Hungerford T.W., Shaw D.J. Contemporary Precalculus: A Graphing Approach
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If b is a real number, then the half-line extending to the right or left of b is also
called an interval. Depending on whether or not b is included, there are four pos-
sibilities.
Some examples are shown in Figure 1–4.
Figure 1–4
In a similar vein, (, ) denotes the set of all real numbers.
NEGATIVE NUMBERS AND NEGATIVES OF NUMBERS
The positive numbers are those to the right of 0 on the number line, that is,
All numbers c with c 0.
The negative numbers are those to the left of 0, that is,
All numbers c with c 0.
The nonnegative numbers are the numbers c with c 0.
The word “negative” has a second meaning in mathematics. The negative of
a number c is the number c. For example, the negative of 5 is 5, and the neg-
ative of 3 is (3) 3. Thus the negative of a negative number is a positive
number. Zero is its own negative, since 0 0. In summary,
−2−3−4−5
[2, ∞)
Interval Picture on the number line
325410−1
−2−3−4−5325410−1
−2−3−4−5325410−1
−2−3−4−5325410−1
(−3.5, ∞)
(−∞, 2.75]
(−∞, −1)
6 CHAPTER 1 Basics
Interval
Notation
[b, ) denotes the set of all real numbers x such that x b.
(b, ) denotes the set of all real numbers x such that x b.
(, b] denotes the set of all real numbers x such that x b.
(, b) denotes the set of all real numbers x such that x b.
NOTE
The symbol is read “infinity,” and we
call the set [b, ) “the interval from b
to infinity.” The symbol does not de-
note a real number; it is simply part of
the notation used to label the first two
sets of numbers defined in the previous
box. Analogous remarks apply to the
symbol , which is read “negative
infinity.”
Negatives
The negative of the number c is c.
If c is a positive number, then c is a negative number.
If c is a negative number, then c is a positive number.
TECHNOLOGY TIP
To enter a negative number, such as
5, on most calculators, you must use
the negation key: () 5. If you use the
subtraction key on such calculators
and enter 5, the display will read
ANS 5
which tells the calculator to subtract 5
from the previous answer.
SCIENTIFIC NOTATION
In mid-2006 the U.S. national debt was 8.4 trillion dollars. Since one trillion is
10
12
(that is, 1 followed by 12 zeros), the national debt is the number
8.4 10
12
.
This is an example of scientific notation. A positive number is said to be in scien-
tific notation when it is written in the form
a 10
n
where 1 a 10 and n is an integer.
You should be able to translate between scientific notation and ordinary notation
and vice versa.
EXAMPLE 1
Express these numbers in ordinary notation:
(a) 1.55 10
9
(b) 2.3 10
8
SOLUTION
(a) 1.55 10
9
1.55 1,000,000,000 1,550,000,000
You can do this multiplication in your head by using the fact that multiplying
by 10
9
is equivalent to moving the decimal point 9 places to the right.
(b) 2.3 10
8
1
2
0
.3
8
.000000023*
This computation can also be done mentally by using the fact that dividing by
10
8
is equivalent to moving the decimal point 8 places to the left. ■
EXAMPLE 2
Write each of these numbers in scientific notation:
(a) 356 (b) 1,564,000 (c) .072 (d) .00000087
SOLUTION In each case, move the decimal point to the left or right to obtain
a number between 1 and 10; then write the original number in scientific notation
as follows.
(a) Move the decimal point in 356 two places to the left to obtain the number
3.56, which is between 1 and 10. You can get the original number back by
multiplying by 100: 356 3.56 100. So the scientific notation form is
356 3.56 10
2
Note that the original decimal point is moved 2 places to the left and 10 is
raised to the power 2.
(b) 1,564,000 1.564 1,000,000 1.564 10
6
[Decimal point is moved
6 places to the left, and 10 is raised to the 6th power.]
(c) .072 7.2
1
1
00
7.2 10
2
[Decimal point is moved 2 places to the
right, and 10 is raised to 2.]
SECTION 1.1 The Real Number System 7
*Negative exponents are explained in the first section of the Algebra Review Appendix.
TECHNOLOGY TIP
To enter 1.55 10
9
on a calculator,
use the EE key (labeled EEX on HP and
10
x
on Casio). Key in
1.55 EE 9
and the display reads
1.55 E 9.
Similarly, 2.3 10
8
is entered
as 2.3 EE 8 and the display reads
2.3 E 8.
(d) .00000087 8.7
10,00
1
0,000
8.7 10
7
[Decimal point is moved 7
places to the right, and 10 is raised to the 7.] ■
Scientific notation is useful for computations with very large or very small
numbers.
