Editing Inequality (mathematics)

{{Short description|Mathematical relation expressed with < or ≤}}
{{hatnote group|
{{about|relations "greater than" and "less than"|the relation "not equal"|Inequation}}
{{redirect-distinguish|Less than|Less Than (song)}}
{{redirect-distinguish|≪|Absolute continuity of measures}}
}}
{{More footnotes|date=May 2017}}
[[File:Linear Programming Feasible Region.svg|frame|The [[feasible region]]s of [[linear programming]] are defined by a set of inequalities.]]

In [[mathematics]], an '''inequality''' is a relation which makes a non-equal comparison between two numbers or other mathematical expressions.<ref name=":0">{{Cite web|url=https://www.mathsisfun.com/definitions/inequality.html|title=Inequality Definition (Illustrated Mathematics Dictionary)|website=www.mathsisfun.com|access-date=2019-12-03}}</ref> It is used most often to compare two numbers on the [[number line]] by their size. The main types of inequality are '''less than''' and '''greater than''' (denoted by {{char|<}} and {{char|>}}, respectively the [[less-than sign|less-than]] and [[greater-than sign|greater-than]] signs).

==Notation==
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''',<ref name=":0" /> meaning that ''a'' is strictly less than or strictly greater than ''b''. Equality 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'' 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'' or  ''a'' ≧ ''b'' means that ''a'' is '''greater than or equal to''' ''b'' (or, equivalently, at least ''b'', or not less than ''b'').

In the 17th and 18th centuries, personal  notations  or  typewriting signs were used to signal inequalities.<ref>{{cite journal |first1=Elena | last1= Halmaghi |first2=Peter | last2= Liljedahl |title=Inequalities in the History of Mathematics: From Peculiarities to a Hard Discipline |journal=Proceedings of the 2012 Annual Meeting of the Canadian Mathematics Education Study Group}}</ref> For example, In 1670, [[John Wallis]] used a single horizontal bar ''above''  rather than below the < and >.
Later in 1734, ≦ and ≧, known as "less than (greater-than) over equal to" or  "less than (greater than) or equal to with double horizontal bars", first appeared in [[Pierre Bouguer]]'s work .<ref>{{cite web |title=Earliest Uses of Symbols of Relation |url=https://mathshistory.st-andrews.ac.uk/Miller/mathsym/relation/ |website=MacTutor |publisher=University of St Andrews, Scotland}}</ref> After that, mathematicians simplified Bouguer's symbol to "less than (greater than) or equal to with one horizontal bar" (≤), or "less than (greater than) or slanted equal to" (⩽).

The relation '''not greater than''' can also be represented by <math>a \ngtr b,</math> the symbol for "greater than" bisected by a slash, "not". The same is true for '''not less than''', <math>a \nless b.</math>

The notation ''a'' ≠ ''b'' means that ''a'' is not equal to ''b''; this ''[[inequation]]'' sometimes is considered a form of strict inequality.<ref name=":1">{{Cite web|url=http://www.learnalberta.ca/content/memg/Division03/Inequality/index.html| title=Inequality| website=www.learnalberta.ca|access-date=2019-12-03}}</ref> 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]].

In engineering sciences, less formal use of the notation is to state that one quantity is "much greater" than another,<ref name="Polyanin2006">{{cite book | last1=Polyanin | first1=A.D. | last2=Manzhirov | first2=A.V. | title=Handbook of Mathematics for Engineers and Scientists | publisher=CRC Press | year=2006 | isbn=978-1-4200-1051-0 | url=https://books.google.com/books?id=ge6nk9W0BCcC&pg=PR29 | access-date=2021-11-19 | page=29}}</ref> normally by several [[Order of magnitude|orders of magnitude]].  
* The notation ''a'' ≪ ''b'' means that ''a'' is '''much less than''' ''b''.<ref>{{Cite web|url=http://mathworld.wolfram.com/MuchLess.html|title=Much Less|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref>
* The notation ''a'' ≫ ''b'' means that ''a'' is '''much greater than''' ''b''.<ref>{{Cite web|url=http://mathworld.wolfram.com/MuchGreater.html|title=Much Greater|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref>
This implies that the lesser value can be neglected with little effect on the accuracy of an [[approximation]] (such as the case of [[ultrarelativistic limit]] 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 following [[Property (philosophy)|properties]]. 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]]s.

