I am trying to solve a problem which I have reduced down to find out one of integer solutions to a number of nonhomogeneous linear inequalities.
Can this be done efficiently ? If so: how?; if not: why?
You are describing integer programming. It's an NP-hard problem, so there are no known efficient (i.e., polynomial-time) solutions in general. However, this does not mean that your particular problem cannot be solved in reasonable time. Try the GLPK.