TITLE: How do you tell if a system of linear inequalities has a solution?
QUESTION [9 upvotes]: A naive solution would be to optimize a dummy variable via linear programming and see if a result is returned.  I imagine there must be a more direct way.

REPLY [9 votes]: Most linear programming solvers check for feasibility first (usually this takes as much time as the second phase, which is actually finding the optimum, and uses the exact same algorithm). I would advise reading a standard textbook on the subject (Luenberger is good, Schrijver great if you are interested in integer/mixed problems).