Question

How do you solve `x/(x-2)>2` using a sign chart?

Answer 1

The answer is `x in ] 2,4 [`

Explanation:

We cannot do crossing over

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

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

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

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

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

Now we can do our sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `2` `color(white)(aaaa)` `4` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x-2` `color(white)(aaaa)` `-` `color(white)(a)` `∥` `color(white)(a)` `+` `color(white)(aa)` `+`

`color(white)(aaaa)` `4-x` `color(white)(aaaa)` `+` `color(white)(a)` `∥` `color(white)(a)` `+` `color(white)(aa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `-` `color(white)(a)` `∥` `color(white)(a)` `+` `color(white)(aa)` `-`

Therefore,

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