Question

How do you solve `x(4-x)(x-6)<=0`?

Answer 1

The answer is `x in [0,4] uu [6,+ oo[`

Explanation:

Let `f(x)=x(4-x)(x-6)`

Let's do a sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `0` `color(white)(aaaaa)` `4` `color(white)(aaaaaa)` `6` `color(white)(aaaaaa)` `+oo`

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

`color(white)(aaaa)` `4-x` `color(white)(aaaa)` `+` `color(white)(aaaa)` `+` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-`

`color(white)(aaaa)` `x-6` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `-` `color(white)(aaaa)` `+`

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

For `f(x)<=0`

The solution is `x in [0,4] uu [6,+ oo[`
graph{x(4-x)(x-6) [-14.56, 13.9, -8.47, 5.77]}