Question

How do you solve `x/(x-1)>2`?

Answer 1

The solution is `x in (2,4)`

Explanation:

You cannot do crossing over.

The inequality is

`x/(x-2)>2`

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

Placing on the same denominator

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

`=>`, `(x-2x+4)/(x-2)>0`

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

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

Let's build a sign chart

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

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

`color(white)(aaaa)` `4-x` `color(white)(aaaaaa)` `+` `color(white)(aaaa)` ###color(white)(a)#`+` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `-` `color(white)(aaaa)` ###color(white)(a)#`+` `color(white)(aaaa)` `-`

Therefore,

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

graph{(4-x)/(x-2) [-20.27, 20.27, -10.14, 10.14]}