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

The notation

a < b !

means that a is less than b and

The notation

a > b !

means that a is greater than b.

These relations are known as strict inequality; in contrast

a le b

means that a is less than or equal to b;

a ge b

means that a is greater than or equal to b;

a

ot> b means that a is not greater than b and

a

ot< b 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.

    Inequality
        Properties
            Trichotomy
            Transitivity
            Reversal
            Addition and subtraction
            Multiplication and division
            Applying a function to both sides
            Ordered Fields
        Chained notation
        Power inequalities
            Examples
        Well-known inequalities
        Mnemonics for students
        Complex numbers and inequalities
        See also

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 mirror images in the sense that:

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 states:

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

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 a × c > b × c and a / c > b / c

If c is positive and a < b, then a × c < b × c and a / c < b / c

If c is negative and a > b, then a × c < b × c and a / c < b / c

If c is negative and a < b, then a × c > b × c and a / c > b / c

Applying a function to both sides
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.

More subtly, if you have a non-strict inequality (a ≤ b, a ≥ b) then:

Applying a monotonically increasing function will preserve the relation (≤ remains ≤, ≥ remains ≥)

Applying a strictly increasing function will make the relation strict (≤ becomes <, ≥ becomes >)

Applying a monotonically decreasing function reverses the relation, but keeps it nonstrict (≤ becomes ≥, ≥ becomes ≤)

Applying a strictly decreasing function reverses the relation and makes it strict (≤ becomes >, ≥ becomes <)

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.

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.

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

x>0 Rightarrow x^x ge ( rac)^ rac

x>0 Rightarrow x^ ge x

x, y, z>0 Rightarrow (x+y)^z + (x+z)^y + (y+z)^x > 2

For any real distinct numbers

a

and

b

rac>e^

If

a,b>0

and

0 then (x+y)^p

For any positive

x

,

y

and

z

then

x^x y^y z^z ge (xyz)^ rac

For any real positive

a

and

b

then

a^b + b^a > 1

. This result was generalized by R. Ozols in 2002 who proved that for any real positive numbers

a_1, a_2,

...,

a_n

is true inequality

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

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 a hungry alligator that is trying to eat the larger number; thus, it opens towards 8 in both 3 < 8 and 8 > 3.* 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--->). This is actually where the greater than and less than signs came from.

Complex numbers and inequalities

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,

i = sqrt

≤ 0 or 0 ≤

i

.

Say 0 ≤

i

. Therefore

0.i

≤

i.i

, thus 0 ≤ -1 which is false.

Say

i

≤

0

. Therefore

i+(-i)

≤

0+(-i)

, thus

0

≤

-i

and

0.(-i)

≤

(-i).(-i)

, hence 0 ≤ -1 which is false.

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:

a ≤ b if

Re(a)

≤

Re(b)

or (

Re(a) = Re(b)

and

Im(a)

≤

Im(b)

)

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

See also

Binary relation

Partially ordered set

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