Linear inequality

From Wikipedia, the free encyclopedia

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

[edit] Linear inequalities in real numbers

[edit] Definitions

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

$a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n < 0$

or

$a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \leq 0$

Sometimes they may be written out in the forms

$a_1 x_1 + a_2 x_2 + \cdots + a_n x_n < b$

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, $a_{0},\ a_{1},\ a_{2},...,\ a_{n}$ 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:

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

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.

[edit] 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 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 Rⁿ.

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

[edit] 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=m and y-intercept=b, you can find m by doing $\frac {y_2 - y_1}{x_2 - x_1}$ . You can solve for b by plugging in x, y, and m, or on a graph by finding where the line crosses the y axis because b is the y-intercept.

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.

[edit] References

Linear inequality

From Wikipedia, the free encyclopedia

Contents

[edit] Linear inequalities in real numbers

[edit] Definitions

[edit] Interpretations and applications

[edit] Linear inequalities in terms of other mathematical objects

[edit] References

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