Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 13COORDINATE SYSTEMS ON A LINE 3EXAMPLES0 1 2 3 4 5. , , . . . ,YI . . I .0 AI A2-3 -2 -1 0 1 2 3 41 1 1 1 I 1 I II 1 U U 1 1 I PYA2 0 AI- = 14 - (41 = 14 4- 31 = 171 = 7 = A1A2A special case of (1.6) is very important. If a is the coordinate of A, thenI a I = distance between A and the originNotice that, for any positive number c,lul I c is equivalent to-c I u I c1 1 1 *Y1 1 1-C 0 CEXAMPLE lul S 3 ifand only if -3 I U I 3.Similarly, lul < c is equivalent to -c < U < c (1.9)EXAMPLE To find a simpler form for the condition I x - 3 I < 5, substitute x - 3 for U in (1.9), obtaining- 5 < x - 3 < 5. Adding 3, we have - 2 < x < 8. From a geometric standpoint, note that I x - 3 I < 5 is equivalentto saying that the distance between the point A having coordinate x and the point having coordinate 3 is less than 5.I 1 II 1 1 Y-2 3 8It follows immediately from the definition of the absolute value that, for any two numbers a and b,-1al I a I lal and -161 I 6 I lbl(In fact, either a = I a I or a = -1 a I.) Adding the inequalities, we obtainand so, by(1.8), with u = a + 6 and c = lal + Ibl,(-14) +(-lbl) I a b 2s I4 + lbl-(lal+ 161) I U + 6 I lal+ lblla + 61 lal+ 161 (1.10)The inequality (1.10) is known as the triangle inequality. In (1.20) the sign c applies if and only if a andb are of opposite signs.EXAMPLE 13 + (-2)( = 11 I = 1, but 131 + 1-21 = 3 + 2 = 5.