=== Converse ===
The relations ≤ and ≥ are each other's [[Converse relation|converse]], meaning that for any [[real number]]s ''a'' and ''b'':
{{block indent|text=''a'' ≤ ''b'' and ''b'' ≥ ''a'' are equivalent.}}

===Transitivity===

The transitive property of inequality states that for any [[real number]]s ''a'', ''b'', ''c'':<ref>{{cite book |last1=Drachman |first1=Bryon C. |last2=Cloud |first2=Michael J. |title=Inequalities: With Applications to Engineering |date=2006 |publisher=Springer Science & Business Media |isbn=0-3872-2626-5 |pages=2–3 |url=https://books.google.com/books?id=sIbfBwAAQBAJ}}</ref>
{{block indent|text=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:
{{block indent|text=If ''a'' ≤ ''b'' and ''b'' < ''c'', then ''a'' < ''c''.}}
{{block indent|text=If ''a'' < ''b'' and ''b'' ≤ ''c'', then ''a'' < ''c''.}}

===Addition and subtraction===
[[File:Translation invariance of less-than-relation.svg|thumb|300px|If ''x'' < ''y'', then ''x'' + ''a'' < ''y'' + ''a''.]]
A common constant ''c'' may be [[addition|added]] to or [[subtraction|subtracted]] from both sides of an inequality.<ref name=":1" /> So, for any [[real number]]s ''a'', ''b'', ''c'':
{{block indent|text=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 [[Partially ordered group|ordered group]] under addition.

===Multiplication and division===
[[File:Invariance of less-than-relation by multiplication with positive number.svg|thumb|If ''x'' < ''y'' and ''a'' > 0, then ''ax'' < ''ay''.]]
[[File:Inversion of less-than-relation by multiplication with negative number.svg|thumb|If ''x'' < ''y'' and ''a'' < 0, then ''ax'' > ''ay''.]]
The properties that deal with [[multiplication]] and [[division (mathematics)|division]] state that for any real numbers, ''a'', ''b'' and non-zero ''c'':
{{block indent|text=If ''a'' ≤ ''b'' and ''c'' > 0, then ''ac'' ≤ ''bc'' and ''a''/''c'' ≤ ''b''/''c''.}}
{{block indent|text=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 field]]. For more information, see ''[[#Ordered fields|§ Ordered fields]]''.

===Additive inverse===

The property for the [[additive inverse]] states that for any real numbers ''a'' and ''b'':
{{block indent|text=If ''a'' ≤ ''b'', then −''a'' ≥ −''b''.}}

===Multiplicative inverse===

If both numbers are positive, then the inequality relation between the [[multiplicative inverse]]s is opposite of that between the original numbers. More specifically, for any non-zero real numbers ''a'' and ''b'' that are both [[Positive number|positive]] (or both [[Negative number|negative]]):
{{block indent|text=If ''a'' ≤ ''b'', then {{sfrac|1|''a''}} ≥ {{sfrac|1|''b''}}.}}

All of the cases for the signs of ''a'' and ''b'' can also be written in [[#Chained notation|chained notation]], as follows:

{{block indent|text=If 0 < ''a'' ≤ ''b'', then {{sfrac|1|''a''}} ≥ {{sfrac|1|''b''}} > 0.}}
{{block indent|text=If ''a'' ≤ ''b'' < 0, then 0 > {{sfrac|1|''a''}} ≥ {{sfrac|1|''b''}}.}}
{{block indent|text=If ''a'' < 0 < ''b'', then {{sfrac|1|''a''}} < 0 < {{sfrac|1|''b''}}.}}

