In

^{n}'' ≤ ''b^{n}''.
: 0 ≤ ''a'' ≤ ''b'' ⇔ ''a''^{−''n''} ≥ ''b''^{−''n''} ≥ 0.
* Taking the

ordered fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, because −1 is the square of ''i'' and would therefore be positive.
Besides from being an ordered field, R also has the

_{1} ≤ ''a''_{2} ≤ ... ≤ ''a''_{''n''} means that ''a''_{''i''} ≤ ''a''_{''i''+1} for ''i'' = 1, 2, ..., ''n'' − 1. By transitivity, this condition is equivalent to ''a''_{''i''} ≤ ''a''_{''j''} for any 1 ≤ ''i'' ≤ ''j'' ≤ ''n''.
When solving inequalities using chained notation, it is possible and sometimes necessary to evaluate the terms independently. For instance, to solve the inequality 4''x'' < 2''x'' + 1 ≤ 3''x'' + 2, it is not possible to isolate ''x'' in any one part of the inequality through addition or subtraction. Instead, the inequalities must be solved independently, yielding ''x'' < 1/2 and ''x'' ≥ −1 respectively, which can be combined into the final solution −1 ≤ ''x'' < 1/2.
Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the _{1} < ''a''_{2} > ''a''_{3} < ''a''_{4} > ''a''_{5} < ''a''_{6} > ... . Mixed chained notation is used more often with compatible relations, like <, =, ≤. For instance, ''a'' < ''b'' = ''c'' ≤ ''d'' means that ''a'' < ''b'', ''b'' = ''c'', and ''c'' ≤ ''d''. This notation exists in a few

_{1}, ''a''_{2}, …, ''a''_{''n''} we have where
:

^{''n''} with the standard inner product, the Cauchy–Schwarz inequality is
: $\backslash left(\backslash sum\_^n\; u\_i\; v\_i\backslash right)^2\backslash leq\; \backslash left(\backslash sum\_^n\; u\_i^2\backslash right)\; \backslash left(\backslash sum\_^n\; v\_i^2\backslash right).$

^{''b''}, where ''a'' and ''b'' are real positive numbers or variable expressions. They often appear in mathematical olympiads exercises.

ordered fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

. To make an ordered fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

, it would have to satisfy the following two properties:
* if , then ;
* if and , then .
Because ≤ is a

Graph of Inequalities

by Ed Pegg, Jr.

AoPS Wiki entry about Inequalities

{{Authority control Inequalities, Elementary algebra Mathematical terminology

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, an inequality is a relation which makes a non-equal comparison between two numbers or other mathematical expressions. It is used most often to compare two numbers on the number line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as abstraction for real numbers, denoted by \mathbb. Every point of a number line is assumed to correspond to a real number, and ever ...

by their size. There are several different notations used to represent different kinds of inequalities:
* The notation ''a'' < ''b'' means that ''a'' is less than ''b''.
* The notation ''a'' > ''b'' means that ''a'' is greater than ''b''.
In either case, ''a'' is not equal to ''b''. These relations are known as strict inequalities, meaning that ''a'' is strictly less than or strictly greater than ''b''. Equivalence is excluded.
In contrast to strict inequalities, there are two types of inequality relations that are not strict:
* The notation ''a'' ≤ ''b'' or ''a'' ⩽ ''b'' means that ''a'' is less than or equal to ''b'' (or, equivalently, at most ''b'', or not greater than ''b'').
* The notation ''a'' ≥ ''b'' or ''a'' ⩾ ''b'' means that ''a'' is greater than or equal to ''b'' (or, equivalently, at least ''b'', or not less than ''b'').
The relation not greater than can also be represented by ''a'' ≯ ''b'', the symbol for "greater than" bisected by a slash, "not". The same is true for not less than and ''a'' ≮ ''b''.
The notation ''a'' ≠ ''b'' means that ''a'' is not equal to ''b''; this ''inequation
In mathematics, an inequation is a statement that an inequality (mathematics), inequality or a non-equality holds between two values. It is usually written in the form of a pair of expression (mathematics), expressions denoting the values in ques ...

'' sometimes is considered a form of strict inequality. It does not say that one is greater than the other; it does not even require ''a'' and ''b'' to be member of an ordered set
Image:Hasse diagram of powerset of 3.svg, upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparab ...

.
In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another, normally by several orders of magnitude
An order of magnitude is an approximation of the logarithm of a value relative to some contextually understood reference value, usually ten, interpreted as the base of the logarithm and the representative of values of magnitude one. Logarithmic ...

