Question

How do solve `(5x-8)/(x-5)>=2` and write the answer as a inequality and interval notation?

Answer 1

The solution is `x in (-oo,-2/3] uu(5, +oo)`

Explanation:

The inequality is

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

We cannot do crossing over, so

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

Putting on the same denominator

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

`(5x-8-2x+10)/(x-5)>=0`

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

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

Build the sign chart

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

`color(white)(aaaa)` `3x+2` `color(white)(aaaaa)` `-` `color(white)(aaa)` `0` `color(white)(aaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x-5` `color(white)(aaaaaa)` `-` `color(white)(aaa)` ###color(white)(aaaa)#`-` `color(white)(aa)` `||` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaaa)` `+` `color(white)(aaa)` `0` `color(white)(aaa)` `-` `color(white)(aa)` `||` `color(white)(aa)` `+`

Therefore,

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

and `x<=-2/3` and `x>5` as an inequality

graph{(5x-8)/(x-5)-2 [-29.8, 35.14, -12.44, 20.05]}