Question

How do you solve `(x + 13) /( x - 2)>0`?

Answer 1

The solution is `x in (-oo,-13) uu (2,+oo)`

Explanation:

We solve this inequality with a sign chart.

Let `f(x)=(x+13)/(x-2)`

We build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-13` `color(white)(aaaaaaa)` `2` `color(white)(aaaaaaa)` `+oo`

`color(white)(aaaa)` `x+13` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `||` `color(white)(aaaa)` `+`

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

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

Therefore,

`f(x)>0` when `x in (-oo,-13) uu (2,+oo)`