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| | | Hello! My name is Michael. <br>It is a little about myself: I live in Australia, my city of Dianella. <br>It's called often Eastern or cultural capital of WA. I've married 3 years ago.<br>I have 2 children - a son (Hans) and the daughter (Cindi). We all like Kiteboarding.<br><br>Also visit my site: [http://transcendentbliss.org/2014/03/19/lets-be-young-and-dumb/ fifa 15 coin Generator] |
| :''Not to be confused with [[Inequation]]. "Less than", "Greater than", and "More than" redirect here. For the use of the "<" and ">" signs as punctuation, see [[Bracket]]. For the UK insurance brand "More Th>n", see [[RSA Insurance Group]].''
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| [[File:Linear Programming Feasible Region.svg|frame|The [[feasible region]]s of [[linear programming]] are defined by a set of inequalities.]]
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| In [[mathematics]], an '''inequality''' is a relation that holds between two values when they are different (see also: [[equality (mathematics)|equality]]).
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| *The notation ''a'' ≠ ''b'' means that ''a'' is '''not equal to''' ''b''.
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| It does not say that one is greater than the other, or even that they can be compared in size.
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| If the values in question are elements of an [[ordered set]], such as the [[integer]]s or the [[real number]]s, they can be compared in size.
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| *The notation ''a'' < ''b'' means that ''a'' is '''less than''' ''b''.
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| *The notation ''a'' > ''b'' means that ''a'' is '''greater than''' ''b''.
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| In either case, ''a'' is not equal to ''b''. These relations are known as '''strict inequalities'''. The notation ''a'' < ''b'' may also be read as "''a'' is strictly less than ''b''".
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| In contrast to strict inequalities, there are two types of inequality relations that are not strict:
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| *The notation ''a'' ≤ ''b'' means that ''a'' is '''less than or equal to''' ''b'' (or, equivalently, '''not greater than''' ''b'', or '''at most''' ''b'').
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| *The notation ''a'' ≥ ''b'' means that ''a'' is '''greater than or equal to''' ''b'' (or, equivalently, '''not less than''' ''b'', or '''at least''' ''b'')
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| An additional use of the notation is to show that one quantity is much greater than another, normally by several [[orders of magnitude]].
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| *The notation ''a'' {{Unicode|≪}} ''b'' means that ''a'' is '''much less than''' ''b''. (In [[measure theory]], however, this notation is used for [[Absolute continuity#Absolute continuity of measures|absolute continuity]], an unrelated concept.)
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| *The notation ''a'' {{Unicode|≫}} ''b'' means that ''a'' is '''much greater than''' ''b''.
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| ==Properties==
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| 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 equalities (< and >) and (in the case of applying a function) monotonic
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| functions are limited to ''strictly'' monotonic functions.
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| ===Transitivity===
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| The Transitive property of inequality states:
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| * For any [[real number]]s ''a'', ''b'', ''c'':
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| ** If ''a'' ≥ ''b'' and ''b'' ≥ ''c'', then ''a'' ≥ ''c''.
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| ** If ''a'' ≤ ''b'' and ''b'' ≤ ''c'', then ''a'' ≤ ''c''.
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| * If ''either'' of the premises is a strict inequality, then the conclusion is a strict inequality.
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| ** E.g. if ''a'' ≥ ''b'' and ''b'' > ''c'', then ''a'' > ''c''
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| * An equality is of course a special case of a non-strict inequality.
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| ** E.g. if ''a'' = ''b'' and ''b'' > ''c'', then ''a'' > ''c''
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| ===Converse===
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| The relations ≤ and ≥ are each other's [[Converse relation|converse]]:
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| * For any [[real number]]s ''a'' and ''b'':
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| **If ''a'' ≤ ''b'', then ''b'' ≥ ''a''.
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| **If ''a'' ≥ ''b'', then ''b'' ≤ ''a''.
