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Inequality

For the socioeconomic meaning, see social inequality.

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 ≤ 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.

Properties

Inequalities are governed by the following properties:

Trichotomy

The trichotomy property states:

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

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", 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

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
• 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's inequality
• Pedoe's inequality
• Triangle inequality

See also

• Binary relation
• Partially ordered set
• Inequation