Download P.1 Real Numbers

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Chapter P
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Prerequisites
P.1 Real Numbers
What you should learn
Real Numbers
Real numbers are used in everyday life to describe quantities such as age, miles
per gallon, and population. Real numbers are represented by symbols such as
⫺5, 9, 0,
4
3,
0.666 . . . , 28.21, 冪2, ␲, and
䊏
䊏
⫺32.
3
冪
Here are some important subsets (each member of subset B is also a member of
set A) of the set of real numbers.
再1, 2, 3, 4, . . .冎
䊏
䊏
䊏
Represent and classify real numbers.
Order real numbers and use inequalities.
Find the absolute values of real numbers
and the distance between two real
numbers.
Evaluate algebraic expressions.
Use the basic rules and properties of
algebra.
Set of natural numbers
Why you should learn it
再0, 1, 2, 3, 4, . . .冎
Set of whole numbers
再. . . , ⫺3, ⫺2, ⫺1, 0, 1, 2, 3, . . .冎
Set of integers
A real number is rational if it can be written as the ratio p兾q of two integers,
where q ⫽ 0. For instance, the numbers
Real numbers are used in every aspect of our
lives, such as finding the surplus or deficit in
the federal budget. See Exercises 83–88 on
page 10.
1
1
125
⫽ 0.3333 . . . ⫽ 0.3, ⫽ 0.125, and
⫽ 1.126126 . . . ⫽ 1.126
3
8
111
are rational. The decimal representation of a rational number either repeats 共as in
⫽ 3.145 兲 or terminates 共as in 12 ⫽ 0.5兲. A real number that cannot be
written as the ratio of two integers is called irrational. Irrational numbers have
infinite nonrepeating decimal representations. For instance, the numbers
173
55
冪2 ⫽ 1.4142135 . . . ⬇ 1.41
␲ ⫽ 3.1415926 . . . ⬇ 3.14
and
are irrational. (The symbol ⬇ means “is approximately equal to.”) Figure P.1
shows subsets of real numbers and their relationships to each other.
Real numbers are represented graphically by a real number line. The point
0 on the real number line is the origin. Numbers to the right of 0 are positive
and numbers to the left of 0 are negative, as shown in Figure P.2. The term
nonnegative describes a number that is either positive or zero.
© Alan Schein Photography/Corbis
Real
numbers
Origin
Negative
direction
Figure P.2
−4
−3
−2
−1
0
1
2
3
Positive
direction
4
− 2.4
−2
− 53
2
−1
0
1
2
3
Every point on the real number line
corresponds to exactly one real number.
Figure P.3
Rational
numbers
The Real Number Line
There is a one-to-one correspondence between real numbers and points on the
real number line. That is, every point on the real number line corresponds to
exactly one real number, called its coordinate, and every real number corresponds to exactly one point on the real number line, as shown in Figure P.3.
−3
Irrational
numbers
One-to-One Correspondence
−3
−2
π
0.75
−1
0
1
2
Integers
Negative
integers
Noninteger
fractions
(positive and
negative)
Whole
numbers
3
Every real number corresponds to
exactly one point on the real number
line.
Natural
numbers
Figure P.1
Zero
Subsets of Real Numbers
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Section P.1
3
Real Numbers
Ordering Real Numbers
One important property of real numbers is that they are ordered.
Definition of Order on the Real Number Line
If a and b are real numbers, a is less than b if b ⫺ a is positive. This order
is denoted by the inequality a < b. This relationship can also be described
by saying that b is greater than a and writing b > a. The inequality a ≤ b
means that a is less than or equal to b, and the inequality b ≥ a means that
b is greater than or equal to a. The symbols <, >, ≤, and ≥, are inequality
symbols.
a
−1
Geometrically, this definition implies that a < b if and only if a lies to the
left of b on the real number line, as shown in Figure P.4.
b
0
1
2
a < b if and only if a lies
to the left of b.
