Inequalities

Properties of Inequalities

Formal Definition of Inequalities

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 .

The Trichotomy Property and the Transitive Properties of Inequality

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 .
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 ageqc .

Property of Squares of Real Numbers:

a ²≥ 0 for all real numbers a .

Properties of Addition and Subtraction

Addition Properties of Inequality:

If a < b , then a + c < b + c
If a > b , then a + c > b + c

Subtraction Properties of Inequality:

If a < b , then a - c < b - c
If a > b , then a - c > b - c

These 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

Properties of Multiplication and Division

Before examining the multiplication and division properties of inequality, note the following:
Inequality Properties of Opposites

If a > 0 , then - a < 0
If a < 0 , then - a > 0

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:

If a < b and c > 0 , then ac < bc and <
If a > b and c > 0 , then ac > bc and >

Multiplication and Division Properties of Inequalities for negative numbers:

If a < b and c < 0 , then ac > bc and >
If a > b and c < 0 , then ac < bc and <

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

Properties of Reciprocals

Note the following properties:

If a > 0 , then > 0
If a < 0 , then < 0

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:

If a > 0 and b > 0 , or a < 0 and b < 0 , and a < b , then >
If a > 0 and b > 0 , or a < 0 and b < 0 , and a > b , then <

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