Checking if a solution exists across two inequalities If I have $ay < x < by$ $cx < y < dx$ With $a,b,c,d$ as known (they are real-valued, can be positive or negative or 0) and $x,y$ unknown, is there a methodical way to see if a solution exists for $x$ and $y$?

If solution points $(x; y)$ exist, linearly scaled points $(kx; ky)$ must also be solutions, as no inequality is violated by multiplying both sides with the same factor.

Therefore, we can assert an arbitrary non-zero value for $x$ and calculate the corresponding solution interval for $y$. If the interval is empty, no solution exists.

Example:

$x = 1$

$ay < 1 < by$

$c < y < d$

With known values for $a, b, c, d$ we can immediately see if there exists a feasible value interval for $y$. It cannot be more than one interval as the intersection of two intervals is either empty or one uninterrupted interval.