.
* The notation ''a'' ≪ ''b'' means that ''a'' is much less than ''b''.
* The notation ''a'' ≫ ''b'' means that ''a'' is much greater than ''b''.
This implies that the lesser value can be neglected with little effect on the accuracy of an approximation
An approximation is anything that is intentionally similar but not exactly equal
Equal or equals may refer to:
Arts and entertainment
* Equals (film), ''Equals'' (film), a 2015 American science fiction film
* Equals (game), ''Equals'' (game), a ...

(such as the case of ultrarelativistic limitIn physics
Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Phys ...

in physics).
In all of the cases above, any two symbols mirroring each other are symmetrical; ''a'' < ''b'' and ''b'' > ''a'' are equivalent, etc.
Properties on the number line

Inequalities are governed by the followingproperties
Property (''latin: Res Privata'') in the abstract is what belongs to or with something, whether as an attribute or as a component of said thing. In the context of this article, it is one or more components (rather than attributes), whether phys ...

. All of these properties also hold if all of the non-strict inequalities (≤ and ≥) are replaced by their corresponding strict inequalities (< and >) and — in the case of applying a function — monotonic functions are limited to ''strictly'' monotonic function
Figure 3. A function that is not monotonic
In mathematics, a monotonic function (or monotone function) is a function (mathematics), function between List of order structures in mathematics, ordered sets that preserves or reverses the given orde ...

s.
Converse

The relations ≤ and ≥ are each other'sconverse
Converse may refer to:
Mathematics and logic
* Converse (logic), the result of reversing the two parts of a categorical or implicational statement
** Converse implication, the converse of a material implication
** Converse nonimplication, a logical ...

, meaning that for any real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s ''a'' and ''b'':
: ''a'' ≤ ''b'' and ''b'' ≥ ''a'' are equivalent.
Transitivity

The transitive property of inequality states that for anyreal number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s ''a'', ''b'', ''c'':
: If ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c''.
If ''either'' of the premises is a strict inequality, then the conclusion is a strict inequality:
: If ''a'' ≤ ''b'' and ''b'' < ''c'', then ''a'' < ''c''.
: If ''a'' < ''b'' and ''b'' ≤ ''c'', then ''a'' < ''c''.
Addition and subtraction

A common constant ''c'' may be to or from both sides of an inequality. So, for anyreal number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s ''a'', ''b'', ''c'':
: If ''a'' ≤ ''b'', then ''a'' + ''c'' ≤ ''b'' + ''c'' and ''a'' − ''c'' ≤ ''b'' − ''c''.
In other words, the inequality relation is preserved under addition (or subtraction) and the real numbers are an ordered group
In abstract algebra, a partially ordered group is a group (''G'', +) equipped with a partial order
Image:Hasse diagram of powerset of 3.svg, 250px, The Hasse diagram of the power set, set of all subsets of a three-element set , ordered by inclus ...

under addition.
Multiplication and division

The properties that deal withmultiplication
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 Operation (mathematics), mathematical operations of arithmetic, with the ...

and division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting o ...

state that for any real numbers, ''a'', ''b'' and non-zero ''c'':
: If ''a'' ≤ ''b'' and ''c'' > 0, then ''ac'' ≤ ''bc'' and ''a''/''c'' ≤ ''b''/''c''.
: If ''a'' ≤ ''b'' and ''c'' < 0, then ''ac'' ≥ ''bc'' and ''a''/''c'' ≥ ''b''/''c''.
In other words, the inequality relation is preserved under multiplication and division with positive constant, but is reversed when a negative constant is involved. More generally, this applies for an ordered fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

. For more information, see '' § Ordered fields''.
Additive inverse

The property for theadditive inverse
In mathematics, the additive inverse of a is the number that, when to , yields . This number is also known as the opposite (number), sign change, and negation. For a , it reverses its : the additive inverse (opposite number) of a is negative, ...

states that for any real numbers ''a'' and ''b'':
: If ''a'' ≤ ''b'', then −''a'' ≥ −''b''.
Multiplicative inverse

If both numbers are positive, then the inequality relation between themultiplicative inverse
Image:Hyperbola one over x.svg, thumbnail, 300px, alt=Graph showing the diagrammatic representation of limits approaching infinity, The reciprocal function: . For every ''x'' except 0, ''y'' represents its multiplicative inverse. The graph forms a r ...

s is opposite of that between the original numbers. More specifically, for any non-zero real numbers ''a'' and ''b'' that are both positive
Positive is a property of Positivity (disambiguation), positivity and may refer to:
Mathematics and science
* Converging lens or positive lens, in optics
* Plus sign, the sign "+" used to indicate a positive number
* Positive (electricity), a po ...

