In mathematics a linear inequality is an
inequality
Inequality may refer to:
Economics
* Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy
* Economic inequality, difference in economic well-being between population groups
* ...
which involves a
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
. A linear inequality contains one of the symbols of inequality:. It shows the data which is not equal in graph form.
* < less than
* > greater than
* ≤ less than or equal to
* ≥ greater than or equal to
* ≠ not equal to
* = equal to
A linear inequality looks exactly like a
linear equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
, with the inequality sign replacing the equality sign.
Linear inequalities of real numbers
Two-dimensional linear inequalities
![Linearineq1](https://upload.wikimedia.org/wikipedia/commons/d/d7/Linearineq1.svg)
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. 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 of ''ax'' + ''by'' at a point (''x''
0, ''y''
0) which is not on the line and observe whether or not the inequality is satisfied.
For example, to draw the solution set of ''x'' + 3''y'' < 9, one first draws the line with equation ''x'' + 3''y'' = 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.
Linear inequalities in general dimensions
In R
n linear inequalities are the expressions that may be written in the form
:
or
where ''f'' is a
linear form
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear map from a vector space to its field of scalars (often, the real numbers or the complex numbers).
If is a vector space over a field , the s ...
(also called a ''linear functional''),
and ''b'' a constant real number.
More concretely, this may be written out as
:
or
:
Here
are called the unknowns, and
are called the coefficients.
Alternatively, these may be written as
:
or
where ''g'' is an
affine function
In Euclidean geometry, an affine transformation or affinity (from the Latin, ''affinis'', "connected with") is a geometric transformation that preserves lines and parallelism, but not necessarily Euclidean distances and angles.
More generally, ...
.
That is
:
or
:
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.
Systems of linear inequalities
A system of linear inequalities is a set of linear inequalities in the same variables:
:
Here
are the unknowns,
are the coefficients of the system, and
are the constant terms.
This can be concisely written as the
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
inequality
:
where ''A'' is an ''m''×''n'' matrix, ''x'' is an ''n''×1
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
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.
Variables can be eliminated from systems of linear inequalities using
Fourier–Motzkin elimination
Fourier–Motzkin elimination, also known as the FME method, is a mathematical algorithm for eliminating variables from a system of linear inequalities. It can output real solutions.
The algorithm is named after Joseph Fourier who proposed the me ...
.
Applications
Polyhedra
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-spaces defined by individual inequalities. It is a
convex set
In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex r ...
, since the half-spaces are convex sets, and the intersection of a set of convex sets is also convex. In the non-
degenerate case
In mathematics, a degenerate case is a limiting case of a class of objects which appears to be qualitatively different from (and usually simpler than) the rest of the class, and the term degeneracy is the condition of being a degenerate case.
T ...
s this convex set is a
convex polyhedron
A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
(possibly unbounded, e.g., a half-space, a slab between two parallel half-spaces or a
polyhedral cone
In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every .
W ...
). It may also be empty or a convex polyhedron of lower dimension confined to an
affine subspace
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related ...
of the ''n''-dimensional space R
''n''.
Linear programming
A linear programming problem seeks to optimize (find a maximum or minimum value) a function (called the
objective function
In mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cos ...
) subject to a number of constraints on the variables which, in general, are linear inequalities.
The list of constraints is a system of linear inequalities.
Generalization
The above definition requires well-defined operations of
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
,
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
and
comparison
Comparison or comparing is the act of evaluating two or more things by determining the relevant, comparable characteristics of each thing, and then determining which characteristics of each are similar to the other, which are different, and t ...
; therefore, the notion of a linear inequality may be extended to
ordered ring
In abstract algebra, an ordered ring is a (usually commutative) ring ''R'' with a total order ≤ such that for all ''a'', ''b'', and ''c'' in ''R'':
* if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''.
* if 0 ≤ ''a'' and 0 ≤ ''b'' then ...
s, and in particular to
ordered field
In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
s.
References
Sources
*
* {{citation, first1=Charles D., last1=Miller, first2=Vern E., last2=Heeren, year=1986, edition=5th, title=Mathematical Ideas, publisher=Scott, Foresman, isbn=0-673-18276-2, url-access=registration, url=https://archive.org/details/mathematicalidea0000mill
External links
Khan Academy: Linear inequalities, free online micro lectures
Linear algebra
Linear programming
Polyhedra