Nonhomogeneous linear theorems of the alternative

Generalizing the classical theorems, theorems of the alternative of new, common forms are obtained for the consistence of nonhomogeneous linear inequality systems. With their help simple proofs for the nonhomogeneous Farkas theorem as well as for the Kuhn–Fourier theorem are described. As particular cases, we get necessary and sufficient conditions for the consistence of linear inequality systems having positivity, semipositivity or nonnegativity requirements for the variables.