(or both negative):
: If ''a'' ≤ ''b'', then ≥ .
All of the cases for the signs of ''a'' and ''b'' can also be written in chained notation
Chained may refer to:
* ''Chained'' (1934 film), starring Joan Crawford and Clark Gable
* ''Chained'' (2012 film), a Canadian film directed by Jennifer Lynch
* ''Chained'' (2020 film), a Canadian film directed by Titus Heckel
* ''Chained'', a 20 ...

, as follows:
: If 0 < ''a'' ≤ ''b'', then ≥ > 0.
: If ''a'' ≤ ''b'' < 0, then 0 > ≥ .
: If ''a'' < 0 < ''b'', then < 0 < .
Applying a function to both sides

Anymonotonic
Figure 3. A function that is ''not'' monotonic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...

ally increasing function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

, by its definition, may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the domain
Domain may refer to:
Mathematics
*Domain of a function, the set of input values for which the (total) function is defined
**Domain of definition of a partial function
**Natural domain of a partial function
**Domain of holomorphy of a function
*Doma ...

of that function). However, applying a monotonically decreasing function to both sides of an inequality means the inequality relation would be reversed. The rules for the additive inverse, and the multiplicative inverse for positive numbers, are both examples of applying a monotonically decreasing function.
If the inequality is strict (''a'' < ''b'', ''a'' > ''b'') ''and'' the function is strictly monotonic, then the inequality remains strict. If only one of these conditions is strict, then the resultant inequality is non-strict. In fact, the rules for additive and multiplicative inverses are both examples of applying a ''strictly'' monotonically decreasing function.
A few examples of this rule are:
* Raising both sides of an inequality to a power ''n'' > 0 (equiv., −''n'' < 0), when ''a'' and ''b'' are positive real numbers:
: 0 ≤ ''a'' ≤ ''b'' ⇔ 0 ≤ ''anatural logarithm
The natural logarithm of a number is its logarithm to the base (exponentiation), base of the mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approximately equal to . The natura ...

on both sides of an inequality, when ''a'' and ''b'' are positive real numbers:
: 0 < ''a'' ≤ ''b'' ⇔ ln(''a'') ≤ ln(''b'').
: 0 < ''a'' < ''b'' ⇔ ln(''a'') < ln(''b'').
: (this is true because the natural logarithm is a strictly increasing function.)
Formal definitions and generalizations

A (non-strict) partial order is abinary relation
In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...

≤ over a set ''P'' which is reflexive, antisymmetric, and transitive
Transitivity or transitive may refer to:
Grammar
* Transitivity (grammar), a property of verbs that relates to whether a verb can take direct objects
* Transitive verb, a verb which takes an object
* Transitive case, a grammatical case to mark arg ...

. That is, for all ''a'', ''b'', and ''c'' in ''P'', it must satisfy the three following clauses:
# ''a'' ≤ ''a'' ( reflexivity)
# if ''a'' ≤ ''b'' and ''b'' ≤ ''a'', then ''a'' = ''b'' (antisymmetry
In linguistics, antisymmetry is a theory of syntax, syntactic linearization presented in Richard Kayne's 1994 monograph ''The Antisymmetry of Syntax''. The crux of this theory is that hierarchical structure in natural language maps universally onto ...

)
# if ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c'' ( transitivity)
A set with a partial order is called a partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

. Those are the very basic axioms that every kind of order has to satisfy. Other axioms that exist for other definitions of orders on a set ''P'' include:
# For every ''a'' and ''b'' in ''P'', ''a'' ≤ ''b'' or ''b'' ≤ ''a'' (total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X:
# a \ ...

).
# For all ''a'' and ''b'' in ''P'' for which ''a'' < ''b'', there is a ''c'' in ''P'' such that ''a'' < ''c'' < ''b'' (dense orderIn mathematics, a partial order or total order < on a Set (mathematics), set $X$ is said to be dense if, for all $x$ and $y$ in $X$ for which $x\; <\; y$, there is a $z$ in $X$ ...

