6 Analytic geometry in two dimensions
148 < x < 152. This is a special instance illustrating the fact that if
a and 8 (delta) are numbers for which a > 0, then the setof numbers x
for which a - 3 < x < a + S is the same (see Figure 1.18) as the set of
a-S a x a+S X
Figure 1.18numbers x for which Ix - al < S. It is often convenient to allow 3, the
Greek d, to make us think of a distance. When S > 0, the set of points
(or numbers) x for which a - 5 < x < a + b or Ix - al < S is the set of
points (or numbers) x having distance from a which is less than the dis-
tance S. This set is an interval. A complete understanding of the nature
of the assertion Ix - al < 8 happens to be particularly important. There
will be times when we shall use e as well as 8. For example, Problem 42
of Problems 1.19 will invite attention to matters relating to the simple
but important fact that if x and a are numbers for which(1.181) Ix-21 +Ix-31 <e,
then a cannot be 0.01.Problems 1.19
Each of the following 40 statements is followed by a question mark, which
indicates that the statement may be true or may be false. By drawing appro-
priate figures or otherwise, determine which of the statements are true and which
are false. The answers (0 for false and 1 for true) arergiven at the end of the list
of statements. i1 2<5? 2 -2>-3?' 3 7<7?D
4 7=7?! 5 7<-7?( 6 2<-5?
7 -2 5 -1?1 8 1-41 = 4? 9 1-31 < 2?(^10) -5 = 5? (^11) 1-21 > 0? (^12) jabi _ lalibi?
(^13) If p is the number of pages in this book, then 75 < p < 85?
(^14) If C is the circumference of a circle of radius r, then 6r < C < 6.3r?
(Remember that C = 27rr, where a = 3.14159-1-.)
(^15) Ifx = 5.4, then 29 <x2 < 30?
(^16) If x = 420, then 20<x<21?
17 Ifx=4,then2<x<9?
(^18) If 55x57,then 4;g x59?
(^19) If 55_x57,then 5 <x <7?
20 If5<x<7,then 5<_x;5 7?
(^21) Ifs 5x <7,then 5 <x <7?
22 Ifa<b,then jai <Ibl?
23 If-2<x<2, then0<x2<4?
24 If x2<4,then IxI5_2?