It's easy to sketch the region given by the inequality $(x-ay)(x-by)<0$ where $a,b\in\mathbb{R}$. A cubic one would take a bit more work but doable nevertheless. But I was trying to find a general way to find the region given by $$(x-a_1y)(x-a_2y)(x-a_3y)\dots<0 \textrm{ where } a_i\in\mathbb{R}$$
To do this, we start by plotting the lines $x=a_iy$. This partitions the Cartesian plane into $2n$ sections where $n$ is the number of $(x-a_iy)$ factors in the inequality. After doing this a few times (on Mathematica), I have noticed the following:
Given that none of the factors are repeated, no two adjacent sections are in the region given by the inequality, producing a dartboard shape.
If one of the factors is repeated, like $(x-3y)^2$, then the two adjacent sections partitioned by this line, in this case $x=3y$, must either be both in the region or both not in the region.
Using the above rules, I could easily sketch the regions by testing one point inside a section, if it works in the inequality then that section is in the region. So now we can find the other sections by using those rules.
However I have NOT proved those rules, and I'm not sure how would I even start the proof so for now I would just like to know if they are correct (this is my question). And if they are, can they be extended to inequality $[f_1(x)-y][f_2(x)-y]\dots<0$?
To prove that of dartboard rule, just take two points not in the lines, in adjacent regions. The sign of each parenthesis is the same for two, except for a single parenthesis. This means that one point is in the region and the other not.
The second rule is obvious since the repeated parenthesis does not change the sign of the product.
If you want to use formulas and equations, sort the slopes $a_1<a_2<\cdots <a_n$. For any point $(x,y)$ the quotient $\frac xy$ is between two of these slopes, or before the first, or after the last. Can you determine the sign of the product, given the interval where $\frac xy$ is?
I find your second question impossible to answer with no conditions about the functions. Even for relatively simple examples, like polynomials, the region can become very complicated.