Question

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

Answer 1

`x in [1, 2] uu [3, oo)`

Explanation:

`f(x) = (x-1)(x-2)(x-3) = x^3-6x^2+11x-6`

is a cubic with positive leading coefficient and zeros of multiplicity `1` at `x=1`, `x=2` and `x=3`.

As a result:

• `f(x)` is continuous.

• `f(x)` is positive for large positive values of `x`.

• `f(x)` is negative for large negative values of `x`.

• `f(x)` changes sign at each of its zeros.

Hence `f(x)` is non-negative in the intervals `[1, 2]` and `[3, oo)`

graph{x^3-6x^2+11x-6 [-3.397, 6.603, -2.24, 2.76]}