There are formal definitions of the inequality relations > , < ,≥,≤ in terms of the familiar notion of equality. We say a is less than b, written a < b if and only if there is a positive number c such that a + c = b. Recall that zero is not a positive number, so this cannot hold if a = b. Similarly, we say a is greater than b and write a > b if b is less than a; alternately, there exists a positive number c such that a = b + c.

Trichotomy Property: For any two real numbers a and b, exactly one of the following is true: a < b, a = b, a > b.

Transitive Properties of Inequality:
Note: These properties also apply to "less than or equal to" and "greater than or equal to":

Property of Squares of Real Numbers:

Addition Properties of Inequality:
Subtraction Properties of Inequality:
These properties also apply to ≤ and ≥:

Before examining the multiplication and division properties of inequality, note the following:
Inequality Properties of Opposites
For example, 4 > 0 and -4 < 0. Similarly, -2 < 0 and 2 > 0. Whenever we multiply an inequality by -1, the inequality sign flips. This is also true when both numbers are non-zero: 4 > 2 and -4 < - 2; 6 < 7 and -6 > - 7; -2 < 5 and 2 > - 5.

In fact, when we multiply or divide both sides of an inequality by any negative number, the sign always flips. For instance, 4 > 2, so 4(- 3) < 2(- 3): -12 < - 6. -2 < 6, so > : 1 > -3. This leads to the multiplication and division properties of inequalities for negative numbers.

Multiplication and Division Properties of Inequalities for positive numbers:
Multiplication and Division Properties of Inequalities for negative numbers:

Note: All the above properties apply to ≤ and ≥.

Note the following properties: When we take the reciprocal of both sides of an equation, something interesting happens--if the numbers on both sides have the same sign, the inequality sign flips. For example, 2 < 3, but > . Similarly, > , but -3 < . We can write this as a formal property:

Note: All the above properties apply to ≤ and ≥.