Question

How do you solve `2w^3-8w<0` using a sign chart?

Answer 1

The solution is `w in (-oo,-2) uu (0,2)`

Explanation:

Let's factorise the inequality

`2w^3-8w<0`

`2w(w^2-4)<0`

`2w(w+2)(w-2)<0`

Let `f(w)=2w(w+2)(w-2)`

Let's build the sign chart

`color(white)(aaaa)` `w` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `-2` `color(white)(aaaa)` `0` `color(white)(aaaaaa)` `2` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `w+2` `color(white)(aaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `w` `color(white)(aaaaaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `w-2` `color(white)(aaaaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

Therefore,

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

graph{2x^3-8x [-14.24, 14.24, -7.12, 7.12]}