Acyclic model: Difference between revisions

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In mathematics  a '''linear inequality''' is an [[inequality (mathematics)|inequality]] which involves a [[linear function]].


==Linear inequalities in real numbers==
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===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
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:<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''&times;''n'' matrix, ''x'' is an ''n''&times;1 [[column vector]] of variables, and ''b'' is an ''m''&times;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]]

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