# Dictionary Definition

inequality n : lack of equality; "the growing
inequality between rich and poor" [ant: equality]

# User Contributed Dictionary

## English

### Etymology

from inégalité, from inæqualitas, from inæqualis, "unequal," from in- "not" & æqualis "equal"### Noun

- An unfair, not
equal, state.
- The inequality in living standards lead to a civil war as the have nots rebelled.

- A statement that of two quantities one is specifically less than (or greater than) another.
- The inequality x is less than y, together with that y<z, allows us to deduce the inequality x<z.

- In the context of "mathematics|more|_|generally": A statement that of two quantities one is specifically less than (or greater than, or not less than, or not greater than) another.

#### Translations

An unfair, not equal, state

- Czech: nerovnost
- Danish: ulighed
- Finnish: epätasa-arvo

A statement that one quantity is less (or
greater) than another

- Danish: ulighed
- Finnish: erisuuruus

A statement that one quantity is less (or
greater, or not less, or not greater) than another

- Danish: ulighed

- ttbc French: inégalité
- ttbc German: Ungleichheit

# Extensive Definition

In mathematics, an inequality
is a statement about the relative size or order of two objects.
(See also: equality)

- The notation a means that a is less than b and
- The notation a > b \!\ means that a is greater than b.

- a \le b means that a is less than or equal to b (or, equivalently, not greater than b);
- a \ge 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 \ll b means that a is much less than b.
- The notation a \gg 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.

## 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 () are replaced with their corresponding non-strict inequality sign (≤ and ≥).### Trichotomy

The trichotomy
property states:

- For any real numbers,
a and b, exactly one of the following is true:
- a b

### Transitivity

The transitivity
of inequalities states:

- For any real numbers,
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
relations:

- For any real numbers,
a and b:
- If a > b then b a

### Addition and subtraction

The properties which deal with addition and subtraction state:

- For any real numbers,
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
state:

- For any real numbers, a, b, c:
- If c is positive 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
- If a > b then −a < −b

### Multiplicative inverse

The properties for the multiplicative
inverse state:

### Applying a function to both sides

We consider two cases of functions: monotonic and strictly monotonic.Any strictly monotonically
increasing 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 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 \mathbb,+,*,≤ and \mathbb,+,*,≤
are ordered
fields.

≤ cannot be defined in order to make
\mathbb,+,*,≤ an ordered
field.

The non-strict inequalities ≤ and ≥ on real
numbers are total orders.
The strict inequalities on real numbers are .

## 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, a1 ≤ a2 ≤ ... ≤ an means that ai ≤ ai+1 for i
= 1, 2, ..., n − 1. By transitivity,
this condition is equivalent to ai ≤ aj 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 4x < 2x + 1
≤ 3x + 2, you won't be able to isolate x in any one part of the
inequality through addition or subtraction. Instead, you can solve
4x < 2x + 1 and 2x + 1 ≤ 3x + 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 c ≤ d means that a c, and c ≤ d. In addition to rare
use in mathematics, this notation exists in a few programming
languages such as
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.## Inequalities between means

There are many inequalities between means. For
example, for any positive numbers a_1, a_2, ..., a_n

- H \le G \le A \le Q, where

- H = \frac (harmonic mean),

- G = \sqrt[n] (geometric mean),

- A = \frac (arithmetic mean),

- Q = \sqrt (quadratic mean).

## Power inequalities

Sometimes with notation "power inequality" understand inequalities which contain a^b type expressions where a and b are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations.### Examples

- If x>0, then x^x \ge \left( \tfrac\right)^ \tfrac
- If x>0, then x^ \ge x
- If x, y, z>0, then (x+y)^z + (x+z)^y + (y+z)^x > 2.
- For any real distinct numbers a and b, \tfrac>e^
- If x,y>0 and 0, then (x+y)^p
- If x, y and z are positive, then x^x y^y z^z \ge (xyz)^ \frac
- If a and b are positive, then a^b + b^a > 1. This result was generalized by R. Ozols in 2002 who proved that if a_1, a_2, ..., a_n are any real positive numbers, then a_1^+a_2^+...+a_n^>1 (result is published in Latvian popular-scientific quarterly The Starry Sky, see references).