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| ===Addition and subtraction===
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| A common constant ''c'' may be [[addition|added to]] or [[subtraction|subtracted from]] both sides of an inequality:
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| * For any [[real number]]s ''a'', ''b'', ''c''
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| **If ''a'' ≤ ''b'', then ''a'' + ''c'' ≤ ''b'' + ''c'' and ''a'' − ''c'' ≤ ''b'' − ''c''.
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| **If ''a'' ≥ ''b'', then ''a'' + ''c'' ≥ ''b'' + ''c'' and ''a'' − ''c'' ≥ ''b'' − ''c''.
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| i.e., the real numbers are an [[ordered group]] under addition.
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| ===Multiplication and division===
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| The properties that deal with [[multiplication]] and [[division (mathematics)|division]] state:
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| * For any real numbers, ''a'', ''b'' and non-zero ''c'':
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| ** If ''c'' is [[positive number|positive]], then multiplying or dividing by ''c'' does not change the inequality:
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| *** If ''a'' ≥ ''b'' and ''c'' > 0, then ''ac'' ≥ ''bc'' and ''a/c'' ≥ ''b/c''.
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| *** If ''a'' ≤ ''b'' and ''c'' > 0, then ''ac'' ≤ ''bc'' and ''a/c'' ≤ ''b/c''.
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| ** If ''c'' is [[negative number|negative]], then multiplying or dividing by ''c'' inverts the inequality:
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| *** If ''a'' ≥ ''b'' and ''c'' < 0, then ''ac'' ≤ ''bc'' and ''a/c'' ≤ ''b/c''.
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| *** If ''a'' ≤ ''b'' and ''c'' < 0, then ''ac'' ≥ ''bc'' and ''a/c'' ≥ ''b/c''.
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| More generally, this applies for an [[ordered field]], see below.
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| ===Additive inverse===
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| The properties for the [[additive inverse]] state:
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| *For any real numbers ''a'' and ''b'', negation inverts the inequality:
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| **If ''a'' ≤ ''b'', then −''a'' ≥ −''b''.
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| **If ''a'' ≥ ''b'', then −''a'' ≤ −''b''.
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| ===Multiplicative inverse===
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| The properties for the [[multiplicative inverse]] state:
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| *For any non-zero real numbers ''a'' and ''b'' that are both [[Positive number|positive]] or both [[Negative number|negative]]:
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| **If ''a'' ≤ ''b'', then 1/''a'' ≥ 1/''b''.
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| **If ''a'' ≥ ''b'', then 1/''a'' ≤ 1/''b''.
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| *If one of ''a'' and ''b'' is positive and the other is negative, then:
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| **If ''a'' < ''b'', then 1/''a'' < 1/''b''.
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| **If ''a'' > ''b'', then 1/''a'' > 1/''b''.
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| These can also be written in [[#Chained notation|chained notation]] as:
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| * For any non-zero real numbers ''a'' and ''b'':
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| ** If 0 < ''a'' ≤ ''b'', then 1/''a'' ≥ 1/''b'' > 0.
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| ** If ''a'' ≤ ''b'' < 0, then 0 > 1/''a'' ≥ 1/''b''.
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| ** If ''a'' < 0 < ''b'', then 1/''a'' < 0 < 1/''b''.
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| ** If 0 > ''a'' ≥ ''b'', then 1/''a'' ≤ 1/''b'' < 0.
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| ** If ''a'' ≥ ''b'' > 0, then 0 < 1/''a'' ≤ 1/''b''.
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| ** If ''a'' > 0 > ''b'', then 1/''a'' > 0 > 1/''b''.
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| ===Applying a function to both sides===
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| [[File:Log.svg|right|thumb|The graph of ''y'' = ln ''x'']]
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| Any [[Monotonic function|monotonic]]ally increasing [[function (mathematics)|function]] may be applied to both sides of an inequality (provided they are in the [[Domain of a function|domain]] of that function) and it will still hold. Applying a monotonically decreasing function to both sides of an inequality means the opposite inequality now holds. The rules for additive and multiplicative inverses are both examples of applying a monotonically decreasing function.
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| 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. The rules for additive and multiplicative inverses are both examples of applying a ''strictly'' monotonically decreasing function.
