Inequality 89489 226036089 2008-07-16T15:22:23Z Paul August 87355 rm inline TeX {{otheruses4|inequalities in mathematics|other senses of this word|inequality (disambiguation)}} {{for|the use of the < and > signs in punctuation|Bracket}} {{for|the not-equals statement|Inequation}} [[Image:Linear programming example graph.png|right|thumb|250px|The [[feasible region]]s of [[linear programming]] are defined by a set of inequalities.]] In [[mathematics]], an '''inequality''' is a statement about the relative size or order of two objects, ''or'' about whether they are the same or not (See also: [[equality (mathematics)|equality]]) *The notation ''a'' < ''b'' means that ''a'' is '''less than''' ''b''. *The notation ''a'' > ''b'' means that ''a'' is '''greater than''' ''b''. *The notation ''a'' ≠ ''b'' means that ''a'' is '''not equal to''' ''b,'' but does not say that one is bigger than the other or even that they can be compared in size – they could be [[wikt:apples and oranges|apples and oranges]] In all these cases, ''a'' is not equal to ''b,'' hence, "inequality". These relations are known as '''strict inequality'''; in contrast *The notation ''a'' ≤ ''b'' means that ''a'' is '''less than or equal to''' ''b'' (or, equivalently, '''not greater than''' ''b''); *The notation ''a'' ≥ ''b'' means that ''a'' is '''greater than or equal to''' ''b'' (or, equivalently, '''not smaller than''' ''b''); An additional use of the notation is to show that one quantity is much greater than another, normally by several [[orders of magnitude]]. *The notation ''a'' ≪ ''b'' means that ''a'' is '''much less than''' ''b''. *The notation ''a'' ≫ ''b'' means that ''a'' is '''much greater than''' ''b''. If the sense of the inequality is the same for all values of the variables for which its members are defined, then the inequality is called an "absolute" or "unconditional" inequality. If the sense of an inequality holds only for certain values of the variables involved, but is reversed or destroyed for other values of the variables, it is called a conditional inequality. == Solving Inequalities == An inequality may appear unsolvable because it only states whether a number is larger or smaller than another number; but it is possible to apply the same operations for equalities to inequalities. For example, to find x for the inequality 10x > 23 one would divide 23 by 10. ==Properties== Inequalities are governed by the following [[properties]]. Note that, for the transitivity, reversal, addition and subtraction, and multiplication and division properties, the property also holds if strict inequality signs (< and >) are replaced with their corresponding non-strict inequality sign (≤ and ≥). ===Trichotomy=== The [[Trichotomy (mathematics)|trichotomy]] property states: * For any [[real number]]s, ''a'' and ''b'', exactly one of the following is true: ** ''a'' < ''b'' ** ''a'' = ''b'' ** ''a'' > ''b'' ===Transitivity=== The [[Transitive relation|transitivity]] of inequalities states: * For any [[real number]]s, ''a'', ''b'', ''c'': **If ''a'' > ''b'' and ''b'' > ''c''; then ''a'' > ''c'' **If ''a'' < ''b'' and ''b'' < ''c''; then ''a'' < ''c'' ===Reversal=== The inequality relations are [[inverse relation]]s: * For any [[real number]]s, ''a'' and ''b'': **If ''a'' > ''b'' then ''b'' < ''a'' **If ''a'' < ''b'' then ''b'' > ''a'' ===Addition and subtraction=== The properties which deal with [[addition]] and [[subtraction]] state: * For any [[real number]]s, ''a'', ''b'', ''c'': **If ''a'' > ''b'', then ''a'' + ''c'' > ''b'' + ''c'' and ''a'' − ''c'' > ''b'' − ''c'' **If ''a'' < ''b'', then ''a'' + ''c'' < ''b'' + ''c'' and ''a'' − ''c'' < ''b'' − ''c'' i.e., the real numbers are an [[ordered group]]. ===Multiplication and division=== The properties which deal with [[multiplication]] and [[division (mathematics)|division]] state: * For any real numbers, ''a'', ''b'', ''c'': ** If ''c'' is [[positive number|positive]] and ''a'' < ''b'', then ''ac'' < ''bc'' ** If ''c'' is [[negative number|negative]] and ''a'' < ''b'', then ''ac'' > ''bc'' More generally this applies for an [[ordered field]], see below. ===Additive inverse=== The properties for the [[additive inverse]] state: *For any real numbers ''a'' and ''b'' **If ''a'' < ''b'' then &minus;''a'' > &minus;''b'' **If ''a'' > ''b'' then &minus;''a'' < &minus;''b'' ===Multiplicative inverse=== The