The solution is `x in ]-oo, -1/2]uu]9,+oo[`

Explanation:

We solve this inequality with a sign chart

Let `f(x)=(2x+1)/(x-9)`

We can now build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-1/2` `color(white)(aaaaaa)` `9` `color(white)(aaaaaaaa)` `+oo`

`color(white)(aaaa)` `2x+1` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-9` `color(white)(aaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `||` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)>=0` when `x in ]-oo, -1/2]uu]9,+oo[`