Figure P.4
x≤2
Example 1 Interpreting Inequalities
x
Describe the subset of real numbers represented by each inequality.
a. x ≤ 2
b. x > ⫺1
0
1
2
3
4
Figure P.5
c. ⫺2 ≤ x < 3
Solution
x > −1
a. The inequality x ≤ 2 denotes all real numbers less than or equal to 2, as shown
in Figure P.5.
b. The inequality x > ⫺1 denotes all real numbers greater than ⫺1, as shown in
Figure P.6.
c. The inequality ⫺2 ≤ x < 3 means that x ≥ ⫺2 and x < 3. The “double
inequality” denotes all real numbers between ⫺2 and 3, including ⫺2 but not
including 3, as shown in Figure P.7.
Now try Exercise 31(a).
x
−2
−1
1
0
1
2
3
2
3
Figure P.6
−2 ≤ x < 3
x
−2
−1
0
1
Figure P.7
Inequalities can be used to describe subsets of real numbers called intervals.
In the bounded intervals below, the real numbers a and b are the endpoints of
each interval.
Bounded Intervals on the Real Number Line
Notation
关a, b兴
Interval Type
Closed
Inequality
Graph
a ≤ x ≤ b
x
a
共a, b兲
Open
b
a < x < b
x
a
关a, b兲
a ≤ x < b
共a, b兴
a < x ≤ b
b
x
a
b
a
b
x
STUDY TIP
The endpoints of a closed
interval are included in the
interval. The endpoints of an
open interval are not included
in the interval.
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Prerequisites
The symbols ⬁, positive infinity, and ⫺ ⬁, negative infinity, do not
represent real numbers. They are simply convenient symbols used to describe the
unboundedness of an interval such as 共1, ⬁兲 or 共⫺ ⬁, 3兴.
Unbounded Intervals on the Real Number Line
Notation
关a, ⬁兲
Interval Type
Inequality
x ≥ a
STUDY TIP
Graph
x
a
共a, ⬁兲
x > a
Open
An interval is unbounded when
it continues indefinitely in one
or both directions.
x
a
共⫺ ⬁, b兴
x ≤ b
x
b
共⫺ ⬁, b兲
x < b
Open
x
b
共⫺ ⬁, ⬁兲
Entire real line
⫺⬁ < x <
⬁
x
Example 2 Using Inequalities to Represent Intervals
Use inequality notation to describe each of the following.
a. c is at most 2.
b. All x in the interval 共⫺3, 5兴
Solution
a. The statement “c is at most 2” can be represented by c ≤ 2.
b. “All x in the interval 共⫺3, 5兴” can be represented by ⫺3 < x ≤ 5.
Now try Exercise 35.
Example 3 Interpreting Intervals
Give a verbal description of each interval.
a. 共⫺1, 0兲
b. 关 2, ⬁兲
c. 共⫺ ⬁, 0兲
Solution
a. This interval consists of all real numbers that are greater than ⫺1 and less
than 0.
b. This interval consists of all real numbers that are greater than or equal to 2.
c. This interval consists of all negative real numbers.
Now try Exercise 47.
The Law of Trichotomy states that for any two real numbers a and b,
precisely one of three relationships is possible:
a ⫽ b, a < b, or a > b.
Law of Trichotomy
Additional Examples
Use inequality notation to describe each of
the following.
a. m is at least ⫺3.
b. All x in the interval [⫺2, 2]
c. w is at least 1 and at most 5.
d. All q in the interval 共⫺12, 0兲
Solution
a. m ≥ ⫺3
b. ⫺2 ≤ x ≤ 2
c. 1 ≤ w ≤ 5
d. ⫺12 < q < 0
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Section P.1
5
Real Numbers
Absolute Value and Distance
The absolute value of a real number is its magnitude, or the distance between the
origin and the point representing the real number on the real number line.
Exploration
Definition of Absolute Value
If a is a real number, the absolute value of a is
ⱍaⱍ ⫽ 冦⫺a,
a,
Absolute value expressions can
be evaluated on a graphing
utility. When evaluating an
expression such as 3 ⫺ 8 ,
parentheses should surround the
expression as shown below.
ⱍ
if a ≥ 0
.
if a < 0
ⱍ
Notice from this definition that the absolute value of a real number is never
negative. For instance, if a ⫽ ⫺5, then ⫺5 ⫽ ⫺ 共⫺5兲 ⫽ 5. The absolute value
of a real number is either positive or zero. Moreover, 0 is the only real number
whose absolute value is 0. So, 0 ⫽ 0.
