Inequality

In mathematics, an inequality is a statement about the relative size or order of two objects. (See also: equality and equality (mathematics)) The notation a < b means that a is less than b and the notation a > b means that a is greater than b. a ≤ b means that a is less than or equal to b and a ≥ b means that a is greater than or equal to 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 "unconditonal" 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.

Inequalities are governed by the following properties:

Table of contents

1 The trichotomy property 2 The addition and subtraction properties of inequalities 3 The multiplication and division properties of inequalities 4 Well-known inequalities 5 See also

The trichotomy property

The trichotomy property states:

• For any real numbers, "a" and "b", only one of the following is true:
  • a < b
  • a = b
  • a > b

The addition and subtraction properties of inequalities

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

The multiplication and division properties of inequalities

The properties which deal with multiplication and division state:

• For any real numbers, "a", "b", and "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

Well-known 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
• Boole's inequality
• Cauchy-Schwarz inequality
• Chebyshev's inequality
• Chernoff's inequality
• Hoeffding's inequality
• Hölder's inequality
• Jensen's inequality
• Markov's inequality
• Minkowski's inequality
• Triangle inequality

See also

Binary relation, partially ordered set, inequation.