## Well-known inequalities

See also list
of 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
- 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

## 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 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
().

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
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 numbers, it is a totally ordered set. However, it is impossible to define ≤ so that \mathbb,+,*,≤ becomes an ordered field. If \mathbb,+,*,≤ 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 ≤ a2; this means
that i^2>0 and 1^2>0; so 1>0 and -1>0,
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 Re(a) Re(b) or (Re(a) = Re(b) and Im(a) ≤ Im(b))

## See also

- Binary relation
- Bracket for the use of the signs as brackets
- Fourier-Motzkin elimination
- Inequation
- Interval (mathematics)
- Partially ordered set
- Relational operators, used in programming languages to denote inequality

## References

- Inequalities
- An Introduction to Inequalities
- Inequalities: With Applications to Engineering
- ">http://www.kalva.demon.co.uk/usa/usa74.html}}
- ">http://www.astr.lu.lv/zvd/stsky.html}}

## External links

- interactive linear inequality & graph at www.mathwarehouse.com
- Solving Inequalities
- WebGraphing.com – Inequality Graphing Calculator.
- Graph of Inequalities by Ed Pegg, Jr., The Wolfram Demonstrations Project.

inequality in Bengali: অসমতা

inequality in Catalan: Inequació

inequality in Czech: Nerovnost
(matematika)

inequality in Danish: Ulighed (matematik)

inequality in German: Ungleichung

inequality in Spanish: Inecuación

inequality in Esperanto: Neegalaĵo (pli granda,
malpli granda)

inequality in Persian: نامساوی

inequality in French: Inégalité
(mathématiques)

inequality in Galician: Inecuación

inequality in Korean: 부등식

inequality in Ido: Ne egaleso

inequality in Indonesian: Pertidaksamaan

inequality in Italian: Disuguaglianza

inequality in Hebrew: אי-שוויון

inequality in Latvian: Nevienādība

inequality in Hungarian: Egyenlőtlenség

inequality in Dutch: Ongelijkheid

inequality in Japanese: 不等式

inequality in Polish: Nierówność

inequality in Portuguese: Desigualdade

inequality in Romanian: Inegalitate

inequality in Russian: Неравенство

inequality in Simple English: Inequality

inequality in Finnish: Epäyhtälö

inequality in Swedish: Olikhet

inequality in Thai: อสมการ

inequality in Vietnamese: Bất đẳng thức

inequality in Yiddish: אומגלייכהייט

inequality in Chinese: 不等

# Synonyms, Antonyms and Related Words

antagonism, argumentation, asperity, bias, bumpiness, capriciousness, changeability, changeableness, choppiness, clashing, conflict, contradiction, contrariety, contrast, controversy, cragginess, dappleness, departure, deviation, difference, differentiation,
disaccord, disaccordance, disagreement, disconformity, discongruity, discord, discordance, discordancy, discrepancy, discreteness, discrimination, disharmony, disorder, disparity, disproportion, dissension, dissent, dissidence, dissimilarity, dissonance, distinction, distinctness, disunion, disunity, divarication, divergence, divergency, diversification,
diversity, faction, far cry, favoritism, granulation, harshness, heterogeneity, hispidity, imbalance, imparity, inaccordance, inclination, incompatibility,
incongruence,
incongruity,
inconsistency,
inconsonance,
inconstancy,
inequity, inharmoniousness,
inharmony, injustice, instability, interest, involvement, irreconcilability,
irregularity,
jaggedness, jarring, jerkiness, joltiness, leaning, mercuriality, mixture, motleyness, mutability, negation, nepotism, nonconformism, nonconformity, nonstandardization,
nonuniformity,
odds, one-sidedness,
opposition, oppugnancy, otherness, parti pris,
partiality, partisanism, partisanship, pluralism, preference, preferential
treatment, prejudice,
raggedness, repugnance, rough air,
roughness, ruggedness, rugosity, scraggliness, separateness, tooth, turbulence, unconformism, unconformity, undetachment, undispassionateness,
unevenness, unfairness, unharmoniousness,
unlikeness, unneutrality, unorthodoxy, unsmoothness, unsteadiness, ununiformity, variability, variance, variation, variegation, variety, variousness, versatility, wavering