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Inequalities

Properties of Inequalities

Problems

Properties of Inequalities, page 2

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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 ab and bc , then ac .
If ab and bc , 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 ab , then a + cb + c
If ab , then a + cb + c
If ab , then a - cb - c
If ab , then a - cb - 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 .

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