Say we have the equation $Ax>b$, where $A$ is an M-by-N matrix, $b$ is a known vector of length N, x is an unknown vector of length N, and the inequality sign means that each element of $Ax$ is greater than the corresponding element of $b$. Is there some linear-algebra-based method, such as e.g. some modification of LU decomposition or something similar, that would be amenable to programming on a computer and which would allow me to generically solve this system for $x$, given numerical values for $A$ and $b$? I will also accept any method of determining merely whether a solution exists, if not any solutions themselves.
EDIT: I should clarify that I'm mainly working with matrices with N and M both less than 10, so an exact method that works analogously to e.g. LU decomposition will probably be faster than an iterative method.