Question

How do you solve `x/(x-2)>9`?

Answer 1

The solution is `x in ]2,9/4[`

Explanation:

We cannot do crossing over.

We rearrange the equation

`x/(x-2)>9`

`x/(x-2)-9>0`

`(x-9(x-2))/(x-2)>0`

`(x-9x+18)/(x-2)>0`

`(18-8x)/(x-2)>0`

`(2(9-4x))/(x-2)>0`

Let `f(x)=(2(9-4x))/(x-2)`

The domain of `f(x)` is `D_f(x)=RR-{2}`

We can build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaaa)` `2` `color(white)(aaaaaaaa)` `9/4` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaa)` `9-4x` `color(white)(aaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`

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

Therefore,

`f(x)>0` when `x in ]2,9/4[`