Download Review of Real Numbers

Objective 1.1A
New Vocabulary
set
elements
natural numbers
prime number
composite number
whole numbers
integers
negative integers
positive integers
rational numbers
irrational numbers
real numbers
graph of a real number
variable
additive inverses
opposites
absolute value
New Symbols
is an element of ()
is not an element of (
is less than (<)
is greater than (>)
absolute value | |
Discuss the Concepts
1. Explain the difference between the set of natural numbers and the set of whole numbers.
2. What is a terminating decimal? What is a repeating decimal? Give examples of each.
3. Explain the difference between rational numbers and irrational numbers.
4. What is the additive inverse of a number?
5. What is the absolute value of a number?
Concept Check
Determine whether the statement is true or false.
1. All integers are rational numbers.
True
2. If a number is an integer, then it is a whole number.
3. The absolute value of a number is positive.
False
False; |0| = 0
4. Given |x| = 3, then x is a positive integer.
False
5. If x < 0, then the absolute value of x is -x.
True
Optional Student Activity
Let S be the set of positive integers that have the following properties:
a. when divided by 6 leave a remainder of 5
b. when divided by 5 leave a remainder of 4
c. when divided by 4 leave a remainder of 3
d. when divided by 3 leave a remainder of 2
e. when divided by 2 leave a remainder of 1
1. Find three elements of set S.
The least common multiple of 2, 3, 4, 5, and 6 is 60; one less than any multiple of 60 will generate an
element of S. The form of these elements is 60k - 1, where k is a natural number. Some possible values
are 59, 119, 179, and 239.
2. Find the minimum value of S.
The minimum value occurs when k = 1. Therefore, the minimum value is 59.
Objective 1.1B
New Vocabulary
roster method
infinite set
finite set
empty set
null set
set-builder notation
New Symbols
empty set (or { } )
null set (or { } )
Discuss the Concepts
1. Give an example of a finite set. Give an example of an infinite set.
2. Give an example of the empty set.
Concept Check
Determine whether the statement is true or false.
1. 7 {2, 3, 5, 7, 9}
2. 4  {-8, -4, 0, 4, 8}
True
False
3. {0, 1, 2, 4}False
4. {a} {a, b, c, d, e}
False
5. 5 {x | x  prime numbers } True
6. 19  {x | x  integers }
7. 3.9  {x | x  real numbers }
True
True
Optional Student Activity
1. List the elements from the set { 30, 22,  22 , 
22,
18 14
,
, 3} whose value is between -3 and 5.
5 3
14
3
2. Let A = {0, 2, 4, 6} and B = {1, 2, 3, 4, 5}. Use the roster method to list the elements in the set that
satisfy {x | x  A and x  B} .
{0, 6}
3. One hundred students were asked if they liked country western music or jazz. The results were that 5
students liked neither, 85 liked country western, and 23 liked jazz. How many students liked both types
of music?
13 students
Objective 1.1C
New Vocabulary
union of two sets
intersection of two sets
interval notation
closed interval
open interval
half-open interval
endpoints of an interval
New Symbols
union (
intersection (
infinity (
negative infinity (-
Discuss the Concepts
1. Explain the difference between the union of two sets and the intersection of two sets.
2. Explain the difference between {x|x < 5} and {x|x ≤ 5}.
3. Explain the similarities and differences between open intervals and closed intervals.
Concept Check
1. Is the intersection of two infinite sets always an infinite set? Why or why not?
2. Is the union of two infinite sets always an infinite set? Why or why not?
3. Find A  B
A  {x | x  whole numbers}
B  {x | x  positive integers}
{x | x whole numbers}
4. Find A  B
A  {x | x  rational numbers}
B  {x | x  real numbers}
{x | x rational numbers}
5. Find A  B
A  {x | x  rational numbers}
B  {x | x  irrational numbers}

