Linear inequality

In mathematics a linear inequality is an inequality which involves a linear function.

Two-dimensional linear inequalities are expressions in two variables of the form: where the inequalities may either be strict or not.

The solution set of such an inequality can be graphically represented by a half-plane (all the points on one "side" of a fixed line) in the Euclidean plane.

[2] The line that determines the half-planes (ax + by = c) is not included in the solution set when the inequality is strict.

A simple procedure to determine which half-plane is in the solution set is to calculate the value of ax + by at a point (x0, y0) which is not on the line and observe whether or not the inequality is satisfied.

For example,[3] to draw the solution set of x + 3y < 9, one first draws the line with equation x + 3y = 9 as a dotted line, to indicate that the line is not included in the solution set since the inequality is strict.

Then, pick a convenient point not on the line, such as (0,0).

Since 0 + 3(0) = 0 < 9, this point is in the solution set, so the half-plane containing this point (the half-plane "below" the line) is the solution set of this linear inequality.

Alternatively, these may be written as where g is an affine function.

[4] That is or Note that any inequality containing a "greater than" or a "greater than or equal" sign can be rewritten with a "less than" or "less than or equal" sign, so there is no need to define linear inequalities using those signs.

A system of linear inequalities is a set of linear inequalities in the same variables: Here

This can be concisely written as the matrix inequality 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.

[citation needed] In the above systems both strict and non-strict inequalities may be used.

[5] 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.

The set of solutions of a system of linear inequalities corresponds to the intersection of the half-spaces defined by individual inequalities.

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 may also be empty or a convex polyhedron of lower dimension confined to an affine subspace of the n-dimensional space Rn.

A linear programming problem seeks to optimize (find a maximum or minimum value) a function (called the objective function) subject to a number of constraints on the variables which, in general, are linear inequalities.

[6] The list of constraints is a system of linear inequalities.

The above definition requires well-defined operations of addition, multiplication and comparison; therefore, the notion of a linear inequality may be extended to ordered rings, and in particular to ordered fields.