Question

How do you solve and write the following in interval notation: `x^2 + 7x + 10 >0`?

Answer 1

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

Explanation:

Let's factorise the inequality

`x^2+7x+10=(x+2)(x+5)`

So,

`(x+2)(x+5)>0`

Let `f(x)=(x+2)(x+5)`

We build a sign chart

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

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

`color(white)(aaaa)` `x+2` `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,-5) uu (-2,+oo)`