===Applying a function to both sides===
[[File:Log.svg|right|thumb|The graph of ''y'' = ln ''x'']]
Any [[Monotonic function|monotonic]]ally increasing [[function (mathematics)|function]], by its definition,<ref>{{Cite web | url=http://www.cs.yale.edu/homes/aspnes/pinewiki/ProvingInequalities.html|title=ProvingInequalities | website=www.cs.yale.edu | access-date=2019-12-03}}</ref> may be applied to both sides of an inequality without breaking the inequality relation (provided that both expressions are in the [[Domain of a function|domain]] 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:<!--
--> {{block indent|text=0 ≤ ''a'' ≤ ''b'' ⇔ 0 ≤ ''a<sup>n</sup>'' ≤ ''b<sup>n</sup>''.}}<!--
--> {{block indent|text=0 ≤ ''a'' ≤ ''b'' ⇔ ''a''<sup>−''n''</sup> ≥ ''b''<sup>−''n''</sup> ≥ 0.}}
* Taking the [[natural logarithm]] on both sides of an inequality, when ''a'' and ''b'' are positive real numbers: <!--
--> {{block indent|0 < ''a'' ≤ ''b'' ⇔ ln(''a'') ≤ ln(''b'').}} <!--
--> {{block indent|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 a [[binary relation]] ≤ over a [[Set (mathematics)|set]] ''P'' which is [[Reflexive relation|reflexive]], [[Antisymmetric relation|antisymmetric]], and [[Transitive relation|transitive]].<ref>{{cite book|title=Mathematical Tools for Data Mining: Set Theory, Partial Orders, Combinatorics|author1=Simovici, Dan A.|author2=Djeraba, Chabane|publisher=Springer|year=2008|isbn=9781848002012|chapter=Partially Ordered Sets|chapter-url=https://books.google.com/books?id=6i-F3ZNcub4C&pg=PA127|name-list-style=amp}}</ref> That is, for all ''a'', ''b'', and ''c'' in ''P'', it must satisfy the three following clauses:

* ''a'' ≤ ''a'' ([[Reflexive relation|reflexivity]])
* if ''a'' ≤ ''b'' and ''b'' ≤ ''a'', then ''a'' = ''b'' ([[Antisymmetric relation|antisymmetry]])
* if ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c'' ([[Transitive relation|transitivity]])

A set with a partial order is called a '''[[partially ordered set]]'''.<ref>{{Cite web|url=http://mathworld.wolfram.com/PartiallyOrderedSet.html|title=Partially Ordered Set|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-12-03}}</ref> Those are the very basic axioms that every kind of order has to satisfy. 

A strict partial order is a relation < that satisfies 
* ''a'' ≮ ''a'' ([[Reflexive relation#irreflexive|irreflexivity]]),
* if ''a'' < ''b'', then ''b'' ≮ ''a'' ([[Asymmetric relation|asymmetry]]),
* if ''a'' < ''b'' and ''b'' < ''c'', then ''a'' < ''c'' ([[Transitive relation|transitivity]]),
where {{char|≮}} means that {{char|<}} does not hold.

Some types of partial orders are specified by adding further axioms, such as:

* [[Total order]]: For every ''a'' and ''b'' in ''P'', ''a'' ≤ ''b'' or ''b'' ≤ ''a'' .
* [[Dense order]]: For all ''a'' and ''b'' in ''P'' for which ''a'' < ''b'', there is a ''c'' in ''P'' such that ''a'' < ''c'' < ''b''.
* [[Least-upper-bound property]]: Every non-empty [[Set (mathematics)|subset]] of ''P'' with an [[upper bound]] has a [[Least upper bound|''least'' upper bound]] (supremum) in ''P''.

=== Ordered fields ===
{{Main|Ordered field}}
If (''F'', +, ×) is a [[Field (mathematics)|field]] and ≤ is a [[total order]] on ''F'', then (''F'', +, ×, ≤) is called an '''[[ordered field]]''' if and only if:
* ''a'' ≤ ''b'' implies ''a'' + ''c'' ≤ ''b'' + ''c'';
* 0 ≤ ''a'' and 0 ≤ ''b'' implies 0 ≤ ''a'' × ''b''.

Both {{tmath|(\mathbb Q, +, \times, \leq)}} and {{tmath|(\mathbb R, +, \times, \leq)}} are [[ordered field]]s, but {{math|≤}} cannot be defined in order to make {{tmath|(\mathbb C, +, \times, \leq)}} an [[ordered field]],<ref>{{Cite web|url=http://www.math.ubc.ca/~feldman/m320/fields.pdf |archive-url=https://ghostarchive.org/archive/20221009/http://www.math.ubc.ca/~feldman/m320/fields.pdf |archive-date=2022-10-09 |url-status=live|title=Fields|last=Feldman|first=Joel|date=2014|website=math.ubc.ca|access-date=2019-12-03}}</ref> because −1 is the square of ''i'' and would therefore be positive.