ⱍ ⱍ
ⱍⱍ
Example 4 Evaluating the Absolute Value of a Number
Evaluate
ⱍxⱍ for (a) x > 0 and (b) x < 0.
Evaluate each expression. What
can you conclude?
x
Solution
ⱍⱍ
a. If x > 0, then x ⫽ x and
ⱍxⱍ ⫽ x ⫽ 1.
ⱍⱍ
x
b. If x < 0, then x ⫽ ⫺x and
ⱍⱍ
ⱍ5 ⫺ 2ⱍ
ⱍ ⱍ
ⱍ
ⱍ
a. 6
b. ⫺1
c.
d. 2 ⫺ 5
x
ⱍxⱍ ⫽ ⫺x ⫽ ⫺1.
x
x
Now try Exercise 53.
Properties of Absolute Value
ⱍ ⱍ ⱍⱍ
a
a
⫽ ⱍ ⱍ, b ⫽ 0
b
ⱍbⱍ
ⱍⱍ
2. ⫺a ⫽ a
ⱍ ⱍ ⱍ ⱍⱍ ⱍ
4.
1. a ≥ 0
3. ab ⫽ a b
ⱍⱍ
Absolute value can be used to define the distance between two points on
the real number line. For instance, the distance between ⫺3 and 4 is
ⱍ⫺3 ⫺ 4ⱍ ⫽ ⱍ⫺7ⱍ ⫽ 7
Distance Between Two Points on the Real Number Line
Let a and b be real numbers. The distance between a and b is
ⱍ ⱍ
−3
−2
−1
Figure P.8
as shown in Figure P.8.
ⱍ
7
ⱍ
d共a, b兲 ⫽ b ⫺ a ⫽ a ⫺ b .
0
1
2
3
4
The distance between ⴚ3
and 4 is 7.
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Prerequisites
Algebraic Expressions
One characteristic of algebra is the use of letters to represent numbers. The
letters are variables, and combinations of letters and numbers are algebraic
expressions. Here are a few examples of algebraic expressions.
2x ⫺ 3,
5x,
4
,
x ⫹2
7x ⫹ y
2
Definition of an Algebraic Expression
An algebraic expression is a combination of letters (variables) and real
numbers (constants) combined using the operations of addition, subtraction,
multiplication, division, and exponentiation.
The terms of an algebraic expression are those parts that are separated by
addition. For example,
x 2 ⫺ 5x ⫹ 8 ⫽ x 2 ⫹ 共⫺5x兲 ⫹ 8
has three terms: x 2 and ⫺5x are the variable terms and 8 is the constant term.
The numerical factor of a variable term is the coefficient of the variable term. For
instance, the coefficient of ⫺5x is ⫺5, and the coefficient of x 2 is 1.
To evaluate an algebraic expression, substitute numerical values for each of
the variables in the expression. Here are two examples.
Expression
⫺3x ⫹ 5
3x 2
⫹ 2x ⫺ 1
Value of
Variable
Substitute
Value of
Expression
x⫽3
⫺3共3兲 ⫹ 5
⫺9 ⫹ 5 ⫽ ⫺4
x ⫽ ⫺1
3共⫺1兲 ⫹ 2共⫺1兲 ⫺ 1
3⫺2⫺1⫽0
2
When an algebraic expression is evaluated, the Substitution Principle is
used. It states, “If a ⫽ b, then a can be replaced by b in any expression involving
a.” In the first evaluation shown above, for instance, 3 is substituted for x in the
expression ⫺3x ⫹ 5.
Basic Rules of Algebra
There are four arithmetic operations with real numbers: addition, multiplication,
subtraction, and division, denoted by the symbols ⫹, ⫻ or ⭈ , ⫺, and ⫼ or 兾.
Of these, addition and multiplication are the two primary operations. Subtraction
and division are the inverse operations of addition and multiplication,
respectively.
Subtraction: Add the opposite of b.
a ⫺ b ⫽ a ⫹ 共⫺b兲
Division: Multiply by the reciprocal of b.
If b ⫽ 0, then a兾b ⫽ a
冢b冣 ⫽ b .
1
a
In these definitions, ⫺b is the additive inverse (or opposite) of b, and 1兾b is the
multiplicative inverse (or reciprocal) of b. In the fractional form a兾b, a is the
numerator of the fraction and b is the denominator.
