Question

How do you solve `(x-1)(x-2)(x-3)>=0` using a sign chart?

Answer 1

The answer is `x in [1 ,2 ] uu [3, +oo [`

Explanation:

Let `f(x)=(x-1)(x-2)(x-3)`

Now, we can establish the sign chart

`color(white)(aaaa)` `x` `color(white)(aaaa)` `-oo` `color(white)(aaaaa)` `1` `color(white)(aaaaaa)` `2` `color(white)(aaaaaaa)` `3` `color(white)(aaaaaa)` `-oo`

`color(white)(aaaa)` `x-1` `color(white)(aaaaa)` `-` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+` `color(white)(aaaaa)` `+`

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

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

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

Therefore,

`f(x)>=0` when `x in [1 ,2 ] uu [3, +oo [`