Question

How do you solve and write the following in interval notation: `x^2 - 2x - 8 > 0`?

Answer 1

The solution is `x in (-oo,-2) uu(4, +oo)`

Explanation:

Factorise the inequality

`x^2-2x-8=(x+2)(x-4)>0`

Let `f(x)=(x+2)(x-4)`

Build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `4` `color(white)(aaaa)` `+oo`

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

`color(white)(aaaa)` `x-4` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

Therefore,

`f(x)>0` when `x in (-oo,-2) uu(4, +oo)`

graph{(x+2)(x-4) [-31.62, 26.12, -9.97, 18.9]}