Common Error
A common error is to use the wrong order
of operations when evaluating expressions.
Remind students that the order of operations
is as follows.
1. First do operations that occur within
symbols of grouping.
2. Then evaluate exponential expressions.
3. Then do multiplications and divisions
from left to right.
4. Finally do additions and subtractions
from left to right.
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Section P.1
Real Numbers
Because the properties of real numbers below are true for variables and
algebraic expressions, as well as for real numbers, they are often called the Basic
Rules of Algebra. Try to formulate a verbal description of each property. For
instance, the Commutative Property of Addition states that the order in which two
real numbers are added does not affect their sum.
Basic Rules of Algebra
Let a, b, and c be real numbers, variables, or algebraic expressions.
Property
Example
Commutative Property of Addition:
a⫹b⫽b⫹a
4x ⫹
Commutative Property of Multiplication:
ab ⫽ ba
共1 ⫺ x兲 x 2 ⫽ x 2共1 ⫺ x兲
Associative Property of Addition:
共a ⫹ b兲 ⫹ c ⫽ a ⫹ 共b ⫹ c兲
共x ⫹ 5兲 ⫹ x 2 ⫽ x ⫹ 共5 ⫹ x 2兲
Associative Property of Multiplication:
共ab兲 c ⫽ a共bc兲
共2x ⭈ 3y兲共8兲 ⫽ 共2x兲共3y ⭈ 8兲
Distributive Properties:
a共b ⫹ c兲 ⫽ ab ⫹ ac
3x共5 ⫹ 2x兲 ⫽ 3x ⭈ 5 ⫹ 3x ⭈ 2x
x2
⫽ x 2 ⫹ 4x
共 y ⫹ 8兲 y ⫽ y ⭈ y ⫹ 8 ⭈ y
共a ⫹ b兲c ⫽ ac ⫹ bc
Additive Identity Property:
a⫹0⫽a
5y 2 ⫹ 0 ⫽ 5y 2
Multiplicative Identity Property:
a
⭈1⫽a
共4x 2兲共1兲 ⫽ 4x 2
Additive Inverse Property:
a ⫹ 共⫺a兲 ⫽ 0
Multiplicative Inverse Property:
a
1
⭈ a ⫽ 1,
a⫽0
6x 3 ⫹ 共⫺6x 3兲 ⫽ 0
共x 2 ⫹ 4兲
冢x
2
冣
1
⫽1
⫹4
Because subtraction is defined as “adding the opposite,” the Distributive
Properties are also true for subtraction. For instance, the “subtraction form” of
a共b ⫹ c兲 ⫽ ab ⫹ ac is a共b ⫺ c兲 ⫽ ab ⫺ ac.
Properties of Negation and Equality
Let a, b, and c be real numbers, variables, or algebraic expressions.
Property
1. 共⫺1兲 a ⫽ ⫺a
Example
共⫺1兲7 ⫽ ⫺7
2. ⫺ 共⫺a兲 ⫽ a
⫺ 共⫺6兲 ⫽ 6
3. 共⫺a兲b ⫽ ⫺ 共ab兲 ⫽ a共⫺b兲
共⫺5兲3 ⫽ ⫺ 共5 ⭈ 3兲 ⫽ 5共⫺3兲
4. 共⫺a兲共⫺b兲 ⫽ ab
共⫺2兲共⫺x兲 ⫽ 2x
5. ⫺ 共a ⫹ b兲 ⫽ 共⫺a兲 ⫹ 共⫺b兲
⫺ 共x ⫹ 8兲 ⫽ 共⫺x兲 ⫹ 共⫺8兲 ⫽ ⫺x ⫺ 8
6. If a ⫽ b, then a ⫹ c ⫽ b ⫹ c.
1
2
⫹ 3 ⫽ 0.5 ⫹ 3
共2兲 ⫽ 16共2兲
7
8. If a ⫹ c ⫽ b ⫹ c, then a ⫽ b. 1.4 ⫺ 1 ⫽ 5 ⫺ 1
7. If a ⫽ b, then ac ⫽ bc.
42
9. If ac ⫽ bc and c ⫽ 0, then a ⫽ b.
3 冪9
⫽
4
4
STUDY TIP
Be sure you see the difference
between the opposite of a
number and a negative number.
