Question

How do you solve and write the following in interval notation: `(x + 4)(x − 5)(x + 6) ≥ 0`?

Answer 1

The solution is `x in [-6,-4] uu [5, +oo)`

Explanation:

Let `f(x)=(x+4)(x-5)(x+6)`

We build a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-6` `color(white)(aaaaa)` `-4` `color(white)(aaaaaaa)` `5` `color(white)(aaaaa)` `+oo`

`color(white)(aaaa)` `x+6` `color(white)(aaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x+4` `color(white)(aaaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `x-5` `color(white)(aaaaa)` `-` `color(white)(aaaaaa)` `-` `color(white)(aaaaa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aaa)` `+` `color(white)(aa)` `0` `color(white)(aa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+`

Therefore,

`f(x)>=0` when `x in [-6,-4] uu [5, +oo)`

graph{(x+4)(x-5)(x+6) [-44.86, 47.6, -31.26, 15]}