properties for the [[multiplicative inverse]] state: *For any real numbers ''a'' and ''b'' that are both [[positive]] or both [[negative]] **If ''a'' < ''b'' then 1/''a'' > 1/''b'' **If ''a'' > ''b'' then 1/''a'' < 1/''b'' ===Applying a function to both sides=== We consider two cases of functions: monotonic and strictly monotonic. Any strictly [[Monotonic function|monotonic]]ally increasing [[function (mathematics)|function]] may be applied to both sides of an inequality and it will still hold. Applying a strictly 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. If you have a non-strict inequality (''a'' ≤ ''b'', ''a'' ≥ ''b'') then: * Applying a monotonically increasing function preserves the relation (≤ remains ≤, ≥ remains ≥) * Applying a monotonically decreasing function reverses the relation (≤ becomes ≥, ≥ becomes ≤) It will never become strictly unequal, since, for example, 3 ≤ 3 does not imply that 3 < 3. ===Ordered fields=== If F,+,* be a [[Field (mathematics)|field]] and ≤ be a [[total order]] on F, then F,+,*,≤ is called an [[ordered field]] if and only if: * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c'' * if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''a b'' Note that both <math>\mathbb{Q}</math>,+,*,≤ and <math>\mathbb{R}</math>,+,*,≤ are [[ordered field]]s. ≤ cannot be defined in order to make <math>\mathbb{C}</math>,+,*,≤ an [[ordered field]]. The non-strict inequalities ≤ and ≥ on real numbers are [[total order]]s. The strict inequalities < and > on real numbers are {{ml|Total_order|Strict_total_order|strict total orders}}. == 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''. 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. But care must be taken so that you really use the same number in all cases, e.g. ''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''&nbsp;&minus;&nbsp;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, you won't be able to isolate ''x'' in any one part of the inequality through addition or subtraction. Instead, you can solve 4''x'' < 2''x'' + 1 and 2''x'' + 1 ≤ 3''x'' + 2 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 [[logical conjunction]] of the inequalities between adjacent terms. For instance, ''a'' < ''b'' > ''c'' ≤ ''d'' means that ''a'' < ''b'', ''b'' > ''c'', and ''c'' ≤ ''d''. In addition to rare use in mathematics, this notation exists in a few [[programming language]]s such as [[Python (programming language)|Python]]. == Representing inequalities on the real number line == Every inequality (except those which involve [[imaginary numbers]]) can be represented on the real [[number line]] showing darkened regions on the line. {{Sectstub|date=May 2008}} ==Inequalities between means== There are many inequalities between means. For example, for any positive numbers <math>a_1</math>, <math>a_2</math>, ..., <math>a_n</math> :<math>H \le G \le A \le Q</math>, where :<math>H = \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}}</math> ([[harmonic mean]]), :<math>G = \sqrt[n]{a_1 \cdot a_2 \cdot ... \cdot a_n} </math> ([[geometric mean]]), :<math>A = \frac{a_1 + a_2 + ... + a_n}{n}</math> ([[arithmetic mean]]), :<math>Q = \sqrt{\frac{a_1^2 + a_2^2 + ... + a_n^2}{n}}</math> ([[Root mean square|quadratic mean]]). ==Power inequalities== Sometimes with notation "'''power inequality'''" understand inequalities which contain <math>a^b</math> type expressions where <math>a</math> and <math>b</math> are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations. ===Examples=== # If <math>x>0</math>, then <math>x^x \ge \left( \tfrac{1}{e}\right)^ \tfrac{1}{e}</math> # If <math>x>0</math>, then <math>x^{x^x} \ge x</math> # If <math>x, y, z>0</math>, then <math>(x+y)^z + (x+z)^y + (y+z)^x > 2</math>. # For any real distinct numbers <math>a</math> and <math>b</math>, <math>\tfrac{e^b-e^a}{b-a}>e^{\frac{a+b}{2}}</math> # If <math>x,y>0</math> and <math>0<p<1</math>, then <math>(x+y)^p<x^p+y^p</math> # If <math>x</math>, <math>y</math> and <math>z</math> are positive, then <math>x^x y^y z^z \ge (xyz)^ \frac{x+y+z}{3}</math> # If <math>a</math> and <math>b</math> are positive, then <math>a^b + b^a > 1</math>. This result was generalized by R. Ozols in 2002 who proved that if <math>a_1</math>, <math>a_2</math>, ..., <math>a_n</math> are any real positive numbers, then <math>a_1^{a_2}+a_2^{a_3}+...