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:
If a < b and b < c, then a < c.Note: These properties also apply to "less than or equal to" and "greater than or equal to":
If a > b and b > c, then a > c.
If a≤b and b≤c, then a≤c.
If a≥b and b≥c, then ageqc.
Property of Squares of Real Numbers:
a2≥ 0 for all real numbers a.
Addition Properties of Inequality:
If a < b, then a + c < b + cSubtraction Properties of Inequality:
If a > b, then a + c > b + c
If a < b, then a - c < b - cThese properties also apply to ≤ and ≥:
If a > b, then a - c > b - c
If a≤b, then a + c≤b + c
If a≥b, then a + c≥b + c
If a≤b, then a - c≤b - c
If a≥b, then a - c≥b - c
Before examining the multiplication and division properties of inequality, note the
Inequality Properties of Opposites
If a > 0, then - a < 0For 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.
If a < 0, then - a > 0