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 .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:
a 2≥ 0 for all real numbers a .
Properties of Addition and Subtraction
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
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 < 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
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 andMultiplication and Division Properties of Inequalities for negative numbers:<
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If a > b and c > 0 , then ac > bc and>
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If a < b and c < 0 , then ac > bc and>
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If a > b and c < 0 , then ac < bc and<
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Note: All the above properties apply to ≤ and ≥ .
Properties of Reciprocals
Note the following properties:
If a > 0 , thenWhen 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> 0
If a < 0 , then< 0
>
. 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>
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If a > 0 and b > 0 , or a < 0 and b < 0 , and a > b , then<
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Note: All the above properties apply to
≤
and
≥
.
<
> 0





