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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.
Properties of Addition and Subtraction
Addition Properties of Inequality:
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 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
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.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 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
> ![]() 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
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 > 0If 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
> ![]() If a > 0 and b > 0, or a < 0 and b < 0, and a > b, then < ![]() Note: All the above properties apply to ≤ and ≥. |
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