If a is already negative, then its
opposite, ⫺a, is positive. For
instance, if a ⫽ ⫺2, then
⫺a ⫽ ⫺ 共⫺2兲 ⫽ 2.
7
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Chapter P
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Prerequisites
Properties of Zero
STUDY TIP
Let a and b be real numbers, variables, or algebraic expressions.
1. a ⫹ 0 ⫽ a and
3.
0
⫽ 0,
a
a⫺0⫽a
2. a
a⫽0
4.
⭈0⫽0
a
is undefined.
0
5. Zero-Factor Property: If ab ⫽ 0, then a ⫽ 0 or b ⫽ 0.
The “or” in the Zero-Factor
Property includes the possibility
that either or both factors may
be zero. This is an inclusive
or, and it is the way the word
“or” is generally used in
mathematics.
Properties and Operations of Fractions
Let a, b, c, and d be real numbers, variables, or algebraic expressions such
that b ⫽ 0 and d ⫽ 0.
1. Equivalent Fractions:
a
c
⫽
b d
if and only if
a ⫺a
a
⫽
2. Rules of Signs: ⫺ ⫽
b
b
⫺b
3. Generate Equivalent Fractions:
and
ad ⫽ bc.
⫺a a
⫽
⫺b b
a ac
⫽ , c⫽0
b bc
a ±c
a c
4. Add or Subtract with Like Denominators: ± ⫽
b b
b
5. Add or Subtract with Unlike Denominators:
6. Multiply Fractions:
7. Divide Fractions:
a
b
c
x
2x 5
⫹
⫽
3
5
ac
⭈ d ⫽ bd
a
c
a
⫼ ⫽
b d b
b.
7 3 7
⫼ ⫽
x
2
x
⭈ x ⫹ 3 ⭈ 2x ⫽ 11x
15
2
ⱍ ⱍ
a c
ad ± bc
± ⫽
b d
bd
d
ad
⭈ c ⫽ bc ,
c⫽0
Example 5 Properties and Operations of Fractions
a.
Activities
1. Use inequality notation to describe the
set of nonnegative numbers.
Answer: x ≥ 0
2. Use interval notation to describe the
inequality ⫺6 < x ≤ 13.
Answer: 共⫺6, 13兴
3. Evaluate: 4 ⫺ ⫺3 .
Answer: 1
4. Find the distance between ⫺41 and 16.
Answer: 57
15
14
⭈ 3 ⫽ 3x
Add fractions with unlike denominators.
Divide fractions.
STUDY TIP
In Property 1 of fractions, the
phrase “if and only if” implies
two statements. One statement
is: If a兾b ⫽ c兾d, then ad ⫽ bc.
The other statement is: If
ad ⫽ bc, where b ⫽ 0 and
d ⫽ 0, then a兾b ⫽ c兾d.
Now try Exercise 113.
If a, b, and c are integers such that ab ⫽ c, then a and b are factors or
divisors of c. A prime number is an integer that has exactly two positive factors:
itself and 1. For example, 2, 3, 5, 7, and 11 are prime numbers. The numbers 4,
6, 8, 9, and 10 are composite because they can be written as the product of two
or more prime numbers. The number 1 is neither prime nor composite. The
Fundamental Theorem of Arithmetic states that every positive integer greater
than 1 can be written as the product of prime numbers. For instance, the prime
factorization of 24 is 24 ⫽ 2 ⭈ 2 ⭈ 2 ⭈ 3.
Point out to students that to add or subtract
fractions with unlike denominators, they
can either use Property 5 of fractions as
in Example 5(a), or they can rewrite the
fractions with like denominators using
the least common denominator (LCD)
of the fractions.
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Section P.1
P.1 Exercises
Real Numbers
9
See www.CalcChat.com for worked-out solutions to odd-numbered exercises.
Vocabulary Check
Fill in the blanks.
1. A real number is _______ if it can be written as the ratio
p
of two integers, where q ⫽ 0.
q
2. _______ numbers have infinite nonrepeating decimal representations.