EXAMPLE 3
(.00000002)(4,300,000,000) (2 10
8
)(4.3 10
9
)
2(4.3)10
89
(8.6)10
1
86. ■
Calculators automatically switch to scientific notation whenever a number
is too large or too small to be displayed in the standard way. If you try to enter
a number with more digits than the calculator can handle, such as
45,000,000,333,222,111, a typical calculator will approximate it using scientific
notation as 4.500000033 E 16, that is, as 45,000,000,330,000,000.
SQUARE ROOTS
A square root of a nonnegative real number d is any number whose square is d.
For instance both 5 and 5 are square roots of 25 because
5
2
25 and (5)
2
25.
The nonnegative square root (in this case, 5) is given a special name and notation.
8 CHAPTER 1 Basics
For example,
25 5 because 5 0 and 5
2
25.
The radical symbol always denotes a nonnegative number. To express the nega-
tive square root of 25 in terms of radicals, we write 5 25.
Although 25 is a real number, the expression 25
is not defined in the
real numbers because there is no real number whose square is 25. In fact, since
the square of every real number is nonnegative,
No negative number has a square root in the real numbers.
Some square roots can be found (or verified) by hand, such as
225
15 and 1.21
1.1.
Usually, however, a calculator is needed to obtain rational approximations of
roots. For instance, we know that 87 is between 9 and 10 because 9
2
81 and
10
2
100. A calculator shows that 879.327379.
†
*The symbol 1
is called a radical.
†
means “approximately equal.”
If d is a nonnegative real number, the principal square root of d is the
nonnegative number whose square is d. It is denoted d
.* Thus,
d
0 and
d
2
d.
TECHNOLOGY TIP
To compute 7
2
51
3 on a
calculator, you must use a pair of
parentheses:
(7
2
5
1)
3.
Otherwise the calculator will not
compute the correct answer, which is:
7
2
51
3 49 5
1
3
100
3
13.
Try it!
We shall often use the following property of square roots:
cd
c
d
for any nonnegative real numbers c and d.
For example, 9 16
9
16 3 4 12. But be careful—there is no similar
property for sums, as the Caution in the margin demonstrates.
EXAMPLE 4
Suppose you are located h feet above the ground. Because of the curvature of the
earth, the maximum distance you can see is approximately d miles, where
d
1.5h
(3.587
10
8
)h
2
.
How far can you see from the 500-foot-high Smith Tower in Seattle and from the
1454-foot-high Sears Tower in Chicago?
SOLUTION For the Smith Tower, substitute 500 for h in the formula, and use
your calculator:
d
1.5(50
0) (
3.587
10
8
)500
2
27.4 miles.
For the Sears Tower, you can see almost 47 miles because
d
1.5(14
54)
(3.587
10
8
)1454
2
46.7 miles. ■
ABSOLUTE VALUE
On an informal level, most students think of absolute value like this:
The absolute value of a positive number is the number itself.
The absolute value of a negative number is found by “erasing the minus sign.”
If c denotes the absolute value of c, then, for example, 5 5 and 4 4.
This informal approach is inadequate, however, for finding the absolute value
of a number such as p 6. It doesn’t make sense to “erase the minus sign” here.
So we must develop a more precise definition. The statement 5 5 suggests that
the absolute value of a nonnegative number ought to be the number itself. For
negative numbers, such as 4, note that 4 4 (4), that is, the absolute
value of the negative number 4 is the negative of 4. These facts are the basis
of the formal definition.
EXAMPLE 5
(a) 3.5 3.5 and 7/2 (7/2) 7/2.
(b) To find p 6, note that p 3.14, so p 6 0. Hence, p 6 is defined
to be the negative of p 6, that is,
p 6 (p 6) p 6.
(c) 5 2
5 2
because 5 2
0. ■
SECTION 1.1 The Real Number System 9
Absolute
Value
The absolute value of a real number c is denoted c and is defined as follows.
If c 0, then c c.
If c 0, then c c.
CAUTION
If c and d are positive real numbers,
then
c d
c
d
.
For example,
9 16
25
5,
but
9
16
3 4 7.
Here are the important facts about absolute value.
EXAMPLE 6
Here are examples of the last three properties in the box.
3. 3 3 and 3 3, so 3 3.
4. If c 6 and d 2, then
cd 6(2) 12 12
and
c
d 6
2 6
2 12,
so cd c
d.
