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*
.

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:

Ifa<bandb<c, thena<c.

Ifa>bandb>c, thena>c.

Ifa≤bandb≤c, thena≤c.

Ifa≥bandb≥c, thenageqc.

Property of Squares of Real Numbers:

a^{2}≥ 0 for all real numbersa.

Addition Properties of Inequality:

IfSubtraction Properties of Inequality:a<b, thena+c<b+c

Ifa>b, thena+c>b+c

IfThese properties also apply to ≤ and ≥ :a<b, thena-c<b-c

Ifa>b, thena-c>b-c

Ifa≤b, thena+c≤b+c

Ifa≥b, thena+c≥b+c

Ifa≤b, thena-c≤b-c

Ifa≥b, thena-c≥b-c

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

Inequality Properties of Opposites

IfFor example, 4 > 0 and -4 < 0 . Similarly, -2 < 0 and 2 > 0 . Whenever we multiply an inequality by -1 , thea> 0 , then -a< 0

Ifa< 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:

IfMultiplication and Division Properties of Inequalities fora<bandc> 0 , thenac<bcand <

Ifa>bandc> 0 , thenac>bcand >

Ifa<bandc< 0 , thenac>bcand >

Ifa>bandc< 0 , thenac<bcand <

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

Note the following properties:

IfWhen 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:a> 0 , then > 0

Ifa< 0 , then < 0

Ifa> 0 andb> 0 , ora< 0 andb< 0 , anda<b, then >

Ifa> 0 andb> 0 , ora< 0 andb< 0 , anda>b, then <

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