3. The distance between a point on the real number line and the origin is the _______ of the real number.
4. Numbers that can be written as the product of two or more prime numbers are called _______ numbers.
5. Integers that have exactly two positive factors, the integer itself and 1, are called _______ numbers.
6. An algebraic expression is a combination of letters called _______ and real numbers called _______ .
7. The _______ of an algebraic expression are those parts separated by addition.
8. The numerical factor of a variable term is the _______ of the variable term.
9. The _______ states: If ab ⫽ 0, then a ⫽ 0 or b ⫽ 0.
In Exercises 1– 6, determine which numbers are (a) natural
numbers, (b) whole numbers, (c) integers, (d) rational
numbers, and (e) irrational numbers.
1. 再 ⫺9, ⫺ 72, 5, 23, 冪2, 0, 1, ⫺4, ⫺1冎
2. 再 冪5, ⫺7, ⫺ 73, 0, 3.12, 54, ⫺2, ⫺8, 3冎
In Exercises 19–24, plot the two real numbers on the real
number line. Then place the correct inequality symbol (< or
>) between them.
19. ⫺4, ⫺8
20. ⫺3.5, 1
3
21. 2, 7
16
22. 1, 3
3. 再2.01, 0.666 . . . , ⫺13, 0.010110111 . . . ,
1, ⫺10, 20冎
23.
4. 再2.3030030003 . . . , 0.7575, ⫺4.63, 冪10, ⫺2,
0.03, ⫺10冎
In Exercises 25–32, (a) verbally describe the subset of real
numbers represented by the inequality, (b) sketch the
subset on the real number line, and (c) state whether the
interval is bounded or unbounded.
1 6 1
5. 再 ⫺ ␲, ⫺ 3, 3, 2冪2, ⫺7.5, ⫺2, 3, ⫺3冎
12
1
6. 再 25, ⫺17, ⫺ 5 , 冪9, 3.12, 2␲, 6, ⫺4, 18冎
In Exercises 7–12, use a calculator to find the decimal form
of the rational number. If it is a nonterminating decimal,
write the repeating pattern.
7.
9.
11.
5
16
41
333
⫺ 100
11
10.
17
4
3
7
12.
218
⫺ 33
8.
5 2
6, 3
8
3
24. ⫺ 7, ⫺ 7
25. x ≤ 5
26. x > 3
27. x < 0
28. x ≥ 4
29. ⫺2 < x < 2
30. 0 ≤ x ≤ 5
31. ⫺1 ≤ x < 0
32. 0 < x ≤ 6
In Exercises 33 –38, use inequality and interval notation to
describe the set.
In Exercises 13 –16, use a calculator to rewrite the rational
number as the ratio of two integers.
13. 4.6
14. 12.3
15. ⫺6.5
16. ⫺1.83
33. x is negative.
34. z is at least 10.
35. y is nonnegative.
36. y is no more than 25.
37. p is less than 9 but no less than ⫺1.
38. The annual rate of inflation r is expected to be at
least 2.5%, but no more than 5%.
In Exercises 17 and 18, approximate the numbers and place
the correct inequality symbol (< or >) between them.
In Exercises 39– 44, use interval notation to describe the
graph.
17.
39.
18.
−2
−7
−1
−6
0
−5
1
−4
2
−3
−2
3
4
−1
0
x
−2 −1
0
1
2
3
4
5
6
−3 −2 −1
0
1
2
3
4
5
40.
x
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Chapter P
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Prerequisites
41.
75. y is at least six units from 0.
x
−4 −2
0
−4
−2
2
4
6
8
10 12
42.
76. y is at most two units from a.
x
0
2
4
77. While traveling on the Pennsylvania Turnpike, you pass
milepost 57 near Pittsburgh, then milepost 236 near
Gettysburg. How many miles do you travel between these
two mileposts?
6
43.
x
−a
a+4
44.
78. The temperature in Bismarck, North Dakota was 60⬚F at
noon, then 23⬚F at midnight. What was the change in
temperature over the 12-hour period?
x
−c + 2
c+1
In Exercises 45 – 48, give a verbal description of the interval.
47. 共⫺ ⬁, 2兴
46. 共⫺ ⬁, 4兴
48. 关1, ⬁兲
In Exercises 49– 54, evaluate the expression.
ⱍ
ⱍ
ⱍⱍ
ⱍ ⱍ ⱍ ⱍ
ⱍx ⫺ 1ⱍ
49. ⫺10
50. 0
51. ⫺3 ⫺3
52. ⫺1 ⫺ ⫺2
53.
