Simple Single Variable Equation

Question

How can we prove that x = 2 or 3 step-by-step:

$$(2-x)(3-x) = 0$$

I know the answer, but how do we get it without plugging in 2 and 3 directly?

Thank you!

Answer 1

A product in a ring with no divisors of zero can only be zero if one of the factors is zero and thus $$(2-x)(3-x)=0$$ implies $2-x=0$ or $3-x=0$.

Answer 2

$$ab=0\implies a=0\lor b=0$$ because $$a\ne0\land b\ne0\implies ab\ne0.$$

Now

$$2-x=0\lor 3-x=0\implies 2=x\lor 3=x,$$ by adding $x$ to the equalities.