A question from one of the comprehension questions I have is:
How would the intervals of the solution set differ between a polynomial inequality and a rational inequality?
I have tried to research the topic but have come up short. I know that rational functions can have vertical asymptotes and restrictions on the domain but I am not sure if this is relevant or what the difference specifically would be. I was thinking it might be because the rational function would have a hole in the intervals and a polynomial would not?
Two differences:
A polynomial inequality (PI) can always be replaced by a PI with one side zero. That is, $p(x) > q(x)$ is the same as $p(x)-q(x) > 0 $. A rational inequality (RI) can not, because you do not know the sign of the denominator; $\frac{a(x)}{b(x)} > \frac{c(x)}{d(x)} $ can not safely be replaced by $\frac{a(x)d(x)-b(x)c(x)}{b(x)d(x)} > 0 $ because $b(x)$ and/or $d(x)$ might change signs unexpectedly.
Of course, as you said, denominators can be zero, which can be annoying.