Solving Inequalities and Proving Farkas's Lemma Made Easy David Avis and Bohdan Kaluzny Summary: Solving Inequalities and Proving Farkas's Lemma Made Easy David Avis and Bohdan Kaluzny March 20, 2003 1 Introduction Every college student has learned how to solve a system of linear equations, but how many would know how to solve Ax  b for x  0 or show that there is no solution? Solving a system of linear inequalities has traditionally been taught only in higher level courses and is given an incomplete treatment in introductory linear algebra courses. For example, the text of Strang [4] presents linear programming and states Farkas's lemma. It does not, however, include any proof of the niteness of the simplex method or a proof of the lemma. Recent developments have changed the situation dramatically. Re nements of the simplex method by Bland [1] in the 1970s lead to simpler proofs of its niteness, and Bland's original proof was simpli ed further by several authors. In this paper we will use a variant of Bland's pivot rule to solve a system of inequalities directly, without any need for introducing linear programming. We give a simple proof of the niteness of the method, based on ideas contained in the paper of Fukuda and Terlaky [3] on the related criss-cross method. Finally, if the system is infeasible, we show how the termination condition of the algorithm gives a certi cate of infeasibility, thus proving the Farkas Lemma. Terminology and notation used here follows that of Chvatal's linear programming book [2]. We consider the following problem. Given a matrix A = [a ij ] 2 < mn , and a column vector Collections: Computer Technologies and Information Sciences