Question

How do you solve `3x-x^2>0` using a sign chart?

Answer 1

The answer is `x in ] 0,3 [`

Explanation:

Let `f(x)=x(3-x)`

Now, we can establish the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaa)` `0` `color(white)(aaaa)` `3` `color(white)(aaaa)` `+oo`

`color(white)(aaaa)` `x` `color(white)(aaaaaaaa)` `-` `color(white)(aaa)` `+` `color(white)(aaaa)` `+`

`color(white)(aaaa)` `3-x` `color(white)(aaaaa)` `+` `color(white)(aaa)` `+` `color(white)(aaaa)` `-`

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

Therefore,

`f(x)>0`, when, `x in ] 0,3 [`