ELEMENTARY MATHEMATICS
W W L CHEN and X T DUONG
c
W W L Chen, X T Duong and Macquarie University, 1999.
This work is available free, in the hope that it will be useful.
Any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including
photocopying, recording, or any information storage and retrieval system, with or without permission from the authors.
Chapter 6
INEQUALITIES AND
ABSOLUTE VALUES
6.1.
Some Simple Inequalities
Basic inequalities concerning the real numbers are simple, provided that we exercise due care. We begin
by studying the effect of addition and multiplication on inequalities.
ADDITION AND MULTIPLICATION RULES.
Suppose that
a, b
∈
R
and
a < b
. Then
(a) for every
c
∈
R
, we have
a
+
c < b
+
c
;
(b) for every
c
∈
R
satisfying
c >
0
, we have
ac < bc
; and
(c) for every
c
∈
R
satisfying
c <
0
, we have
ac > bc
.
In other words, addition by a real number
c
preserves the inequality. On the other hand, multipli-
cation by a real number
c
preserves the inequality if
c >
0 and reverses the inequality if
c <
0.
Remark.
We can deduce some special rules for positive real numbers. Suppose that
a, b, c, d
∈
R
are
all positive. If
a < b
and
c < d
, then
ac < bd
. To see this, note simply that by part (b) above, we have
ac < bc
and
bc < bd
.
SQUARE AND RECIPROCAL RULES.
Suppose that
a, b
∈
R
and
0
< a < b
. Then
(a)
a
2
< b
2
; and
(b)
a
−
1
> b
−
1
.
Proof.
Part (a) is a special case of our Remark if we take
c
=
a
and
d
=
b
. To show part (b), note
that
a
−
1
−
b
−
1
=
1
a
−
1
b
=
b
−
a
ab
>
0
.
♣
†
This chapter was written at Macquarie University in 1999.