Optional Student Activity
Some search engines make use of the operators “AND” and “OR.” For instance, in Windows XP, the
instructions for a full-text search state that entering “computer and monitor” returns topics containing
both words. Entering a search for “computers or monitors” returns topics containing either word or both
words.
a. Explain the search described above in the context of the intersection of two sets.
b. Explain how entering a search for “inequality and symbol” differs from a search for “inequality or symbol.”
Optional Student Activity
Given A  {0, 2, 4, 6}, B  {1, 2, 3, 4}, C  {x | x  3 , x  integers}, and D  {x | x  4 , x  integers}, find
each union or intersection.
1. A  C
{4, 6}
2. B  C
{4}
3. B  C
{0, 2, 4}
4. B  D
{1, 2, 3, 4}
5. B  D
{x| x ≤ 4, x  integers}
6. B  D
{x| x ≥ 1, x  integers}
Answers to Writing Exercises
108. -3 > x > 5 means the numbers that are less than -3 and greater than 5. There is no number that is both
less than -3 and greater than 5. Therefore, this is incorrect.
Objective 1.2A
New Vocabulary
multiplicative inverse
reciprocal
Discuss the Concepts
1. Explain the meanings of the words minus and negative.
2. Is subtraction a commutative operation?
No
3. When is a ÷ b = b ÷ a? When a = b and a ≠ 0, b ≠ 0
Concept Check
Determine whether the statement is always true, sometimes true, or never true.
1. The sum of two numbers with opposite signs is negative.
Sometimes true
2. The product of two numbers with the same sign is positive.
3. The sum of a number and its additive inverse is zero.
Always true
Always true
4. The sum of two integers is greater than either of the two integers.
Sometimes true
5. The difference between two integers is smaller than either of the two integers.
Sometimes true
6. If two integers are multiplied, the product is greater than either of the two integers.
7. If the value of -5y is a positive integer, then y is a positive integer.
8. If -8x = 0, then x is a negative integer.
Never true
Optional Student Activity
1. Simplify:
1 – 2 + 3 – 4 + ∙ ∙ ∙ – 98 + 99
50
2. Simplify:
– 1 + 2 – 3 + 4 – 5 + ∙ ∙ ∙ + 98 – 99
Objective 1.2B
Vocabulary to Review
rational number
integer
[1.1A]
[1.1A]
-50
Never true
Sometimes true
New Vocabulary
least common multiple (LCM) of the denominators
greatest common factor (GCF)
Discuss the Concepts
1. Can a number be both a rational and an irrational number?
2. Are there any integers that are not rational numbers? Are there any rational numbers that are not
integers?
3. What are the real numbers?
4. Is there a smallest positive rational number? Is there a largest positive rational number?
5. Given any two distinct rational numbers, is it always possible to find a rational number between the two
given numbers? If so, explain how to find one.
Yes. For example, add the two numbers and then divide by 2.
Concept Check
1. When two rational numbers are multiplied, it is possible for the product to be less than either factor,
greater than either factor, or a number between the two factors. Give examples of each of these
occurrences.
2. Suppose the numerator of a fraction is a fixed number—for instance, 5. How does the value of the
fraction change as the denominator increases?
The fraction decreases in value.
Optional Student Activity
1. If b has a value between 3 and 5 and c has a value between 0.5 and 1, what values is
b
between?
c
3 and 10
2. Express the sum as a single fraction in simplest form.
1
1
1
1


  
1 2 2  3 3  4
9 10
9
10
3. If x and y are positive real numbers and x + y =10, what is the minimum value of
1
1

x
y
?
2
5
Objective 1.2C
New Vocabulary
exponent
base
factored form
exponential form
power
Discuss the Concepts
For the expression 45, which is the base and which is the exponent? What does the base represent? What
does the exponent represent?
Concept Check
Rewrite each expression as an exponential expression.
1. seven to the fourth power
45
2. x to the third power
x3
3. six to the nth power
6n
4. b ∙ b ∙ b ∙ b ∙ b ∙ b ∙ b
b7
Optional Student Activity
1. To express 10 as a sum of different powers of 2, we could write 10 = 23 + 21. The sum of the exponents
of these powers is 4. When 100 is expressed as a sum of different powers of 2, what is the sum of the
exponents of these powers?
26 + 25 + 22; 13
2. Given that n is a positive integer, find the smallest value of n for which 12n is divisible by 29.
5
Objective 1.2D
New Vocabulary
grouping symbols
complex fraction
main fraction bar
New Procedures
Order of Operations Agreement
Discuss the Concepts
You may want to use Exercises 93 and 94 on page 27 for a class discussion of the Order of Operations
Agreement.
Concept Check
Arrange the expressions in order from the least value to the greatest value.
a. 3[(12 ÷ 3) – (-3)] + 5
 15  5 
3
2
 2  9 
b. 30  6 
13
2 24 1
c. 