54.
ⱍ ⱍ
ⱍx ⫹ 2ⱍ
x⫹2
x⫺1
In Exercises 55 – 60, evaluate the expression for the given
values of x and y. Then use a graphing utility to verify your
result.
ⱍ
ⱍ
ⱍy ⫺ 4xⱍ for x ⫽ 2 and y ⫽ ⫺3
ⱍxⱍ ⫺ 2ⱍyⱍ for x ⫽ 2 and y ⫽ ⫺1
ⱍ2xⱍ ⫺ 3ⱍyⱍ for x ⫽ ⫺4 and y ⫽ 1
55. 2x ⫹ y for x ⫽ ⫺2 and y ⫽ 3
56.
57.
58.
59.
60.
ⱍ ⱍ ⱍⱍ ⱍ ⱍ
ⱍ ⱍ
3 x ⫺ 2y
for x ⫽ ⫺2 and y ⫽ 2
2x ⫹ y
In Exercises 61–66, place the correct symbol 冇<, >, or ⴝ冈
between the pair of real numbers.
ⱍ ⱍ䊏⫺ ⱍ⫺3ⱍ
⫺5䊏⫺ ⱍ5ⱍ
⫺ ⱍ⫺2ⱍ䊏⫺ ⱍ2ⱍ
ⱍ ⱍ䊏ⱍ4ⱍ
⫺ ⱍ⫺6ⱍ䊏ⱍ⫺6ⱍ
62. ⫺4
63.
64.
65.
Budgeted
Expense, b
79. Wages
$112,700
$113,356
80. Utilities
$9400
$9772
81. Taxes
$37,640
$37,335
82. Insurance
$2575
$2613
69. a ⫽
71. a ⫽
⫺ 52, b ⫽ 0
16
112
5 , b ⫽ 75
66. ⫺(⫺2)䊏⫺2
2000
1800
䊏
䊏
䊏
䊏
0.05b
䊏
䊏
䊏
䊏
1579.3
1600
1400
1351.8
1200
1000
1995
1
4,
b⫽
Receipts
11
4
72. a ⫽ 9.34, b ⫽ ⫺5.65
1997
1999
2001
2003
2005
Year
68. a ⫽ ⫺126, b ⫽ ⫺75
70. a ⫽
ⱍa ⫺ bⱍ
2153.9
1991.2
1827.5
1782.3
2200
In Exercises 67– 72, find the distance between a and b.
67. a ⫽ 126, b ⫽ 75
Actual
Expense, a
Federal Deficit In Exercises 83–88, use the bar graph,
which shows the receipts of the federal government (in
billions of dollars) for selected years from 1995 through
2005. In each exercise you are given the expenditures of the
federal government. Find the magnitude of the surplus or
deficit for the year. (Source: U.S. Office of Management and
Budget)
3x ⫹ 2y
for x ⫽ 4 and y ⫽ 1
x
61. ⫺3
Budget Variance In Exercises 79–82, the accounting
department of a company is checking to determine whether
the actual expenses of a department differ from the
budgeted expenses by more than $500 or by more than 5%.
Fill in the missing parts of the table, and determine whether
the actual expense passes the “budget variance test.”
Receipts
(in billions of dollars)
45. 共⫺6, ⬁兲
83. 1995
84. 1997
In Exercises 73 – 76, use absolute value notation to describe
the situation.
85. 1999
73. The distance between x and 5 is no more than 3.
87. 2003
74. The distance between x and ⫺10 is at least 6.
88. 2005
86. 2001
䊏
䊏
䊏
䊏
䊏
䊏
Expenditures
$1515.8 billion
$1601.3 billion
$1701.9 billion
$1863.9 billion
$2157.6 billion
$2472.2 billion
ⱍ
Receipts ⫺
Expenditures
䊏
䊏
䊏
䊏
䊏
䊏
ⱍ
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Page 11
Section P.1
In Exercises 89 – 94, identify the terms. Then identify the
coefficients of the variable terms of the expression.
1
90. 2x ⫺ 9
n
91. 冪3x2 ⫺ 8x ⫺ 11
92. 7冪5x2 ⫹ 3
5兾n
x
⫺5
2
5
In Exercises 95–98, evaluate the expression for each
value of x. (If not possible, state the reason.)