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| As an example, consider the application of the [[natural logarithm]] to both sides of an inequality when <math>a</math> and <math>b</math> are positive real numbers:
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| :<math>a \leq b \Leftrightarrow \ln(a) \leq \ln(b).</math>
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| :<math>a < b \Leftrightarrow \ln(a) < \ln(b).</math>
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| This is true because the natural logarithm is a strictly increasing function.
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| ==Ordered fields==
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| 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:
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| * ''a'' ≤ ''b'' implies ''a'' + ''c'' ≤ ''b'' + ''c'';
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| * 0 ≤ ''a'' and 0 ≤ ''b'' implies 0 ≤ ''a'' × ''b''.
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| Note that both ('''Q''', +, ×, ≤) and ('''R''', +, ×, ≤) are [[ordered field]]s, but ≤ cannot be defined in order to make ('''C''', +, ×, ≤) an [[ordered field]], because −1 is the square of ''i'' and would therefore be positive.
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| The non-strict inequalities ≤ and ≥ on real numbers are [[total order]]s. The strict inequalities < and > on real numbers are [[Total_order#Strict_total_order|strict total orders]].
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| == Chained notation ==
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| 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''. Obviously, by the above laws, one can add/subtract the same number to all three terms, or multiply/divide all three terms by same nonzero number and reverse all inequalities according to sign. Hence, for example, ''a'' < ''b'' + ''e'' < ''c'' is equivalent to ''a'' − ''e'' < ''b'' < ''c'' − ''e''.
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| 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''.
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| 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.
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| 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 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]].
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| <!--
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| == Representing inequalities on the real number line ==
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| Every inequality involving real numbers can be represented on the real [[number line]] showing darkened regions on the line. A "<" or ">" is graphed by an open circle on the number. A "≤" or "≥" is graphed with a closed or black circle.
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| {{Expand section|date=May 2008}} -->
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| ==Inequalities between means==
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| {{see also|Inequality of arithmetic and geometric means}}
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| 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 {{nowrap|''H'' ≤ ''G'' ≤ ''A'' ≤ ''Q'',}} where
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| :{| style="height:200px"
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| |-
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| |<math>H = \frac{n}{1/a_1 + 1/a_2 + \cdots + 1/a_n}</math>   || ([[harmonic mean]]),
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| |-
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| |<math>G = \sqrt[n]{a_1 \cdot a_2 \cdots a_n} </math> || ([[geometric mean]]),
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| |-
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| |<math>A = \frac{a_1 + a_2 + \cdots + a_n}{n}</math> || ([[arithmetic mean]]),
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| |-
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| |<math>Q = \sqrt{\frac{a_1^2 + a_2^2 + \cdots + a_n^2}{n}}</math> || ([[Root mean square|quadratic mean]]).
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| |}
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| ==Power inequalities==
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| A "'''Power inequality'''" is an inequality containing ''a''<sup>''b''</sup> terms, where ''a'' and ''b'' are real positive numbers or variable expressions. They often appear in [[mathematical olympiads]] exercises.
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| ===Examples===
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| * For any real ''x'',
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| :: <math>e^x \ge 1+x.\,</math>
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| * If ''x'' > 0, then
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| :: <math>x^x \ge \left( \frac{1}{e}\right)^{1/e}.\,</math>
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| * If ''x'' ≥ 1, then
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| :: <math>x^{x^x} \ge x.\,</math>
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| * If ''x'', ''y'', ''z'' > 0, then
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| :: <math>(x+y)^z + (x+z)^y + (y+z)^x > 2.\,</math>
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| * For any real distinct numbers ''a'' and ''b'',
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| :: <math>\frac{e^b-e^a}{b-a} > e^{(a+b)/2}.</math>
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| * If ''x'', ''y'' > 0 and 0 < ''p'' < 1, then
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| :: <math>(x+y)^p < x^p+y^p.\,</math>
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| * If ''x'', ''y'', ''z'' > 0, then
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| :: <math>x^x y^y z^z \ge (xyz)^{(x+y+z)/3}.\,</math>
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| * If ''a'', ''b'' > 0, then
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| :: <math>a^a + b^b \ge a^b + b^a.\,</math>
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| : This inequality was solved by I.Ilani in JSTOR,AMM,Vol.97,No.1,1990.
