Linear inequality

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

Formal definitions

When operating in terms of real numbers, linear inequalities are the ones written in the forms

:

where $f(x)$ is a linear functional in real numbers and "b" is a constant real number. Alternatively, these may be viewed as

:

where $g(x)$ is an affine function.

The above are commonly written out as:

or

:$a\_0\; +\; a\_1\; x\_1\; +\; a\_2\; x\_2\; +\; cdots\; +\; a\_n\; x\_n\; leq\; 0.$

Sometimes they may be written out in the forms

:

or

:$a\_1\; x\_1\; +\; a\_2\; x\_2\; +\; cdots\; +\; a\_n\; x\_n\; leq\; b.$

Here $x\_1,\; x\_2,dots,x\_n$ are called the unknowns, $a\_\{1\},\; a\_\{2\},dots,\; a\_\{n\}$ are called the coefficients, and $b$ is the constant term.

A linear inequality looks exactly like a linear equation, with the inequality sign replacing the equality sign.

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

:

Here $x\_1,\; x\_2,dots,x\_n$ are the unknowns, $a\_\{11\},\; a\_\{12\},dots,\; a\_\{mn\}$ are the coefficients of the system, and $b\_1,\; b\_2,dots,b\_m$ are the constant terms.

This can be concisely written as the matrix inequality:

:$Ax\; leq\; b$

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.

In the above systems both strict and non-strict inequalities may be used.

Not all systems of linear inequalities have solutions.

Linear inequalities in terms of other mathematical objects

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, in, particular, to ordered fields.

Linear inequalities in real numbers

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-planes defined by individual inequalities. It is a convex set, since the half-planes are convex sets, and the intersection of a set of convex sets is also convex. In the non-degenerate cases this convex set if 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 R^"n".

Sets of linear inequalities (called constraints) are used in the definition of linear programming.

References