5. If c 5 and d 4, then
d
c
4
5
5
4
5
4
and
d
c
4
5
5
4
,
so
d
c
d
c
. ■
When c is a positive number, then
c
2
c, but when c is negative, this is
false. For example, if c 3, then
c
2
(3)
2
9
3(not 3),
so
c
2
c. In this case, however, c 3 3, so
c
2
c. The same thing
is true for any negative number c. It is also true for positive numbers (since
c c when c is positive). In other words,
10 CHAPTER 1 Basics
Properties of
Absolute Value
Property Description
1. c 0 The absolute value of a number is
nonnegative.
2. If c 0, then c 0. The absolute value of a nonzero number
is positive.
3. c c A number and its negative have the same
absolute value.
4. cd c
d The absolute value of the product of two
numbers is the product of their absolute
values.
5.
d
c
d
c
(d 0) The absolute value of the quotient of two
numbers is the quotient of their absolute
values.
Square Roots
of Squares
For every real number c,
c
2
c.
TECHNOLOGY TIP
To find 9 3p on a calculator, key in
Abs (9 3p).
The Abs key is located in this
menu/submenu:
TI: MATH/NUM
Casio: OPTN/NUM
HP-39gs: Keyboard
SECTION 1.1 The Real Number System 11
When dealing with long expressions inside absolute value bars, do the com-
putations inside first, and then take the absolute value.
EXAMPLE 7
(a) 5(2 4) 7 5(2) 7 10 7 3 3.
(b) 4 3 9 4 6 4 6 2. ■
The caution shows that c d c d when c 3 and d 5. In the general
case, we have the following fact.
DISTANCE ON THE NUMBER LINE
Observe that the distance from 5 to 3 on the number line is 8 units:
Figure 1–5
This distance can be expressed in terms of absolute value by noting that
(5) 3 8.
That is, the distance is the absolute value of the difference of the two numbers.
Furthermore, the order in which you take the difference doesn’t matter;
3 (5) is also 8. This reflects the geometric fact that the distance from 5 to
3 is the same as the distance from 3 to 5. The same thing is true in the general
case.
−2−3−4−5−6325410
8 units
−1
The Triangle
Inequality
For any real numbers c and d,
c d c d.
Distance on the
Number Line
The distance between c and d on the number line is the number
c d d c.
CAUTION
When c and d have opposite signs,
c d
is not equal to
c
d
. For example,
when c
3 and d 5, then
c d
3 5
2,
but
c
d
3
5
3 5 8.
EXAMPLE 8
The distance from 4.2 to 9 is 4.2 9 4.8 4.8, and the distance from
6 to 2
is 6 2
. ■
When d 0, the distance formula shows that c 0 c. Hence,
Algebraic problems can sometimes be solved by translating them into equiv-
alent geometric problems. The key is to interpret statements involving absolute
value as statements about distance on the number line.
EXAMPLE 9
Solve the equation x 5 3 geometrically.
SOLUTION We rewrite it as x (5) 3. In this form it states that
The distance between x and 5 is 3 units.*
Figure 1–6 shows that 8 and 2 are the only two numbers whose distance to 5
is 3 units:
Figure 1–6
Thus x 8 and x 2 are the solutions of x 5 3. ■
EXAMPLE 10
The solutions of x 1 2 are all numbers x such that
The distance between x and 1 is greater than or equal to 2.
Figure 1–7 shows that the numbers 2 or more units away from 1 are the numbers
x such that
x 1orx 3.
So these numbers are the solutions of the inequality. ■
Figure 1–7
−2−3−4−5325410−1
2 2
−2−3−4−5−6−7−8−9−10 3210
3
−1
3
12 CHAPTER 1 Basics
Distance
to Zero
c is the distance between c and 0 on the number line.
*It’s necessary to rewrite the equation first because the distance formula involves the difference of two
numbers, not their sum.
EXAMPLE 11
The solutions of x 7 2.5 are all numbers x such that
The distance between x and 7 is less than 2.5.
Figure 1–8 shows that the solutions of the inequality, that is, the numbers within
2.5 units of 7, are the numbers x such that 4.5 x 9.5, that is, the interval
(4.5, 9.5). ■
Figure 1–8
109116587324
2.5
9.5
4.5
2.5
SECTION 1.1 The Real Number System 13
Year Percentage change
1996 3.0
1997 2.3
1998 1.6
1999 2.2
2000 3.4
2001 2.8
2002 1.6
2003 2.3
2004 2.7
2005 2.5
*U.S. Bureau of Labor Statistics; data for 2005 is for the first half.
