Linear inequality

Two-dimensional linear inequalities, are expressions in two variables of the form:

${\displaystyle ax+by<c{\text{ and }}ax+by\geq c,}$

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.

In Rⁿ linear inequalities are the expressions that may be written in the form

${\displaystyle f({\bar {x}})<b}$ or ${\displaystyle f({\bar {x}})\leq b,}$

where f is a linear form (also called a linear functional), ${\displaystyle {\bar {x}}=(x_{1},x_{2},\ldots ,x_{n})}$ and b a constant real number.

More concretely, this may be written out as

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<b}$

or

${\displaystyle a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq b.}$

Here ${\displaystyle x_{1},x_{2},...,x_{n}}$ are called the unknowns, and ${\displaystyle a_{1},a_{2},...,a_{n}}$ are called the coefficients.

Alternatively, these may be written as

${\displaystyle g(x)<0\,}$ or ${\displaystyle g(x)\leq 0,}$

where g is an affine function.^[4]

That is

${\displaystyle a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}<0}$

or

${\displaystyle a_{0}+a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq 0.}$

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:

${\displaystyle {\begin{alignedat}{7}a_{11}x_{1}&&\;+\;&&a_{12}x_{2}&&\;+\cdots +\;&&a_{1n}x_{n}&&\;\leq \;&&&b_{1}\\a_{21}x_{1}&&\;+\;&&a_{22}x_{2}&&\;+\cdots +\;&&a_{2n}x_{n}&&\;\leq \;&&&b_{2}\\\vdots \;\;\;&&&&\vdots \;\;\;&&&&\vdots \;\;\;&&&&&\;\vdots \\a_{m1}x_{1}&&\;+\;&&a_{m2}x_{2}&&\;+\cdots +\;&&a_{mn}x_{n}&&\;\leq \;&&&b_{m}\\\end{alignedat}}}$

Here ${\displaystyle x_{1},\ x_{2},...,x_{n}}$ are the unknowns, ${\displaystyle a_{11},\ a_{12},...,\ a_{mn}}$ are the coefficients of the system, and ${\displaystyle b_{1},\ b_{2},...,b_{m}}$ are the constant terms.

This can be concisely written as the matrix inequality

${\displaystyle Ax\leq b,}$

where A is an m×n matrix of constants, x is an n×1 column vector of variables, b is an m×1 column vector of constants and the inequality relation is understood row-by-row.

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

• Not all systems of linear inequalities have solutions.

Variables can be eliminated from systems of linear inequalities using Fourier–Motzkin elimination.^[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. It is a convex set, since the half-spaces are convex sets, and the intersection of a set of convex sets is also convex. 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.

• Angel, Allen R.; Porter, Stuart R. (1989), A Survey of Mathematics with Applications (3rd ed.), Addison-Wesley, ISBN 0-201-13696-1
• Miller, Charles D.; Heeren, Vern E. (1986), Mathematical Ideas (5th ed.), Scott, Foresman, ISBN 0-673-18276-2