Besides being an ordered field, '''R''' also has the [[Least-upper-bound property]]. In fact, '''R''' can be defined as the only ordered field with that quality.<ref>{{cite book |last1=Stewart |first1=Ian |title=Why Beauty Is Truth: The History of Symmetry |date=2007 |publisher=Hachette UK |isbn=978-0-4650-0875-9 |page=106 |url=https://books.google.com/books?id=1ek3DgAAQBAJ&pg=PT106}}</ref>

== 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''<sub>1</sub> ≤ ''a''<sub>2</sub> ≤ ... ≤ ''a''<sub>''n''</sub>''' means that ''a''<sub>''i''</sub> ≤ ''a''<sub>''i''+1</sub> for ''i'' = 1, 2, ..., ''n'' − 1.  By transitivity, this condition is equivalent to ''a''<sub>''i''</sub> ≤ ''a''<sub>''j''</sub> 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'' < {{sfrac|1|2}} and ''x'' ≥ −1 respectively, which can be combined into the final solution −1 ≤ ''x'' < {{sfrac|1|2}}.

Occasionally, chained notation is used with inequalities in different directions, in which case the meaning is the [[logical conjunction]] of the inequalities between adjacent terms.  For example, the defining condition of a [[zigzag poset]] is written as ''a''<sub>1</sub> < ''a''<sub>2</sub> > ''a''<sub>3</sub> < ''a''<sub>4</sub> > ''a''<sub>5</sub> < ''a''<sub>6</sub> > ... .   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 [[programming language]]s such as [[Python (programming language)|Python]].  In contrast, in programming languages that provide an ordering on the type of comparison results, such as [[C (programming language)|C]], even homogeneous chains may have a completely different meaning.<ref>{{cite book | isbn=0131103628 | author=Brian W. Kernighan and Dennis M. Ritchie | title=The C Programming Language | edition=2nd | location=Englewood Cliffs/NJ | publisher=Prentice Hall | series=Prentice Hall Software Series | date=Apr 1988 }} Here: Sect.A.7.9 ''Relational Operators'', p.167: Quote: "a<b<c is parsed as (a<b)<c"</ref>

==Sharp inequalities==

An inequality is said to be ''sharp'' if it cannot be ''relaxed'' and still be valid in general. Formally, a [[universally quantified]] inequality ''φ'' is called sharp if, for every valid universally quantified inequality ''ψ'', if {{nowrap|''ψ'' [[material conditional|⇒]] ''φ''}} holds, then {{nowrap|''ψ'' [[equivalence (logic)|⇔]] ''φ''}} also holds. For instance, the inequality {{nowrap|[[universal quantification|∀]]''a'' ∈ [[real number|'''R''']]. ''a''<sup>2</sup> ≥ 0}} is sharp, whereas the inequality {{nowrap|∀''a'' ∈ '''R'''. ''a''<sup>2</sup> ≥ −1}} is not sharp.{{citation needed|date=May 2017}}

==Inequalities between means==
{{see also|Inequality of arithmetic and geometric means}}

There are many inequalities between means. For example, for any positive numbers ''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''n''</sub> we have

: <math>
H\le G\le A\le Q,
</math>

where they represent the following means of the sequence:

* [[Harmonic mean]] : <math>H = \frac{n}{\frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_n}}</math>
* [[Geometric mean]] : <math>G = \sqrt[n]{a_1 \cdot a_2 \cdots a_n} </math>
* [[Arithmetic mean]] : <math>A = \frac{a_1 + a_2 + \cdots + a_n}{n}</math>
* [[Root mean square|Quadratic mean]] : <math>Q = \sqrt{\frac{a_1^2 + a_2^2 + \cdots + a_n^2}{n}}</math>

==Cauchy–Schwarz inequality==
{{see also|Cauchy–Schwarz inequality}}

The Cauchy–Schwarz inequality states that for all vectors ''u'' and ''v'' of an [[inner product space]] it is true that
<math display="block">|\langle \mathbf{u},\mathbf{v}\rangle| ^2 \leq \langle \mathbf{u},\mathbf{u}\rangle \cdot \langle \mathbf{v},\mathbf{v}\rangle,</math>
where <math>\langle\cdot,\cdot\rangle</math> is the [[inner product]]. Examples of inner products include the real and complex [[dot product]]; In [[Euclidean space]] ''R''<sup>''n''</sup> with the standard inner product, the Cauchy–Schwarz inequality is
<math display="block">\biggl(\sum_{i=1}^n u_i v_i\biggr)^2\leq \biggl(\sum_{i=1}^n u_i^2\biggr) \biggl(\sum_{i=1}^n v_i^2\biggr).</math>

==Power inequalities==
A '''power inequality''' is an inequality containing terms of the form ''a''<sup>''b''</sup>, where ''a'' and ''b'' are real positive numbers or variable expressions. They often appear in [[mathematical olympiads]] exercises.

