The book tells about the relation of systems of linear inequalities to convex polyhedra, gives a description of the set of all solutions of a system of linear inequalities, analyses the questions of compatibility and incompatibility; finally, it gives an insight into linear programming as one of the topics in the theory of systems of linear inequalities. The last section but one gives a proof of duality theorem of linear programming. The book is intended for senior pupils and all amateur mathematicians.