Question

How do you solve `(5x-8)/(x-5)>=2` using a sign chart?

Answer 1

The answer is `x in ]-oo, -2/3 ] uu ] 5, +oo[`

Explanation:

Let's do some simplifications

`(5x-8)/(x-5)>=2`

`(5x-8)/(x-5)-2>=0`

`((5x-8)-2(x-5))/(x-5)>=0`

`(3x+2)/(x-5)>=0`

Let `f(x)=(3x+2)/(x-5)`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaa)` `-2/3` `color(white)(aaaa)` `5` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `3x+2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-5` `color(white)(aaaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

Therefore,

`f(x)>=0` when `x in ]-oo, -2/3 ] uu ] 5, +oo[`