Question

How do you solve `n^2+2n-24<=0`?

Answer 1

The answer is `n in [-6,4]`

Explanation:

Let's factorise the inequality

`n^2+2n-24=(n+6)(n-4)`

Let `f(n)=(n+6)(n-4)`

Now, we can build the sign chart

`color(white)(aaaa)` `n` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-6` `color(white)(aaaaa)` `4` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `n+6` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

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

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

Therefore,

`f(n)<=0` when `n in [-6,4]`