Inequality: Wikis

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

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

For the not-equals statement, see Inequation.

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, or about whether they are the same or not (See also: 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 greater than the other or even that they can be compared in size.

In each statement above, 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".

In contrast to strict inequalities, there are two types of inequality statements that are not strict:

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.

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

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
- If a < b and b = c; then a < c

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, 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
- 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 1/b
- If a > b then 1/a < 1/b

If either a or b is negative then

- 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 however 10-3x<4 means that the answers can be 1, 2, or 3 but not 4

Ordered fields

If (F, +, ×) is a field and ≤ is a total order on F, then (F, +, ×, ≤) is called an ordered field if and only if:

a ≤ b implies a + c ≤ b + c;
0 ≤ a and 0 ≤ b implies 0 ≤ a × b.

Note that both (Q, +, ×, ≤) and (R, +, ×, ≤) are ordered fields, but ≤ cannot be defined in order to make (C, +, ×, ≤) an ordered field, because −1 is the square of i and would therefore be positive.

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, 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 ≤ ... ≤ 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.

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 < 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. A < or > is graphed by an open circle on the number. A ≤ or ≥ is graphed with a closed or black circle.

Inequalities between means

There are many inequalities between means. For example, for any positive numbers a₁, a₂, …, a_n we have H ≤ G ≤ A ≤ Q, where

$H = \frac{n}{1/a_1 + 1/a_2 + \cdots + 1/a_n}$	(harmonic mean),
$G = \sqrt[n]{a_1 \cdot a_2 \cdots a_n}$	(geometric mean),
$A = \frac{a_1 + a_2 + \cdots + a_n}{n}$	(arithmetic mean),
$Q = \sqrt{\frac{a_1^2 + a_2^2 + \cdots + a_n^2}{n}}$	(quadratic mean).

Logarithms of inequalities

When manipulating inequalities it is sometimes useful to take the logarithm of both sides of the inequality. To do this the following result is needed.

$a>b \Leftrightarrow \begin{cases} \log_c a> \log_c b, & \mbox{if } c>1 \\ \log_c a< \log_c b, & \mbox{if } 0<c < 1. \end{cases}$

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

$x^x \ge \left( \frac{1}{e}\right)^{1/e}.\,$

If x > 0, then

$x^{x^x} \ge x.\,$

If x, y, z > 0, then

$(x+y)^z + (x+z)^y + (y+z)^x > 2.\,$

For any real distinct numbers a and b,

$\frac{e^b-e^a}{b-a} > e^{(a+b)/2}.$

If x, y > 0 and 0 < p < 1, then

$(x+y)^p < x^p+y^p.\,$

If x, y, z > 0, then

$x^x y^y z^z \ge (xyz)^{(x+y+z)/3}.\,$

If a, b, then

$a^b + b^a > 1.\,$

This result was generalized by R. Ozols in 2002 who proved that if a₁, ..., a_n, then

$a_1^{a_2}+a_2^{a_3}+\cdots+a_n^{a_1}>1$

(result is published in Latvian popular-scientific quarterly The Starry Sky, see references).

Well-known inequalities

Complex numbers and inequalities

The set of complex numbers $\mathbb{C}$ with its operations of addition and multiplication is a field, but it is impossible to define any relation ≤ so that $\mathbb{C}$ ,+,*,≤ becomes an ordered field. To make ( $\mathbb{C}$ ,+,*,≤) an ordered field, it would have 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, either 0 ≤ a or a ≤ 0 (in which case the first property above implies that 0 ≤ $- a$ ). In either case 0 ≤ a²; this means that $i 2 > 0$ and $12 > 0$ ; so $- 1 > 0$ and $1 > 0$ , which means $( - 1 + 1) > 0$ ; contradiction.

However, an operation ≤ can be defined so as to satisfy only the first property (namely, "if a ≤ b then a + c ≤ b + c"). A definition which is sometimes used is the lexicographical order:

a ≤ b if $R e (a)$ < $R e (b)$ or ( $R e (a) = R e (b)$ and $I m (a)$ ≤ $I m (b)$ )

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

Vector inequalities

Inequality relationships similar to those defined above can also be defined for column vector. If we let the vectors $x,y\in\mathbb{R}^n$ (meaning that $x = \left(x_1,x_2,\ldots,x_n\right)^T$ and $y = \left(y_1,y_2,\ldots,y_n\right)^T$ where $x i$ and $y i$ are real numbers for $i=1,\ldots,n$ ), we can define the following relationships.

$x = y \$ if $x_i = y_i\$ for $i=1,\ldots,n$
$x < y \$ if $x_i < y_i\$ for $i=1,\ldots,n$
$x \leq y$ if $x_i \leq y_i$ for $i=1,\ldots,n$ and $x \neq y$
$x \leqq y$ if $x_i \leq y_i$ for $i=1,\ldots,n$

Similarly, we can define relationships for $x > y$ , $x \geq y$ , and $x \geqq y$ . We note that this notation is consistent with that used by Matthias Ehrgott in Multicriteria Optimization (see References).

We observe that the property of Trichotomy (as stated above) is not valid for vector relationships. We consider the case where $x = \left[ 2, 5 \right]^T$ and $y = \left[ 3, 4 \right]^T$ . 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.

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). Math Strategies. http://www.pims.math.ca/pi/issue7/page26-29.pdf.
Arthur Lohwater (1982). "Introduction to Inequalities". Online e-book in PDF format. http://www.mediafire.com/?1mw1tkgozzu.
Harold Shapiro (2005,1972–1985). "Mathematical Problem Solving". The Old Problem Seminar. Kungliga Tekniska högskolan. http://www.math.kth.se/math/TOPS/index.html.
"3rd USAMO". Archived from the original on 2008-02-03. http://web.archive.org/web/20080203070350/www.kalva.demon.co.uk/usa/usa74.html.
Ehrgott, Matthias (2005). Multicriteria Optimization. Springer-Berlin. ISBN 3-540-21398-8.
Steele, J. Michael (2004). The Cauchy-Schwarz Master Class: An Introduction to the Art of Mathematical Inequalities. Cambridge University Press. ISBN 978-0521546775. http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_index.html.

External links

interactive linear inequality & graph at www.mathwarehouse.com
Solving Inequalities
Graph of Inequalities by Ed Pegg, Jr., Wolfram Demonstrations Project.

Categories: Inequalities | Elementary algebra

Simple English

Inequality is when one object is:

smaller than the other ( $a means that$ a is smaller than b)
bigger than the other ( $a>b$ means that a is bigger than b)
not smaller than the other ( $a\geq b$ means that a is not smaller than b, that is, it is either bigger, or equal to b)
not bigger than the other ( $a\leq b$ means that a is not bigger than b, or it is smaller or equal to b)

Inequality is sometimes used to name a statement that one expression is smaller, greater, not smaller or not greater than the other.

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