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