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| * If ''a'', ''b'' > 0, then
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| :: <math>a^{ea} + b^{eb} \ge a^{eb} + b^{ea}.\,</math>
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| : This inequality was solved by S.Manyama in AJMAA,Vol.7,Issue 2,No.1,2010 and by V.Cirtoaje in JNSA,Vol.4,Issue 2,130-137,2011.
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| * If ''a'', ''b'', ''c'' > 0, then
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| :: <math>a^{2a} + b^{2b} + c^{2c} \ge a^{2b} + b^{2c} + c^{2a}.\,</math>
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| * If ''a'', ''b'' > 0, then
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| :: <math>a^b + b^a > 1.\,</math>
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| : This result was generalized by R. Ozols in 2002 who proved that if ''a''<sub>1</sub>, ..., ''a''<sub>''n''</sub> > 0, then
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| :: <math>a_1^{a_2}+a_2^{a_3}+\cdots+a_n^{a_1}>1</math>
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| : (result is published in Latvian popular-scientific quarterly ''The Starry Sky'', see references).
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| == Well-known inequalities ==
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| {{see also|List of inequalities}}
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| [[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:
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| {{div col}}
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| * [[Azuma's inequality]]
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| * [[Bernoulli's inequality]]
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| * [[Boole's inequality]]
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| * [[Cauchy–Schwarz inequality]]
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| * [[Chebyshev's inequality]]
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| * [[Chernoff's inequality]]
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| * [[Cramér–Rao inequality]]
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| * [[Hoeffding's inequality]]
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| * [[Hölder's inequality]]
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| * [[Inequality of arithmetic and geometric means]]
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| * [[Jensen's inequality]]
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| * [[Kolmogorov's inequality]]
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| * [[Markov's inequality]]
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| * [[Minkowski inequality]]
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| * [[Nesbitt's inequality]]
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| * [[Pedoe's inequality]]
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| * [[Poincaré inequality]]
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| * [[Triangle inequality]]
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| {{div col end}}
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| ==Complex numbers and inequalities==
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| 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 ≤ so that <math>(\mathbb{C},+,\times,\le)</math> becomes an [[ordered field]]. To make <math>(\mathbb{C},+,\times,\le)</math> an [[ordered field]], it would have to satisfy the following two properties:
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| * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''
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| * if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''a b''
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| Because ≤ is a [[total order]], for any number ''a'', either 0 ≤ ''a'' or ''a'' ≤ 0 (in which case the first property above implies that 0 ≤ <math>-a</math>). In either case 0 ≤ ''a''<sup>2</sup>; this means that <math>i^2>0</math> and <math>1^2>0</math>; so <math>-1>0</math> and <math>1>0</math>, which means <math>(-1+1)>0</math>; contradiction.
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| However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''"). Sometimes the [[lexicographical order]] definition is used:
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| * a ≤ b if <math> Re(a)</math> < <math>Re(b)</math> or (<math>Re(a) = Re(b)</math> and <math>Im(a)</math> ≤ <math>Im(b)</math>)
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| It can easily be proven that for this definition ''a'' ≤ ''b'' implies ''a'' + ''c'' ≤ ''b'' + ''c''.
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| ==Vector inequalities==
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| Inequality relationships similar to those defined above can also be defined for [[column vector]]. If we let the vectors <math>x,y\in\mathbb{R}^n</math> (meaning that <math>x = \left(x_1,x_2,\ldots,x_n\right)^\mathsf{T}</math> and <math>y = \left(y_1,y_2,\ldots,y_n\right)^\mathsf{T}</math> where <math>x_i</math> and <math>y_i</math> are real numbers for <math>i=1,\ldots,n</math>), we can define the following relationships.