Values
(a) x ⫽ 3
(b) x ⫽ ⫺ 12
96. 4 ⫺ 3x
(a) x ⫽ 2
(b) x ⫽ ⫺ 56
97. x2 ⫺ 4
x2
98.
x⫹4
(a) x ⫽ 2
(b) x ⫽ ⫺2
(a) x ⫽ 1
(b) x ⫽ ⫺4
0.000001
124. (a) Use a calculator to complete the table.
1
10
100
10,000
100,000
5兾n
(b) Use the result from part (a) to make a conjecture
about the value of 5兾n as n increases without bound.
True or False? In Exercises 125 and 126, determine
whether the statement is true or false. Justify your answer.
99. x ⫹ 9 ⫽ 9 ⫹ x
100. 2共 12 兲 ⫽ 1
125. Let a > b, then
1
共h ⫹ 6兲 ⫽ 1, h ⫽ ⫺6
h⫹6
126. Because
102. 共x ⫹ 3兲 ⫺ 共x ⫹ 3兲 ⫽ 0
103. 2共x ⫹ 3兲 ⫽ 2x ⫹ 6
1 1
> , where a ⫽ 0 and b ⫽ 0.
a b
a⫹b a b
c
c
c
⫽ ⫹ , then
⫽ ⫹ .
c
c
c
a⫹b a b
In Exercises 127 and 128, use the real numbers A, B, and C
shown on the number line. Determine the sign of each
expression.
104. 共z ⫺ 2兲 ⫹ 0 ⫽ z ⫺ 2
105. x ⫹ 共 y ⫹ 10兲 ⫽ 共x ⫹ y兲 ⫹ 10
1
0.0001
Synthesis
In Exercises 99–106, identify the rule(s) of algebra
illustrated by the statement.
106. 7共7
0.01
(b) Use the result from part (a) to make a conjecture
about the value of 5兾n as n approaches 0.
n
95. 2x ⫺ 5
101.
0.5
2x3
94. 3x 4 ⫹
Expression
11
123. (a) Use a calculator to complete the table.
89. 7x ⫹ 4
93. 4x 3 ⫹
Real Numbers
⭈ 12兲 ⫽ 共 17 ⭈ 7兲12 ⫽ 1 ⭈ 12 ⫽ 12
C B
A
0
In Exercises 107–116, perform the operation(s). (Write
fractional answers in simplest form.)
107.
109.
111.
5
3
16 ⫹ 16
5
1
5
8 ⫺ 12 ⫹ 6
108.
110.
x
3x
⫹
6
4
112.
12 1
⫼
113.
x
8
115.
6
4
7 ⫺ 7
10
6
11 ⫹ 33
⫺ 13
66
2x
x
⫹
5
10
11 3
⫼
114.
x
4
共25 ⫼ 4兲 ⫺ 共4 ⭈ 38 兲
116.
共 35 ⫼ 3兲 ⫺ 共6 ⭈ 48 兲
In Exercises 117–122, use a calculator to evaluate the
expression. (Round your answer to two decimal places.)
117. 14共⫺3 ⫹
3
7
兲
11.46 ⫺ 5.37
119.
3.91
121.
2
3 共⫺2
⫺ 6兲
⫺25
118. 3共
5
⫺ 12
⫹
3
8
兲
12.24 ⫺ 8.4
120.
2.5
122.
1
5 共⫺8
⫺ 9兲
⫺ 13
127. (a) ⫺A
128. (a) ⫺C
(b) B ⫺ A
129. Exploration
(b) A ⫺ C
ⱍ
ⱍ
ⱍⱍ ⱍⱍ
Consider u ⫹ v and u ⫹ v .
(a) Are the values of the expressions always equal? If not,
under what conditions are they not equal?
(b) If the two expressions are not equal for certain values
of u and v, is one of the expressions always greater
than the other? Explain.
130. Think About It Is there a difference between saying that
a real number is positive and saying that a real number is
nonnegative? Explain.
131. Writing Describe the differences among the sets of
whole numbers, natural numbers, integers, rational
numbers, and irrational numbers.
ⱍⱍ
132. Writing Can it ever be true that a ⫽ ⫺a for any real
number a? Explain.