EXERCISES 1.1
1. Draw a number line and mark the location of each of these
numbers: 0, 7, 8/3, 10, 1, 4.75, 1/2, 5, and 2.25.
2. Use your calculator to determine which of the following
rational numbers is the best approximation of the irrational
number p.
2
7
2
,
3
1
5
13
5
,
1
3
0
3
3
,
,
1
9
0
9
2
3
,
2
7
,5
9
0
8
8
,4
,4
5
2
8
9
,0
,7
0
8
0
7
.
If your calculator says that one of these numbers equals p,
it’s lying. All you can conclude is that the number agrees
with p for as many decimal places as your calculator can
handle (usually 12–14).
In Exercises 3–8, b, c, and d are real numbers such that b 0,
c 0, and d 0. Determine whether the given number is posi-
tive or negative.
3. b 4. c 5. bcd
6. b c 7. bc bd 8. b
2
c c
2
d
In Exercises 9 and 10, use a calculator and list the given num-
bers in order from smallest to largest.
9.
1
3
8
7
9
,
4
6
5
9
8
1
7
,
47
, 6.735,
27
,
2
5
0
2
4
3
0
10.
3
1
8
7
5
7
,
10
,
1
6
8
3
7
, p,
85
, 2.9884
In Exercises 11–19, express the given statement in symbols.
11. 4 is greater than 8.
12. 17 is less than 6.
13. p is less than 100.
14. x is nonnegative.
15. z is greater than or equal to 4.
16. t is negative.
17. d is not greater than 7.
18. c is at most 3.
19. z is at least 17.
In Exercises 20–24, fill the blank with , , or so that the
resulting statement is true.
20. 6 2 21. 5 3
22. 34 .75 23. 3.1416 p
24. 13 .33
The consumer price index for urban consumers (CPI-U) meas-
ures the cost of consumer goods and services such as food,
housing, transportation, medical costs, etc. The table shows the
yearly percentage increase in the CPI-U over a decade.*
14 CHAPTER 1 Basics
In Exercises 25–29, let p denote the yearly percentage increase
in the CPI-U. Find the number of years in this period which
satisfied the given inequality.
25. p 2.8 26. p 2.6 27. p 2.3
28. p 3.0 29. p 3.4
In Exercises 30–36, fill the blank so as to produce two equiva-
lent statements. For example, the arithmetic statement “a is
negative” is equivalent to the geometric statement “the point a
lies to the left of the point 0.”
Arithmetic Statement Geometric Statement
30. a b
31. a lies c units to the right of b
32. a lies between b and c
33. a b 0
34. a is positive
35. a lies to the left of b
36. a b c (b 0)
In Exercises 37–42, draw a picture on the number line of the
given interval.
37. (0, 8] 38. (0, ) 39. [2, 1]
40. (1, 1) 41. (, 0] 42. [2, 7)
In Exercises 43–48, use interval notation to denote the set of
all real numbers x that satisfy the given inequality.
43. 5 x 10 44. 2 x 7
45. 3 x 14 46. 7 x 77
47. x 9 48. x 12
In Exercises 49–53, express the given numbers (based on 2006
estimates) in scientific notation.
49. Population of the world: 6,506,000,000
50. Population of the United States: 298,400,000
51. Average distance from Earth to Pluto: 5,910,000,000,000
meters
52. Radius of a hydrogen atom: .00000000001 meter
53. Width of a DNA double helix: .000000002 meter
In Exercises 54–57, express the given number in normal deci-
mal notation.
54. Speed of light in a vacuum: 2.9979 10
8
miles per second
55. Average distance from the earth to the sun: 1.50 10
11
meters
56. Electron charge: 1.602 10
27
coulomb
57. Proton mass: 1.6726 10
19
kilogram
58. One light-year is the distance light travels in a 365-day year.
The speed of light is about 186,282.4 miles per second.
(a) How long is 1 light-year (in miles)? Express your an-
swer in scientific notation.
(b) Light from the North Star takes 680 years to reach the
earth. How many miles is the North Star from the earth?
59. The gross federal debt was about 8365 billion dollars
in 2006, when the U.S. population was approximately 298.4
million people.
(a) Express the debt and the population in scientific notation.
(b) At that time, what was each person’s share of the fed-
eral debt?
60. Apple reported that it had sold 28 million iPods through the
end of 2005 and that 14 million iPods were sold in the first
quarter of 2006. If the rate in the first quarter of 2006 con-
tinues through the end of 2008, how many iPods will be
sold? Express your answer in scientific notation.
In Exercises 61–68, simplify the expression without using a cal-
culator. Your answer should not have any radicals in it.
