Begin2.DVI

Inequalities can be defined in terms of addition or subtraction. For example, one

can define

a < b if and only ifa−b < 0

a > b if and only ifa−b > 0 , or alternatively

a > b if and only if there exists a positive numberxsuch that b+x=a.

In dealing with inequalities be sure to observe the following properties associated

with real numbersa,b,c,...

1. A constant can be added to both sides of an inequality without changing the
   inequality sign.

Ifa < b, thena+c < b+cfor all numbers c

2. Both sides of an inequality can be multiplied or divided bya positive constant
   without changing the inequality sign.

Ifa < bandc > 0 , thenac < bc or a/c < b/c

3. If both sides of an inequality are multiplied or divided bya negative quantity,
   then the inequality sign changes.

Ifb > aandc < 0 , thenbc < ac or b/c < a/c

4. The transitivity law
   Ifa < b, andb < c, thena < c
   Ifa=bandb=c,thena=c
   Ifa > b, andb > c, thena > c


5. Ifa > 0 andb > 0 , thenab > 0


6. Ifa < 0 andb < 0 , thenab > 0 or 0 < ab


7. Ifa > 0 andb > 0 witha < b, then

√

a <

√

b

A negative times a negative is a positive

To prove that a real negative number multiplied by another real negative number

gives a positive number start by assumingaandbare real numbers satisfyinga < 0

andb < 0 , then one can write

−a+a <−a or 0 <−a and −b+b <−b or 0 <−b