By Jerrold Marsden, Alan Weinstein

Undergraduate Texts in arithmetic

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Note that the distance from 4 to m is ( 4 - ? ) So we have x ~ ( 4 , 9 )if and only if Ix - y I < 3. (See Fig. ) A The most important algebraic properties of absolute values are listed below. 4. The interval (4,9) may be described by the inequality x - +? I < 3. Example 7 Solution Show by example that Ix + YI is not always equal to 1x1 + I yl. Let x = 3 and y = - 5. Then Ix + yl = 13 - 51 = 2, while 1x1 + lyl = 3 + 5 = 8. (Many other numbers will work as well. ) A Copyright 1985 Springer-Verlag.

Copyright 1985 Springer-Verlag. All rights reserved. 1 Basic Algebra: Real Numbers and Inequalities '19 number line and since - 4T is only half as far from zero as - T, it lies to the right of -T. If x is any real number, we know that x 2 > 0. If x #=0, we can make the stronger statement that x2 > 0. 2. Possible positions of a and b when they have the same sign (ab > 0) or opposite signs (ab < 0). 0000025 > - 100,000 since a positive number is always to the right of a negative number. (b) 2 <$ since 2 = and $ = g .

Another approach is to use the definition We observe that the part of the graph of g for x > 0 is a line through (0,0) with slope 1 , while the part for x < 0 is a line through (0,O) with slope - 1. It follows that the graph of g is as drawn in Fig. 4. 4. Some points on the graph of y = 1x1. 5. " 2 If the domain of a function consists of finitely many points, then the graph consists of isolated dots; there is no line to be filled in. For instance, Fig. 5 shows the graph of a function 1 whose domain is { 1,2,3,4,5,6,7) and for which I(x) is the number of letters in the English name for x.