Question

How do you solve `9-x^2<0` using a sign chart?

Answer 1

The answer is `x in ] -oo,-3 [ uu ] 3, oo[`

Explanation:

Let `f(x)=9-x^2=(3-x)(3+x)`

Now we can do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-3` `color(white)(aaaa)` `3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `3+x` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `3-x` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-`

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

Therefore,

`f(x)<0`, when `x in ] -oo,-3 [ uu ] 3, oo[`