Examples:
* For any real  ''x'', <math display="block">e^x \ge 1+x.</math>
* If ''x'' > 0 and ''p'' > 0, then <math display="block">\frac{x^p - 1}{p} \ge \ln(x) \ge \frac{1 - \frac{1}{x^p}}{p}.</math> In the limit of ''p'' → 0, the upper and lower bounds converge to ln(''x''). 
* If ''x'' > 0, then <math display="block">x^x \ge \left( \frac{1}{e}\right)^\frac{1}{e}.</math>
* If ''x'' > 0, then <math display="block">x^{x^x} \ge x.</math>
* If ''x'', ''y'', ''z'' > 0, then <math display="block">\left(x+y\right)^z + \left(x+z\right)^y + \left(y+z\right)^x > 2.</math>
* For any real distinct numbers ''a'' and ''b'', <math display="block">\frac{e^b-e^a}{b-a} > e^{(a+b)/2}.</math>
* If ''x'', ''y'' > 0 and 0 < ''p'' < 1, then <math display="block">x^p+y^p > \left(x+y\right)^p.</math>
* If ''x'', ''y'', ''z'' > 0, then <math display="block">x^x y^y z^z \ge \left(xyz\right)^{(x+y+z)/3}.</math>
* If ''a'', ''b'' > 0, then<ref>{{Cite journal |jstor = 2324012|last1 = Laub|first1 = M.|last2 = Ilani|first2 = Ishai|title = E3116|journal = The American Mathematical Monthly|year = 1990|volume = 97|issue = 1|pages = 65–67|doi = 10.2307/2324012}}</ref> <math display="block">a^a + b^b \ge a^b + b^a.</math>
* If ''a'', ''b'' > 0, then<ref>{{cite journal|first=S.|last=Manyama|title=Solution of One Conjecture on Inequalities with Power-Exponential Functions|journal=Australian Journal of Mathematical Analysis and Applications|url=https://ajmaa.org/searchroot/files/pdf/v7n2/v7i2p1.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://ajmaa.org/searchroot/files/pdf/v7n2/v7i2p1.pdf |archive-date=2022-10-09 |url-status=live|volume=7|issue=2|page=1|date=2010}}</ref> <math display="block">a^{ea} + b^{eb} \ge a^{eb} + b^{ea}.</math>
* If ''a'', ''b'', ''c'' > 0, then <math display="block">a^{2a} + b^{2b} + c^{2c} \ge a^{2b} + b^{2c} + c^{2a}.</math>
* If ''a'', ''b'' > 0, then <math display="block">a^b + b^a > 1.</math>

== Well-known inequalities ==
{{see also|List of inequalities}}

[[Mathematician]]s often use inequalities to bound quantities for which exact formulas cannot be computed easily.  Some inequalities are used so often that they have names:
{{div col}}
* [[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]]
* [[Sobolev inequality]]
* [[Triangle inequality]]
{{div col end}}

==Complex numbers and inequalities==
The set of [[complex number]]s <math>\mathbb{C}</math> with its operations of [[addition]] and [[multiplication]] is a [[field (mathematics)|field]], but it is impossible to define any relation {{math|≤}} so that <math>(\Complex, +, \times, \leq)</math> becomes an [[ordered field]]. To make <math>(\mathbb{C}, +, \times, \leq)</math> an [[ordered field]], it would have to satisfy the following two properties:
* if {{nowrap|''a'' ≤ ''b''}}, then {{nowrap|''a'' + ''c'' ≤ ''b'' + ''c''}};
* if {{nowrap|0 ≤ ''a''}} and {{nowrap|0 ≤ ''b''}}, then {{nowrap|0 ≤ ''ab''}}.

Because ≤ is a [[total order]], for any number ''a'', either {{nowrap|0 ≤ ''a''}} or {{nowrap|''a'' ≤ 0}} (in which case the first property above implies that {{nowrap|0 ≤ −''a''}}). In either case {{nowrap|0 ≤ ''a''<sup>2</sup>}}; this means that {{nowrap|''i''<sup>2</sup> > 0}} and {{nowrap|1<sup>2</sup> > 0}}; so {{nowrap|−1 > 0}} and {{nowrap|1 > 0}}, which means (−1 + 1) > 0; contradiction.

