Suppose I have a rational inequality, like
$$ \frac{(x-4)(x+4)}{(x-1)^2} < 0 $$
I can manually check between the roots (4,4) and asymptote (1) by plugging in value like 3 to see that:
$$ −4<x<1 \text{ and } 1<x<4 $$ Or in interval terms:
$$ (−4,1) \text{ and } (1,4) $$
Now, is there an algorithm to simply look at the roots/aymptotes, in this case (4,4,1), and know without plugging anything in what the interval or inequality of x is?
I know it's trivial in this case but imagine you had more massive polynomials causing lots of sign switching. Is there a technique for getting the inequalities/intervals from just looking at the roots/asymptotes of the rational function?
Note: I've been following along here https://tutorial.math.lamar.edu/Classes/Alg/SolveRationalInequalities.aspx Basically want to see if there's a way I can do these rational inequalities faster, and deal with bigger ones quickly.