).
# Every non-empty subset
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of ''P'' with an upper bound
In mathematics, particularly in order theory
Order theory is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), s ...

has a ''least'' upper bound (supremum) in ''P'' (least-upper-bound property
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

).
Ordered fields

If (''F'', +, ×) is afield
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

and ≤ is a total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X:
# a \ ...

on ''F'', then (''F'', +, ×, ≤) is called an ordered fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

if and only if:
* ''a'' ≤ ''b'' implies ''a'' + ''c'' ≤ ''b'' + ''c'';
* 0 ≤ ''a'' and 0 ≤ ''b'' implies 0 ≤ ''a'' × ''b''.
Both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fieldIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s, but ≤ cannot be defined in order to make (C, +, ×, ≤) an Least-upper-bound property
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

. In fact, R can be defined as the only ordered field with that quality.
Chained notation

The notation ''a'' < ''b'' < ''c'' stands for "''a'' < ''b'' and ''b'' < ''c''", from which, by the transitivity property above, it also follows that ''a'' < ''c''. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative. Hence, for example, ''a'' < ''b'' + ''e'' < ''c'' is equivalent to ''a'' − ''e'' < ''b'' < ''c'' − ''e''. This notation can be generalized to any number of terms: for instance, ''a''logical conjunction
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, ...

of the inequalities between adjacent terms. For example, the defining condiction of a zigzag poset
In mathematics, a fence, also called a zigzag poset, is a partially ordered set in which the order relations form a path graph, path with alternating orientations:
:''a'' < ''b'' > ''c'' < ''d'' > ''e'' < ''f'' > ''h'' < ''i'' . ...

is written as ''a''programming language
A programming language is a formal language
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calcu ...

s such as Python
PYTHON was a Cold War contingency plan of the Government of the United Kingdom, British Government for the continuity of government in the event of Nuclear warfare, nuclear war.
Background
Following the report of the Strath Committee in 1955, the ...

. In contrast, in programming languages that provide an ordering on the type of comparison results, such as C, even homogeneous chains may have a completely different meaning.
Sharp inequalities

An inequality is said to be ''sharp'', if it cannot be ''relaxed'' and still be valid in general. Formally, auniversally quantified
Universality most commonly refers to:
* Universality (philosophy)
* Universality (dynamical systems)
Universality principle may refer to:
* In statistics, Random matrix#Bulk statistics, universality principle, a property of systems that can be ...

inequality ''φ'' is called sharp if, for every valid universally quantified inequality ''ψ'', if holds, then also holds. For instance, the inequality is sharp, whereas the inequality is not sharp.
Inequalities between means

There are many inequalities between means. For example, for any positive numbers ''a''Cauchy–Schwarz inequality

The Cauchy–Schwarz inequality states that for all vectors ''u'' and ''v'' of aninner product space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

it is true that
: $,\; \backslash langle\; \backslash mathbf,\backslash mathbf\backslash rangle,\; ^2\; \backslash leq\; \backslash langle\; \backslash mathbf,\backslash mathbf\backslash rangle\; \backslash cdot\; \backslash langle\; \backslash mathbf,\backslash mathbf\backslash rangle,$
where $\backslash langle\backslash cdot,\backslash cdot\backslash rangle$ is the inner product
In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbert space is a vector space with a binary operation called an inner product. This operation associates each pair of vectors in the space with a Scalar (mathematics), ...

. Examples of inner products include the real and complex dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...

; In Euclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

''R''Power inequalities

A "power inequality" is an inequality containing terms of the form ''a''Examples

* For any real ''x'', :: $e^x\; \backslash ge\; 1+x.$ * If ''x'' > 0 and ''p'' > 0, then :: $\backslash frac\; \backslash ge\; \backslash ln(x)\; \backslash ge\; \backslash frac.$ : In the limit of ''p'' → 0, the upper and lower bounds converge to ln(''x''). * If ''x'' > 0, then :: $x^x\; \backslash ge\; \backslash left(\; \backslash frac\backslash right)^\backslash frac.$ * If ''x'' > 0, then :: $x^\; \backslash ge\; x.$ * If ''x'', ''y'', ''z'' > 0, then :: $\backslash left(x+y\backslash right)^z\; +\; \backslash left(x+z\backslash right)^y\; +\; \backslash left(y+z\backslash right)^x\; >\; 2.$ * For any real distinct numbers ''a'' and ''b'', :: $\backslash frac\; >\; e^.$ * If ''x'', ''y'' > 0 and 0 < ''p'' < 1, then :: $x^p+y^p\; >\; \backslash left(x+y\backslash right)^p.$ * If ''x'', ''y'', ''z'' > 0, then :: $x^x\; y^y\; z^z\; \backslash ge\; \backslash left(xyz\backslash right)^.$ * If ''a'', ''b'' > 0, then :: $a^a\; +\; b^b\; \backslash ge\; a^b\; +\; b^a.$ * If ''a'', ''b'' > 0, then :: $a^\; +\; b^\; \backslash ge\; a^\; +\; b^.$ * If ''a'', ''b'', ''c'' > 0, then :: $a^\; +\; b^\; +\; c^\; \backslash ge\; a^\; +\; b^\; +\; c^.$ * If ''a'', ''b'' > 0, then :: $a^b\; +\; b^a\; >\; 1.$Well-known inequalities

