When do the equations $f(x)=g(x)$ and $f(x)h(x)=g(x)h(x)$ have the same solution sets?

Question

Let $x$ be a real number and $f$, $g$, and $h$ be functions of $x$. When do the equations $f(x)=g(x)$ and $f(x)h(x)=g(x)h(x)$ have the same solution sets? (i.e. when are they equivalent?)

Here is what I am thinking. If $S$ is the set of all $x$ such that $f(x)=g(x)$, then $h$ must be defined for all $x \in S$ and $h(x) \ne 0$$ \forall x \in S$. These two conditions would then allow us to manipulate $f(x)h(x)=g(x)h(x)$ (divide both sides by $h(x)$) to get $f(x)=g(x)$.

Answer 1

Since $f(x) h(x) - g(x) h(x) = (f(x) - g(x)) h(x)$, and assuming $f$, $g$ and $h$ are all defined on whatever domain you're interested in, the solutions of $f(x) h(x) = g(x) h(x)$ are the solutions of $f(x) = g(x)$ together with the solutions of $h(x) = 0$.