+a_n^{a_1}>1</math> (result is published in Latvian popular-scientific quarterly ''The Starry Sky'', see references). == 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 style="-moz-column-count:2; column-count:2;"> * [[Azuma's inequality]] * [[Bernoulli'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]] * [[Kolgomorov's inequality]] * [[Markov's inequality]] * [[Minkowski inequality]] * [[Nesbitt's inequality]] * [[Pedoe's inequality]] * [[Triangle inequality]] </div> == Mnemonics for students == Young students sometimes confuse the less-than and greater-than signs, which are mirror images of one another. A commonly taught mnemonic is that the sign represents the mouth of a hungry [[alligator]] that is trying to eat the larger number; thus, it opens towards 8 in both 3&nbsp;<&nbsp;8 and 8&nbsp;>&nbsp;3.[http://mathforum.org/library/drmath/view/58428.html] Another method is noticing the larger quantity points to the smaller quantity and says, "ha-ha, I'm bigger than you." Also, on a horizontal number line, the greater than sign is the arrow that is at the larger end of the number line. Likewise, the less than symbol is the arrow at the smaller end of the [[number line]] ('''<'''---0--1--2--3--4--5--6--7--8--9---'''>'''). The symbols may also be interpreted directly from their form - the side with a large vertical separation indicates a large(r) quantity, and the side which is a point indicates a small(er) quantity. In this way the inequality symbols are similar to the musical [[Dynamics (music)#Gradual changes| crescendo and decrescendo]]. The symbols for equality, less-than-or-equal-to, and greater-than-or-equal-to can also be interpreted with this perspective. ==Complex numbers and inequalities== By introducing a [[lexicographical order]] on the [[complex number]]s, it is a [[totally ordered set]]. However, it is impossible to define ≤ so that <math>\mathbb{C}</math>,+,*,≤ becomes an [[ordered field]]. If <math>\mathbb{C}</math>,+,*,≤ were an [[ordered field]], it has to satisfy the following two properties: * if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c'' * if 0 ≤ ''a'' and 0 ≤ ''b'' then 0 ≤ ''a b'' Because ≤ is a [[total order]], for any number ''a'', ''a'' ≤ 0 or 0 ≤ ''a''. In both cases 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>, contradiction. However ≤ can be defined in order to satisfy the first property, i.e. if ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c''. A definition which is sometimes used is the lexicographical order: * 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>) It can easily be proven that for this definition ''a'' ≤ ''b'' then ''a'' + ''c'' ≤ ''b'' + ''c'' ==See also== *[[Binary relation]] *[[Bracket]] for the use of the < and > signs as brackets *[[Fourier-Motzkin elimination]] *[[Inequation]] *[[Interval (mathematics)]] *[[Partially ordered set]] *[[Relational operator]]s, used in programming languages to denote inequality ==References== *{{cite book | author=Hardy, G., Littlewood J.E., Polya, G.| title=Inequalities| publisher=Cambridge Mathematical Library, Cambridge University Press | year=1999 | id=ISBN 0-521-05206-8}} *{{cite book | author=Beckenbach, E.F., Bellman, R.| title=An Introduction to Inequalities| publisher=Random House Inc | year=1975 | id=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 | id=ISBN 0-387-98404-6}} *{{cite paper|title="Quickie" inequalities|author=Murray S. Klamkin|url=http://www.pims.math.ca/pi/issue7/page26-29.pdf|format=PDF|work=Math Strategies}} *{{cite web|title=Mathematical Problem Solving|url=http://www.math.kth.se/math/TOPS/index.html|author=Harold Shapiro|date=missingdate|publisher=Kungliga Tekniska högskolan|work=The Old Problem Seminar}} *{{cite web|title=3rd USAMO|url=http://www.kalva.demon.co.uk/usa/usa74.html}} *{{cite paper|title=The Starry Sky|url=http://www.astr.lu.lv/zvd/stsky.html}} == External links == * [http://www.mathwarehouse.com/algebra/linear_equation/interactive-linear-inequality.php interactive linear inequality & graph] at www.mathwarehouse.com * [http://www.purplemath.com/modules/ineqsolv.htm Solving Inequalities] * [http://www.webgraphing.com/inequality_1d.jsp WebGraphing.com] &ndash; Inequality Graphing Calculator. * [http://demonstrations.wolfram.com/GraphOfInequalities/ Graph of Inequalities] by [[Ed Pegg, Jr.]], [[The Wolfram Demonstrations Project]]. 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