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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*.

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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*.

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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*

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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.