Question

How do you solve `x^3<=4x^2` using a sign chart?

Answer 1

The solution is `x in (-oo, 4]`

Explanation:

The inequality is

`x^3<=4x^2`

Rearranging

`x^3-4x^2<=0`

`x^2(x-4)<=0`

Let `f(x)=x^2(x-4)`

Let's build the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaaa)` `-oo` `color(white)(aaaaa)` `0` `color(white)(aaaaaaaaa)` `4` `color(white)(aaaaaa)` `+oo`

`color(white)(aaaa)` `x^2` `color(white)(aaaaaaa)` `+` `color(white)(aaa)` `0` `color(white)(aaaa)` `+` `color(white)(aaaaaa)` `+`

`color(white)(aaaa)` `x-4` `color(white)(aaaaa)` `-` `color(white)(aaa)` ###color(white)(aaaa)#`-` `color(white)(aaa)` `0` `color(white)(aaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaaa)` `-` `color(white)(aaa)` `0` `color(white)(aaa)` `-` `color(white)(aaa)` `0` `color(white)(aaa)` `+`

Therefore,

`f(x)<=0` when `x in (-oo, 4]`

graph{x^3-4x^2 [-14.29, 14.18, -10.41, 3.83]}