61. 2
8
62. 123
63.
3
5
1
5
2
64.
1
2
1
6
1
1
2
65. 6
2
2
3
66. 12
3
27
67. u
4
(u any real number) 68. 3x
75x
3
(x 0)
In Exercises 69–78, simplify, and write the given number with-
out using absolute values.
69. 3 14 70. (2)3 71. 3 2 5
72. 2 2 73. (13)
2
74. 5
2
75. p 2
76. 2
2 77. 3 p 3
78. 4 2
5
In Exercises 79–84, fill the blank with , , or so that the
resulting statement is true.
79. 25 80. 5 2
81. 3 4 82. 3 0
83. 7 1 84. 4 0
In Exercises 85–92, find the distance between the given
numbers.
85. 3 and 4 86. 7 and 107
87. 7 and 15/2 88. 3/4 and 10
89. p and 3 90. p and 3
91. 2
and 3
92. p and 2
93. Galileo discovered that the period of a pendulum depends
only on the length of the pendulum and the acceleration of
gravity. The period T of a pendulum (in seconds) is
T 2p
g
l
,
SECTION 1.1 The Real Number System 15
where l is the length of the pendulum in feet and g
32.2 ft/sec
2
is the acceleration due to gravity. Find the
period of a pendulum whose length is 4 feet.
94. Suppose you are k miles (not feet) above the ground. The
radius of the earth is approximately 3960 miles. At the point
where your line of sight meets the earth, it is perpendicular to
the radius of the earth, as shown in the figure.
(a) Use the Pythagorean Theorem (see the Geometry
Review Appendix) to show that
d
(3960
k)
2
396
0
2
.
(b) Show that the equation in part (a) simplifies to
d
7920k
k
2
.
(c) If you are h feet above the ground, then you are h/5280
miles high (why?). Use this fact and the equation in
part (b) to obtain the formula used in Example 4.
95. According to data from the Center for Science in the Public
Interest, the healthy weight range for a person depends on
the person’s height. For example,
Height Healthy Weight Range (lb)
5 ft 8 in. 143 21
6 ft 0 in. 163 26
Express each of these ranges as an absolute value inequality
in which x is the weight of the person.
96. At Statewide Insurance, each department’s expenses are
reviewed monthly. A department can fail to pass the budget
variance test in a category if either (i) the absolute value of
the difference between actual expenses and the budget is
more than $500 or (ii) the absolute value of the difference
between the actual expenses and the budget is more than 5%
of the budgeted amount. Which of the following items fail
the budget variance test? Explain your answers.
k
d
The wind-chill factor, shown in the table, calculates how a
given temperature feels to a person’s skin when the wind is
taken into account. For example, the table shows that a
temperature of 20°in a 40 mph wind feels like 1°.*
Budgeted Actual
Item Expense ($) Expense ($)
Wages 220,750 221,239
Overtime 10,500 11,018
Shipping and
Postage 530 589
Temperature (°F)
Wind (mph)
Calm5 10 1520 2530 3540
40 36 34 32 30 29 28 28 27
30 25 21 19 17 16 15 14 13
20139643101
10 1 4 7 9 11 12 14 15
0 11 16 19 22 24 26 27 29
10 22 28 32 35 37 39 41 43
20 34 41 45 48 51 53 55 57
30 46 53 58 61 64 67 69 71
40 57 66 71 74 78 80 82 84
In Exercises 97–99, find the absolute value of the difference of
the two given wind-chill factors. For example, the difference
between the wind-chill at 30°with a 15 mph wind and one at
10°with a 10 mph wind is 19(28) 47° or, equiva-
lently, 28 19 47°.
97. 10° with a 25 mph wind and 20° with a 20 mph wind
98. 30° with a 10 mph wind and 10° with a 30 mph wind
99. 30° with a 5 mph wind and 0° with a 10 mph wind
100. The graph shows the number of Schedule C and C-EZ
forms (in millions) that were filed with the IRS over a six-
year period.
†
In what years was the following statement true:
x 19,500,000
600,000,
where x is the number of Schedule C and C-EZ forms in
that year?
In Exercises 101–108, write the given expression without using
absolute values.
101. t
2
102. 2 y
2
103. b 3 if b 3 104. a 5 if a 5
18
19
21
20
’01 ’02 ’03 ’04 ’05
’06
18.3
18.9
19.7
19.8
20.2
20.6
*Table from the Joint Action Group for Temperature Indices, 2001.
†
Internal Revenue Service. These schedules are for self-employed
individuals.