Mathematicians often use inequalities to bound quantities for which exact formulas cannot be computed easily. Some inequalities are used so often that they have names: * Azuma's inequality * Bernoulli's inequality * Bell's inequality * Boole's inequality * Cauchy–Schwarz inequality * Chebyshev's inequality * Chernoff's inequality * Cramér–Rao inequality * Hoeffding's inequality * Hölder's inequality * Inequality of arithmetic and geometric means * Jensen's inequality * Kolmogorov's inequality * Markov's inequality * Minkowski inequality * Nesbitt's inequality * Pedoe's inequality * Poincaré inequality * Samuelson's inequality * Triangle inequalityComplex numbers and inequalities

The set of complex numbers ℂ with its operations of addition andmultiplication
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 Operation (mathematics), mathematical operations of arithmetic, with the ...

is a field (mathematics), field, but it is impossible to define any relation ≤ so that becomes an total order
In mathematics, a total or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some Set (mathematics), set X, which satisfies the following for all a, b and c in X:
# a \ ...

, for any number ''a'', either or (in which case the first property above implies that ). In either case ; this means that and ; so and , which means (−1 + 1) > 0; contradiction.
However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if , then "). Sometimes the lexicographical order definition is used:
* , if
** , or
** and
It can easily be proven that for this definition implies .
Vector inequalities

Inequality relationships similar to those defined above can also be defined for column vectors. If we let the vectors $x,\; y\; \backslash in\; \backslash mathbb^n$ (meaning that $x\; =\; (x\_1,\; x\_2,\; \backslash ldots,\; x\_n)^\backslash mathsf$ and $y\; =\; (y\_1,\; y\_2,\; \backslash ldots,\; y\_n)^\backslash mathsf$, where $x\_i$ and $y\_i$ are real numbers for $i\; =\; 1,\; \backslash ldots,\; n$), we can define the following relationships: * $x\; =\; y$, if $x\_i\; =\; y\_i$ for $i\; =\; 1,\; \backslash ldots,\; n$. * $x\; <\; y$, if $x\_i\; <\; y\_i$ for $i\; =\; 1,\; \backslash ldots,\; n$. * $x\; \backslash leq\; y$, if $x\_i\; \backslash leq\; y\_i$ for $i\; =\; 1,\; \backslash ldots,\; n$ and $x\; \backslash neq\; y$. * $x\; \backslash leqq\; y$, if $x\_i\; \backslash leq\; y\_i$ for $i\; =\; 1,\; \backslash ldots,\; n$. Similarly, we can define relationships for $x\; >\; y$, $x\; \backslash geq\; y$, and $x\; \backslash geqq\; y$. This notation is consistent with that used by Matthias Ehrgott in ''Multicriteria Optimization'' (see References). The trichotomy property (as stated #Ordered fields, above) is not valid for vector relationships. For example, when $x\; =\; (2,\; 5)^\backslash mathsf$ and $y\; =\; (3,\; 4)^\backslash mathsf$, there exists no valid inequality relationship between these two vectors. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.Systems of inequalities

Systems of linear inequalities can be simplified by Fourier–Motzkin elimination. The cylindrical algebraic decomposition is an algorithm that allows testing whether a system of polynomial equations and inequalities has solutions, and, if solutions exist, describing them. The complexity of this algorithm is double exponential function, doubly exponential in the number of variables. It is an active research domain to design algorithms that are more efficient in specific cases.See also

*Binary relation *Bracket (mathematics), for the use of similar ‹ and › signs as brackets *Inclusion (set theory) *Inequation *Interval (mathematics) *List of inequalities *List of triangle inequalities *Partially ordered set *Relational operators, used in programming languages to denote inequalityReferences

Sources

* * * * * * * * * * *External links

*Graph of Inequalities

by Ed Pegg, Jr.

AoPS Wiki entry about Inequalities

{{Authority control Inequalities, Elementary algebra Mathematical terminology