Question

How do you solve `16p^2-81p^6<=0` using a sign chart?

Answer 1

The answer is `p in ]-oo, -2/3] uu [2/3, +oo [`

Explanation:

We need

`a^2-b^2=(a+b)(a-b)`

Let's factorise the inequality

`16p^2-81p^6=p^2(16-81p^4)`

`=p^2(4-9p^2)(4+9p^2)`

`=p^2(2+3p)(2-3p)(4+9p^2)`

Let `f(p)=p^2(2+3p)(2-3p)(4+9p^2)`

`p^2>=0 , AA p in RR`

`4+9p^2>0 , AA p in RR`

Now we can build the sign chart

`color(white)(aaaa)` `p` `color(white)(aaaaaa)` `-oo` `color(white)(aaaa)` `-2/3` `color(white)(aaaaaa)` `2/3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `2+3p` `color(white)(aaaaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`

`color(white)(aaaa)` `2-3p` `color(white)(aaaaaaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `-`

`color(white)(aaaa)` `f(p)` `color(white)(aaaaaaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `-`

Therefore,

`f(p)<=0` when `p in ]-oo, -2/3] uu [2/3, +oo [`