Consistency examination of linear inequality system

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An investigation of the consistency of a linear inequality system is considered. It is proven that the system of linear inequalities Ax≥b is consistent if and only if for any generalized inverse A− of a matrix A the system of equations (I−AA−)v=−(I−AA−)b has a nonnegative solution for the vector v. Consistency of the above system does not depend on the choice of the matrix A−. The paper also presents methods for investigating the existence of nonnegative solutions of systems of linear equations.