Domain Restrictions of Rational Functions: And vs. Or

I've seen some of this recently:

I see the convenience of setting the denominator "not equal" to zero.  However, the math represented here is a little flawed.

What we know is the zero product property:

We also know that the inverse of this property holds:

In other words, when the product is not equal to zero, we need both factors to be different from zero.

So, when we teach domains of rational functions like the example above, we're getting a more restrictive domain than necessary.

That's my nitpicky rant for the day.