Question

How do you solve `32p^2-2p^6<0` using a sign chart?

Answer 1

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

Explanation:

The inequality is

`32p^2-2p^6<0`

Factorising

`2p^2(16-p^4)<0`

`2p^2(4-p^2)(4+p^2)<0`

`2p^2(2+p)(2-p)(4+p^2)<0`

Let's build the sign chart

`(4+p^2)>0`

Let `f(p)=2p^2(2+p)(2-p)(4+p^2)`

`color(white)(aaaa)` `p` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaaaaa)` `0` `color(white)(aaaaaaaa)` `2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `p^2` `color(white)(aaaaaaaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `2+p` `color(white)(aaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+` `color(white)(aaaaaaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `2-p` `color(white)(aaaaaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aa)` ###color(white)(aaaaaa)#`+` `color(white)(a)` `0` `color(white)(aa)` `-`

`color(white)(aaaa)` `f(p)` `color(white)(aaaaaaa)` `-` `color(white)(aa)` `0` `color(white)(aa)` `+` `color(white)(aaa)` `0` `color(white)(aaaa)` `+` `color(white)(a)` `0` `color(white)(aa)` `-`

Therefore,

`f(p)<0` when `p in (-oo,-2) uu(2,+oo)`

graph{32x^2-2x^6 [-7.554, 6.49, -3.425, 3.595]}