# Linear inequality

In mathematics a linear inequality is an inequality which involves a linear function. A linear inequality contains one of the symbols of inequality:. It shows the data which is not equal in graph form.

• < is less than
• > is greater than
• ≤ is less than or equal to
• ≥ is greater than or equal to
• ≠ is not equal to
• = is equal to

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

## Linear inequalities of real numbers

### Two-dimensional linear inequalities

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

$ax+by 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. 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, 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.

### Linear inequalities in general dimensions

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

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

More concretely, this may be written out as

$a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n} or

$a_{1}x_{1}+a_{2}x_{2}+\cdots +a_{n}x_{n}\leq b.$ Here $x_{1},x_{2},...,x_{n}$ are called the unknowns, and $a_{1},a_{2},...,a_{n}$ are called the coefficients.

Alternatively, these may be written as

$g(x)<0\,$ or $g(x)\leq 0,$ where g is an affine function.

That is

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

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

### Systems of linear inequalities

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

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

### Applications

#### Polyhedra

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.

#### Linear programming

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. The list of constraints is a system of linear inequalities.

## Generalization

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.

## Sources

• 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