Question

How do you solve `\frac { - x + 8} { x - 2} \geq 5`?

Answer 1

The solution is `x in ]2,3]`

Explanation:

We cannot do crossing over.

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

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

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

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

`((18-6x))/(x-2)>=0`

`(6(3-x))/(x-2)>=0`

Let `f(x)=(6(3-x))/(x-2)`

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

We can build the sign chart

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

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

`color(white)(aaaa)` `3-x` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `||` `color(white)(aaa)` `+` `color(white)(aaaa)` `-`

Therefore,

`f(x)>=0` when `x in ]2,3]`