^ In the 2-dimensional case, both linear forms and affine functions are historically called linear functions because their graphs are lines.

In that situation, the solution set is either empty or the entire plane.

^ Technically, for this statement to be correct both a and b can not simultaneously be zero.

It may also be empty or a convex polyhedron of lower dimension confined to an affine subspace of the n-dimensional space R n. In the non- degenerate cases this convex set is a convex polyhedron (possibly unbounded, e.g., a half-space, a slab between two parallel half-spaces or a polyhedral cone). It is a convex set, since the half-spaces are convex sets, and the intersection of a set of convex sets is also convex.

The set of solutions of a system of linear inequalities corresponds to the intersection of the half-spaces defined by individual inequalities. The set of solutions of a real linear inequality constitutes a half-space of the 'n'-dimensional real space, one of the two defined by the corresponding linear equation. Variables can be eliminated from systems of linear inequalities using Fourier–Motzkin elimination. Not all systems of linear inequalities have solutions.In the above systems both strict and non-strict inequalities may be used. Where A is an m× n matrix, x is an n×1 column vector of variables, and b is an m×1 column vector of constants. This can be concisely written as the matrix inequality Two-dimensional linear inequalities are expressions in two variables of the form:Ī x + b y < c and a x + b y ≥ c, are the constant terms.

Linear inequalities khan academy