However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if {{nowrap|''a'' ≤ ''b''}}, then {{nowrap|''a'' + ''c'' ≤ ''b'' + ''c''}}"). Sometimes the [[lexicographical order]] definition is used:
* {{nowrap|''a'' ≤ ''b''}}, if
** {{nowrap|Re(''a'') < Re(''b'')}}, or
** {{nowrap|Re(''a'') {{=}} Re(''b'')}} and {{nowrap|Im(''a'') ≤ Im(''b'')}}
It can easily be proven that for this definition {{nowrap|''a'' ≤ ''b''}} implies {{nowrap|''a'' + ''c'' ≤ ''b'' + ''c''}}.

== Systems of inequalities ==
Systems of [[linear inequalities]] can be simplified by [[Fourier–Motzkin elimination]].<ref>{{Cite Gartner Matousek 2006}}</ref>

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 [[bracket]]s
*[[Inclusion (set theory)]]
*[[Inequation]]
*[[Interval (mathematics)]]
*[[List of inequalities]]
*[[List of triangle inequalities]]
*[[Partially ordered set]]
*[[Relational operator]]s, used in programming languages to denote inequality
<!--
==Notes==  
{{Reflist}}-->

==References==
<references/>

== Sources ==
* {{cite book | author=Hardy, G., Littlewood J. E., Pólya, G.| title=Inequalities| publisher=Cambridge Mathematical Library, Cambridge University Press | year=1999 | isbn=0-521-05206-8}}
* {{cite book | author=Beckenbach, E. F., Bellman, R.| title=An Introduction to Inequalities| publisher=Random House Inc | year=1975 | isbn=0-394-01559-2}}
* {{cite book| author=Drachman, Byron C., Cloud, Michael J.| title=Inequalities: With Applications to Engineering| publisher=Springer-Verlag| year=1998| isbn=0-387-98404-6| url-access=registration| url=https://archive.org/details/inequalitieswith0000clou}}
* {{Citation | last1=Grinshpan | first1=A. Z. | title=General inequalities, consequences, and applications | doi=10.1016/j.aam.2004.05.001 | year=2005 | 
journal=Advances in Applied Mathematics | volume=34 | issue=1 | pages=71–100 | doi-access= }}
* {{cite journal |title='Quickie' inequalities |author=Murray S. Klamkin |url=https://www.math.ualberta.ca/pi/issue7/page26-29.pdf |archive-url=https://ghostarchive.org/archive/20221009/https://www.math.ualberta.ca/pi/issue7/page26-29.pdf |archive-date=2022-10-09 |url-status=live |journal=Math Strategies}}
* {{cite web |title=Introduction to Inequalities |url=http://www.mediafire.com/file/1mw1tkgozzu |author=Arthur Lohwater |year=1982 |publisher=Online e-book in PDF format}}
* {{cite web |title=Mathematical Problem Solving |url=http://www.math.kth.se/math/TOPS/index.html |author=Harold Shapiro |date=2005 |publisher=Kungliga Tekniska högskolan |work=The Old Problem Seminar}}
* {{cite web |title=3rd USAMO |url=http://www.kalva.demon.co.uk/usa/usa74.html |archive-url=https://web.archive.org/web/20080203070350/http://www.kalva.demon.co.uk/usa/usa74.html |archive-date=2008-02-03 |url-status=dead }}
* {{cite book
 | last = Pachpatte
 | first = B. G.
 | title = Mathematical Inequalities
 | publisher = [[Elsevier]]
 | series = North-Holland Mathematical Library
 | volume = 67
 | edition = first
 | year = 2005
 | location = Amsterdam, the Netherlands
 | isbn = 0-444-51795-2
 | issn = 0924-6509
 | mr = 2147066
 | zbl = 1091.26008}}
* {{cite book | author=Ehrgott, Matthias| title=Multicriteria Optimization| publisher=Springer-Berlin| year=2005| isbn=3-540-21398-8}}
* {{cite book | last=Steele | first=J. Michael | author-link=J. Michael Steele | title=The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities | publisher=Cambridge University Press | year=2004 | isbn=978-0-521-54677-5 | url=http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html}}

== External links ==
{{commons category|Inequalities (mathematics)}}
* {{springer|title=Inequality|id=p/i050790}}
* [https://demonstrations.wolfram.com/GraphOfInequalities/ Graph of Inequalities] by [[Ed Pegg, Jr.]]
* [https://artofproblemsolving.com/wiki/index.php/Inequality AoPS Wiki entry about Inequalities]

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[[Category:Inequalities (mathematics)| ]]
[[Category:Elementary algebra]]
[[Category:Mathematical terminology]]