Linear inequality

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

Contents

Linear inequalities in real numbers

Definitions

When two expressions are connected by 'greater than' or 'less than' sign, we get an inequation.

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

f(x) < b \, or f(x) \leq b,

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

g(x) < 0 \, or g(x) \leq 0,

where g(x) is an affine function.

The above are commonly written out as

a_0 %2B a_1 x_1 %2B a_2 x_2 %2B \cdots %2B a_n x_n < 0

or

a_0 %2B a_1 x_1 %2B a_2 x_2 %2B \cdots %2B a_n x_n \leq 0

Sometimes they may be written out in the forms

a_1 x_1 %2B a_2 x_2 %2B \cdots %2B a_n x_n < b

or

a_1 x_1 %2B a_2 x_2 %2B \cdots %2B a_n x_n \leq b

Here x_1,\ x_2,...,x_n are called the unknowns, a_{0},\ a_{1},\ a_{2},...,\ 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:

\begin{alignat}{7} a_{11} x_1 &&\; %2B \;&& a_{12} x_2 &&\; %2B \cdots %2B \;&& a_{1n} x_n &&\; \leq \;&&& b_1 \\ a_{21} x_1 &&\; %2B \;&& a_{22} x_2 &&\; %2B \cdots %2B \;&& a_{2n} x_n &&\; \leq \;&&& b_2 \\ \vdots\;\;\; && && \vdots\;\;\; && && \vdots\;\;\; && &&& \;\vdots \\ a_{m1} x_1 &&\; %2B \;&& a_{m2} x_2 &&\; %2B \cdots %2B \;&& a_{mn} x_n &&\; \leq \;&&& b_m \\ \end{alignat}

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.

Interpretations and applications

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 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.

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

Linear inequalities in terms of other mathematical objects

When you graph a linear inequality, it will be on one side of a line. Also, when you mark points where the line crosses where the x and y axis cross each other you can make the rise over run, which will help you find slope. If slope is denoted by m and y-intercept by b, you can find m = \frac {y_2 - y_1}{x_2 - x_1} and b = \frac {x_2y_1 - x_1y_2} {x_2-x_1}, so long as x_1 \neq x_2. Such line is described by the equation y = mx %2B b.

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.

References