Inequality

2008/9 Schools Wikipedia Selection. Related subjects: Mathematics

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 $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;
$a \not> b$ means that a is not greater than b and
$a \not< b$ 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 $a \gg b$ means that a is much greater than b.
The notation $a \ll b$ 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

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 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
- 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

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 $\mathbb{Q}$ ,+,*,≤ and $\mathbb{R}$ ,+,*,≤ are ordered fields.

≤ cannot be defined in order to make $\mathbb{C}$ ,+,*,≤ 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.

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.

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

$H = \frac{n}{\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}}$ ( harmonic mean),

$G = \sqrt[n]{a_1 \cdot a_2 \cdot ... \cdot a_n}$ ( geometric mean),

$A = \frac{a_1 + a_2 + ... + a_n}{n}$ (arithmetic mean),

$Q = \sqrt{\frac{a_1^2 + a_2^2 + ... + a_n^2}{n}}$ ( 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 $x^x \ge \left( \tfrac{1}{e}\right)^ \tfrac{1}{e}$
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 $b$ , $\tfrac{e^b-e^a}{b-a}>e^{\frac{a+b}{2}}$
If $x, y > 0$ and $0 < p < 1$ , then $(x + y) p < x p + y p$
If $x$ , $y$ and $z$ are positive, then $x^x y^y z^z \ge (xyz)^ \frac{x+y+z}{3}$
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 $a_1^{a_2}+a_2^{a_3}+...+a_n^{a_1}>1$ (result is published in Latvian popular-scientific quarterly The Starry Sky, see references).

Well-known inequalities

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. 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 $\mathbb{C}$ ,+,*,≤ becomes an ordered field. If $\mathbb{C}$ ,+,*,≤ 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 $12 > 0$ ; 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 $R e (a)$ < $R e (b)$ 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 then a + c ≤ b + c

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