Acyclic model - Revision history

en>Chris the speller: sp

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-==Linear inequalities in real numbers==
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-When operating in terms of [[real number]]s, linear inequalities are the ones written in the forms
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en>Brad7777: −Category:Algebraic topology; −Category:Theorems in topology; ±Category:Theorems in abstract algebra→Category:Theorems in algebraic topology using HotCat

2012-01-31T20:24:44Z

−Category:Algebraic topology; −Category:Theorems in topology; ±Category:Theorems in abstract algebra→Category:Theorems in algebraic topology using HotCat

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{{Unreferenced|date=October 2008}}
In mathematics a '''linear inequality''' is an [[inequality (mathematics)|inequality]] which involves a [[linear function]].

==Linear inequalities in real numbers==
===Definitions===
When the two expressions are connected by 'greater than' or 'less than' sign, we get an inequality.

When operating in terms of [[real number]]s, linear inequalities are the ones written in the forms

:<math> f(x) < b \,</math> or <math> f(x) \leq b</math>,
where <math> f(x) </math> is a [[linear functional]] in real numbers and ''b'' is a constant real number. Alternatively, these may be viewed as

:<math> g(x) < 0 \,</math> or <math> g(x) \leq 0</math>,
where <math> g(x) </math> is an [[affine function]].

The above are commonly written out as
:<math>a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n < 0</math>
or
:<math>a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \leq 0</math>

Sometimes they may be written out in the forms
:<math>a_1 x_1 + a_2 x_2 + \cdots + a_n x_n < b </math>
or
:<math>a_1 x_1 + a_2 x_2 + \cdots + a_n x_n \leq b</math>

Here <math>x_1,\ x_2,...,x_n</math> are called the unknowns, <math>a_{0},\ a_{1},\ a_{2},...,\ a_{n}</math> are called the coefficients, and <math>b</math> 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:

:<math>\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}</math>
Here <math>x_1,\ x_2,...,x_n</math> are the unknowns, <math>a_{11},\ a_{12},...,\ a_{mn}</math> are the coefficients of the system, and <math>b_1,\ b_2,...,b_m</math> are the constant terms.

This can be concisely written as the [[matrix (mathematics)|matrix]] inequality:

:<math>Ax \leq b</math>

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 (geometry)|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 case]]s 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 '''R'''<sup>''n''</sup>.

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

==Linear inequalities in terms of other mathematical objects==

When you are about to 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 axis|x]] and [[y axis|y]] axis cross each other you can make the [[slope|rise over run]], which will help you find [[slope]]. If [[slope]] is denoted by ''m'' and [[y-intercept]] by ''b'', you can find ''m'' = <math>\frac {y_2 - y_1}{x_2 - x_1}</math> and ''b'' = <math>\frac {x_2y_1 - x_1y_2} {x_2-x_1}</math>, so long as <math>x_1 \neq x_2</math>. Such line is described by the equation <math>y = mx + b</math>.

The above definition requires well-defined operations of [[addition]], [[multiplication]] and [[Comparison (mathematics)|comparison]], therefore the notion of a linear inequality may be extended to [[ordered ring]]s, in, particular, to [[ordered field]]s.

==References==
{{reflist}}

==External links==
* [https://www.khanacademy.org/math/algebra/linear_inequalities Khan Academy: Linear inequalities, free online micro lectures]

[[Category:Linear algebra]]
[[Category:Linear programming]]
[[Category:Polyhedra]]