In mathematics alinear inequality is aninequality which involves alinear function. A linear inequality contains one of the symbols of inequality:^[1]

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

A linear inequality looks exactly like alinear equation, with the inequality sign replacing theequality sign.

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 ofax +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 ofx + 3y < 9, one first draws the line with equationx + 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.

InRⁿ linear inequalities are the expressions that may be written in the form

or${\displaystyle f(\overline{x})\le b,}$

wheref is alinear form (also called alinear functional),${\displaystyle \overline{x}=(x_1,x_2,\dots ,x_n)}$ andb a constant real number.

More concretely, this may be written out as

or

${\displaystyle a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n\le b.}$

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

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

whereg is anaffine function.^[4]

That is

or

${\displaystyle a_0+a_1{x}_1+a_2{x}_2+\cdots +a_n{x}_n\le 0.}$

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:

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 thematrix inequality

${\displaystyle Ax\le b,}$

whereA is anm×n matrix of constants,x is ann×1column vector of variables,b is anm×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 usingFourier–Motzkin elimination.^[5]

The set of solutions of a real linear inequality constitutes ahalf-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 aconvex 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 aconvex polyhedron (possibly unbounded, e.g., a half-space, a slab between two parallel half-spaces or apolyhedral cone). It may also be empty or a convex polyhedron of lower dimension confined to anaffine subspace of then-dimensional spaceRⁿ.

A linear programming problem seeks to optimize (find a maximum or minimum value) a function (called theobjective 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 ofaddition,multiplication andcomparison; therefore, the notion of a linear inequality may be extended toordered rings, and in particular toordered 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

• Linear algebra
• Linear programming
• Polyhedra

• Articles with short description
• Short description matches Wikidata