Question

How do you solve `(2x)/(x-2)<=3` using a sign chart?

Answer 1

The solution is `x in (-oo,2)uu [6,+oo)`

Explanation:

We cannot do crossing over

Let's rearrange the inequality

`(2x)/(x-2)<=3`

`(2x)/(x-2)-3<=0`

`(2x-3(x-2))/(x-2)<=0`

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

`(6-x)/(x-2)<=0`

Let `f(x)=(6-x)/(x-2)`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaaaa)` `2` `color(white)(aaaaaaa)` `6` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `x-2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(a)` `0` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `6-x` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(a)` `0` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(a)` `0` `color(white)(aaaa)` `-`

Therefore,

`f(x)<=0` when `x in (-oo,2)uu [6,+oo)`