Question

How do you solve `x^2<=8x` using a sign chart?

Answer 1

The answer is `x in [0,8]`

Explanation:

Let's rearrange the equation

`x^2<=8x`

`x^2-8x<=0`

`x(x-8)<=0`

Let `f(x)=x(x-8)`

Let's do the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `0` `color(white)(aaaa)` `8` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaa)` `-` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `x-8` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `f(x)` `color(white)(aaaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

So, `f(x)<=0`, when `x in [0,8]`

graph{x(x-8) [-41.1, 41.1, -20.56, 20.56]}