inequality

inequality

[in-i-kwol-i-tee]
inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved. The symbols > (less than), < (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to) are used in place of the equals sign in expressions of inequalities. As in the case of equations, inequalities can be transformed in various ways. The direction of the inequality remains unchanged if some number is added to both sides or subtracted from both sides or if both sides are multiplied or divided by some positive number; e.g., subtracting 10 from both sides of the inequality x > 8 gives x - 10 > -2, and multiplying the inequality by 2 gives 2x > 16. Multiplication or division by a negative number reverses the sign of the inequality; e.g., if -2x > 8, then dividing both sides by -2 results in the inequality x < -4.

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 ab means that a is not equal to b, but does not say that one is bigger than the other or even that they can be compared in size.

In all these cases, a is not equal to b, hence, "inequality".

These relations are known as strict inequality; in contrast

  • The notation ab means that a is less than or equal to b (or, equivalently, not greater than b);
  • The notation ab 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 ab means that a is much less than b.
  • The notation ab 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.

Solving Inequalities

An inequality may appear unsolvable because it only states whether a number is larger or smaller than another number; but it is possible to apply the same operations for equalities to inequalities. For example, to find x for the inequality 10x > 23 one would divide 23 by 10.

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

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 ac > bc
    • If a < b, then a + c < b + c and ac < bc

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

  • ab implies a + cb + 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 .

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 ae < b < ce.

This notation can be generalized to any number of terms: for instance, a1a2 ≤ ... ≤ an means that aiai+1 for i = 1, 2, ..., n − 1. By transitivity, this condition is equivalent to aiaj for any 1 ≤ ijn.

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 < b > cd means that a < b, b > c, and cd. 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 a1, a2, …, an we have 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).

Power inequalities

Sometimes with notation "power inequality" understand inequalities which contain ab 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, ..., an, 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

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:

Student Learning Techniques

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 ab then a + cb + 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 ab then a + cb + 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))

It can easily be proven that for this definition ab then a + cb + c

See also

References

External links

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