Question

How do you solve `3w^4-27w^2>0` using a sign chart?

Answer 1

You can factorize into `3w^2(w^2-9)>0`

Explanation:

Since `3w^2>=0` you may divide by it without changing the `>`sign.
If `w=0` it won't fit the bill, so `w!=0`

Which leaves us with `w^2-9>0->w^2>9`

So either `w<-3orw>+3`
graph{3x^4-27x^2 [-16.04, 16, -8.03, 8]}

Answer 2

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

Explanation:

Let `f(w)=3w^4-27w^2`

`=3w^2(w+3)(w-3)`

`w^2 >0, AA win RR`

Let's do a sign chart

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

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

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

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

Therefore,

`f(w)>0`, when `w in ] -oo,-3 [ uu ] 3,+ oo[`