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| * <math>x = y \ </math> if <math>x_i = y_i\ </math> for <math>i=1,\ldots,n</math>
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| * <math>x < y \ </math> if <math>x_i < y_i\ </math> for <math>i=1,\ldots,n</math>
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| * <math>x \leq y </math> if <math>x_i \leq y_i </math> for <math>i=1,\ldots,n</math> and <math>x \neq y</math>
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| * <math>x \leqq y </math> if <math>x_i \leq y_i </math> for <math>i=1,\ldots,n</math>
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| Similarly, we can define relationships for <math> x > y </math>, <math> x \geq y </math>, and <math> x \geqq y </math>. We note that this notation is consistent with that used by Matthias Ehrgott in ''Multicriteria Optimization'' (see References).
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| The property of Trichotomy (as stated above) is not valid for vector relationships. For example, when <math>x = \left[ 2, 5 \right]^\mathsf{T} </math> and <math>y = \left[ 3, 4 \right]^\mathsf{T} </math>, there exists no valid inequality relationship between these two vectors. Also, a [[multiplicative inverse]] would need to be defined on a vector before this property could be considered. However, for the rest of the aforementioned properties, a parallel property for vector inequalities exists.
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| ==See also==
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| *[[Binary relation]]
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| *[[Bracket (mathematics)]], for the use of similar ‹ and › signs as [[bracket]]s
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| *[[Fourier-Motzkin elimination]]
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| *[[Inclusion (set theory)]]
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| *[[Inequation]]
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| *[[Interval (mathematics)]]
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| *[[List of inequalities]]
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| *[[Partially ordered set]]
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| *[[Relational operator]]s, used in programming languages to denote inequality
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| ==References==
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| {{Reflist}}
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| *{{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}}
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| *{{cite book | author=Beckenbach, E.F., Bellman, R.| title=An Introduction to Inequalities| publisher=Random House Inc | year=1975 | isbn=0-394-01559-2}}
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| *{{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}}
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| *{{cite journal|title="Quickie" inequalities|author=Murray S. Klamkin|url=http://ua-mirror.pims.math.ca/pi/issue7/page26-29.pdf|format=PDF|work=Math Strategies}}
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| *{{cite web|title=Introduction to Inequalities|url=http://www.mediafire.com/?1mw1tkgozzu |author=Arthur Lohwater|year=1982|publisher=Online e-book in PDF format}}
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| *{{cite web|title=Mathematical Problem Solving|url=http://www.math.kth.se/math/TOPS/index.html|author=Harold Shapiro|date=2005,1972–1985|publisher=Kungliga Tekniska högskolan|work=The Old Problem Seminar}}
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| *{{cite web|title=3rd USAMO|url=http://www.kalva.demon.co.uk/usa/usa74.html|archiveurl=http://web.archive.org/web/20080203070350/www.kalva.demon.co.uk/usa/usa74.html|archivedate=2008-02-03}}
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| *{{cite book
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| | last = Pachpatte
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| | first = B.G.
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| | title = Mathematical Inequalities
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| | publisher = [[Elsevier]]
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| | series = North-Holland Mathematical Library
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| | volume = 67
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| | edition = first
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| | year = 2005
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| | location = Amsterdam, The Netherlands
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| | isbn = 0-444-51795-2
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| | issn = 0924-6509
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| | mr = 2147066
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| | zbl = 1091.26008}}
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| *{{cite book | author=Ehrgott, Matthias| title=Multicriteria Optimization| publisher=Springer-Berlin| year=2005| isbn=3-540-21398-8}}
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| *{{cite book | last=Steele | first=J. Michael | authorlink=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}}
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| == External links ==
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| * {{springer|title=Inequality|id=p/i050790}}
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| * [http://demonstrations.wolfram.com/GraphOfInequalities/ Graph of Inequalities] by [[Ed Pegg, Jr.]], [[Wolfram Demonstrations Project]].
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| * [http://www.artofproblemsolving.com/Wiki/index.php/Inequality AoPS Wiki entry about Inequalities]
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| [[Category:Inequalities| ]]
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| [[Category:Elementary algebra]]
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