Begin2.DVI

since equals can be added to both sides of an inequality without changing the in-

equality sign. Using the fact that both sides of an inequality can be multiplied by a

positive number without changing the inequality sign, one can write

0 <(−a)(−b) or (−a)(−b)> 0

Another way to show a negative times a negative

is a positive is as follows. Think of a number line

with the number 0 separating the positive num-

bers and negative numbers. By agreement, if a

number on this number line is multiplied by -1,

then the number is to be rotated counterclockwise 180 degrees. If the positive num-

berxis multiplied by -1, then it is rotated counterclockwise 180degrees to produce

the number−x. If the number−xis multiplied by -1, then it is to be rotated 180

degrees counterclockwise to produce the positive numberx. Ifa > 0 andb > 0 , then

the producta(−b)scales the number−bto produce the negative number−ab. If the

number−abis multiplied by − 1 , which is equivalent to the product(−a)(−b), one

obtains by rotation the number+ab.

Absolute Value

The absolute value of a numberxis defined

|x|=

{ x, ifx≥ 0

−x, ifx < 0

The symbol⇐⇒is often used to represent equivalence of two equations. Forexample,

ifaandbare real numbers the statements

|x−a|≤b ⇐⇒−b≤x−a≤b ⇐⇒ a−b≤x≤a+b

are all equivalent statements involving restrictions on the real numberx.

An important inequality known as the triangle inequality iswritten

|x+y|≤|x|+|y| (1.1)

wherexandyare real numbers. To prove this inequality observe that|x|satisfies

−|x|≤x≤|x|and also−|y|≤y≤|y|, so that by adding these results one obtains

−(|x|+|y|)≤x+y≤|x|+|y| or |x+y|≤|x|+|y| (1.2)

Related to the inequality (1.2) is the reverse triangle inequality

|x−y|≥|x|−|y| (1.3)

a proof of which is left as an exercise.