Inequality

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Information about Inequality

This article is about inequalities in mathematics. For other senses of this word, see inequality (disambiguation).

For the use of the < and > signs in punctuation, see .

The feasible regions 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. (See also: equality)

The notation $\text{[math]}$ means that a is less than b and
The notation $\text{[math]}$ means that a is greater than b.

These relations are known as strict inequality; in contrast

$\text{[math]}$ means that a is less than or equal to b;
$\text{[math]}$ means that a is greater than or equal to b;
$\text{[math]}$ means that a is not greater than b and
$\text{[math]}$ means that a is not less 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 greater than b.
The notation a << b means that a is much less 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. The sense of an inequality is not changed if both sides are increased or decreased by the same number, or if both sides are multiplied or divided by a positive number; the sense of an inequality is reversed if both members are multiplied or divided by a negative number.

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 property states:

For any real numbers, a and b, exactly one of the following is true:
a < b
a = b
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
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
If c is 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 -a > -b
If a > b then -a < -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 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 $\text{[math]}$ ,+,*,≤ and $\text{[math]}$ ,+,*,≤ are ordered fields.

≤ cannot be defined in order to make $\text{[math]}$ ,+,*,≤ an ordered field.

The non-strict inequalities ≤ and ≥ on real numbers are total orders. The strict inequalities < and > on real numbers are 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, eg. 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 ≤ ... ≤ 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.

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 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 $\text{[math]}$ , $\text{[math]}$ , ..., $\text{[math]}$

$\text{[math]}$ , where

$\text{[math]}$ (harmonic mean),

$\text{[math]}$ (geometric mean),

$\text{[math]}$ (arithmetic mean),

$\text{[math]}$ (quadratic mean).

Power inequalities

Sometimes with notation "power inequality" understand inequalities which contain $\text{[math]}$ type expressions where $\text{[math]}$ and $\text{[math]}$ are real positive numbers or expressions of some variables. They can appear in exercises of mathematical olympiads and some calculations.

Examples

If $\text{[math]}$ , then $\text{[math]}$
If $\text{[math]}$ , then $\text{[math]}$
If $\text{[math]}$ , then $\text{[math]}$ .
For any real distinct numbers $\text{[math]}$ and $\text{[math]}$ , $\text{[math]}$
If $\text{[math]}$ and $\text{[math]}$ , then $\text{[math]}$
If $\text{[math]}$ , $\text{[math]}$ and $\text{[math]}$ are positive, then $\text{[math]}$
If $\text{[math]}$ and $\text{[math]}$ are positive, then $\text{[math]}$ . This result was generalized by R. Ozols in 2002 who proved that if $\text{[math]}$ , $\text{[math]}$ , ..., $\text{[math]}$ are any real positive numbers, then $\text{[math]}$ (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 < 8 and 8 > 3.[1] 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 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 $\text{[math]}$ ,+,*,≤ becomes an ordered field. If $\text{[math]}$ ,+,*,≤ 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 ≤ a²; this means that $\text{[math]}$ and $\text{[math]}$ ; so $\text{[math]}$ and $\text{[math]}$ , 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 $\text{[math]}$ < $\text{[math]}$ or ( $\text{[math]}$ and $\text{[math]}$ ≤ $\text{[math]}$ )

It can easily be proven that for this definition a ≤ b then a + c ≤ b + c

References

Hardy, G., Littlewood J.E., Polya, G. (1999). Inequalities. Cambridge Mathematical Library, Cambridge University Press. ISBN 0-521-05206-8.
Beckenbach, E.F., Bellman, R. (1975). An Introduction to Inequalities. Random House Inc. ISBN 0-394-01559-2.
Drachman, Byron C., Cloud, Michael J. (1998). Inequalities: With Applications to Engineering. Springer-Verlag. ISBN 0-387-98404-6.
Murray S. Klamkin. ""Quickie" inequalities" (PDF).
Harold Shapiro (missingdate). Mathematical Problem Solving. The Old Problem Seminar. Kungliga Tekniska högskolan.
3rd USAMO.
. "The Starry Sky".
Problem 6 solution.

External links

interactive linear inequality & graph at www.mathwarehouse.com
Solving Inequalities
WebGraphing.com – Inequality Graphing Calculator.

Inequality may refer to:

Inequality (mathematics)
Social inequality
Economic inequality
International inequality
Inequalities (1934) is the title of a mathematics book by G. H. Hardy, J. E. Littlewood, and G. Polya.

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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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equal if and only if they are precisely the same in every way. The complementary notion is distinctness. This defines a binary relation, equality, denoted by the sign of equality "=" in such a way that the statement "x = y" means that x
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An order of magnitude is the class of scale or magnitude of any amount, where each class contains values of a fixed ratio to the class preceding it. The ratio most commonly used is 10.
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A negative number is a number that is less than zero, such as −3. A positive number is a number that is greater than zero, such as 3. Zero itself is neither positive nor negative.
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In law and political theory, property refers to an ownership interest in land or other resources.

A property of an object is some intrinsic or extrinsic quality of that object, where the nature of the "object" in question will depend on the field, as, for example, indicated
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a binary relation R over a set X is transitive if it holds for all a, b, and c in X, that if a is related to b and b is related to c, then a is related to c.
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In mathematics, the inverse relation of a binary relation is the relation taken 'backwards', as in changing the relation 'child of' to 'parent of'. In formal terms, if

is a binary relation with

then the inverse relation is

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Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. Addition also provides a model for related processes such as joining two collections of objects into one collection.
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Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. Subtraction is denoted by a minus sign in infix notation.

The traditional names for the parts of the formula

c − b = a

are
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In abstract algebra, an ordered group is a group G equipped with a partial order "≤" which is translation-invariant; in other words, "≤" has the property that, for all a, b, and g in G, if a ≤
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Multiplication is the mathematical operation of adding together multiple copies of the same number. For example, four multiplied by three is twelve, since three sets of four make twelve:

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In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication.

Specifically, if c times b equals a, written:

where b is not zero, then a
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In mathematics, an ordered field is a field together with a total ordering of its elements that agrees in a certain sense with the field operations. This concept was introduced by Emil Artin in 1927.
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In mathematics, the additive inverse, or opposite, of a number n is the number that, when added to n, yields zero. The additive inverse of n is denoted −n.
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multiplicative inverse for a number x, denoted by ¹⁄_x or x⁻¹, is a number which when multiplied by x yields the multiplicative identity, 1.
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Positive may refer to:

Mathematics and science

Positive number, a number that is greater than 0
Positive operator, in functional analysis, a bounded linear operator whose spectrum consists of positive real numbers
Positive electric charge, in physics

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Negative may refer to:

Negative and non-negative numbers
Photographic negative, an image with inverted luminance or a strip of film with such an image

* Film negative, the film in a motion picture camera which captures the original image

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monotonic function (or monotone function) is a function which preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
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function expresses dependence between two quantities, one of which is given (the independent variable, argument of the function, or its "input") and the other produced (the dependent variable, value of the function, or "output").
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field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
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In mathematics, a total order, linear order, simple order, or (non-strict) ordering on a set X is any binary relation on X that is antisymmetric, transitive, and total.
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Inequality

Information about Inequality

Properties

Trichotomy

Transitivity

Reversal

Addition and subtraction

Multiplication and division

Additive inverse

Multiplicative inverse

Applying a function to both sides

Ordered fields

Chained notation

Representing Inequalities on the real number line

Inequalities between means

Power inequalities

Examples

Well-known inequalities

Mnemonics for students

Complex numbers and inequalities

See also

References

External links

Mathematics and science