3 3 1 8
3
d. 0.3 1.3  2.1  4.7
2
 23 
 , d. (4.892), b. (17), and a. (26)
 24 
c.  
Optional Student Activity
1. Simplify:
1
 1  1
 1

2 1    3 1    4 1      10 1  
 2  3
 4
 10 
45
2. Define a @ b as a ∙ b + b. Use this definition to find the value of x @ (y @ z) when x = 1.7, y = 2.3, and
z = -1.8.
-16.038
Answers to Writing Exercises
1. a. Students should paraphrase the rule: Add the absolute values of the numbers; then attach the sign of
the addends.
b. Students should paraphrase the rule: Find the absolute value of each number; subtract the smaller
of the two numbers from the larger; then attach the sign of the number with the larger absolute value.
2. To rewrite 8 – (-12) as addition of the opposite, change the subtraction to addition and change -12 to the
opposite of -12: 8 – (-12) = 8 + 12.
Answers to Writing Exercises
39. a. The least common multiple of two numbers is the smallest number that is a multiple of each of those
numbers.
b. The greatest common factor of two numbers is the largest integer that divides evenly into both
numbers.
40. To divide two fractions, change the division sign to a multiplication sign, write the reciprocal of the
second fraction, and then multiply the two fractions.
Answers to Writing Exercises
93. We need an Order of Operations Agreement to ensure that there is only one way in which an
expression can be correctly simplified.
94. Students should describe the steps in the Order of Operations Agreement:
Step 1: Perform operations inside grouping symbols.
Step 2: Simplify exponential expressions.
Step 3: Do multiplication and division as they occur from left to right.
Step 4: Do addition and subtraction as they occur from left to right.
Objective 1.3A
New Properties
Commutative Property of Addition
Commutative Property of Multiplication
Associative Property of Addition
Associative Property of Multiplication
Addition Property of Zero
Multiplication Property of Zero
Multiplication Property of One
Inverse Property of Addition
Inverse Property of Multiplication
Distributive Property
Vocabulary to Review
additive inverse [1.1A]
multiplicative inverse [1.2A]
reciprocal [1.2A]
Concept Check
Classify each statement below as illustrating the Commutative Property of Addition, the Associative
Property of Addition, the Commutative Property of Multiplication, or the Associative Property of
Multiplication.
a. 13 + 8 = 8 + 13
Comm. Prop. of Add.
b. (6 + 2) + 4 = 6 + (2 + 4)
Assoc. Prop. of Add.
c. 10(5) = 5(10)
Comm. Prop. of Mult.
d. (3 ∙ 8)9 = 3(8 ∙ 9)
e. x + 27 = 27 + x
Assoc. Prop. of Mult.
Comm. Prop. of Add.
f. y + (1 + 6) = (y + 1) + 6
g. pq = qp
Assoc. Prop. of Add.
Comm. Prop. of Mult.
h. x(yz) = (xy)z
Assoc. Prop. of Mult.
Optional Student Activity
A, B, C, and D are four distinct real numbers such that
A+B=A
B∙A=B
C+D=B
C(B + A) = A
C–D=A
Find the values of a. A, b. B, c. C, and d. D.
a. A = 2, b. B = 0, c. C = 1, and d. D = -1.
Optional Student Activity
In Objective 1.2D, complex fractions were simplified by rewriting the numerator and denominator of the
complex fraction as single fractions and then dividing the numerator by the denominator. However, a
different approach is to multiply the numerator and denominator by the LCM of the denominators. Now that
the Distributive Property has been presented, you might have students simplify a complex fraction by this
alternative method. Here is the simplification of the complex fraction on page 23.
3 1
3 1
3
1

60(  ) 60( )  60( )
4 3
4 3 
4
3
1
1
1
 2 60(  2) 60( )  60(2)
5
5
5
45  20
25
25



12  120 108
108
Objective 1.3B
New Vocabulary
variable expression
terms
variable terms
constant term
numerical coefficient
variable part of a variable term
evaluating a variable expression
Discuss the Concepts
Can the value of a variable ever equal 0? If not, explain why not. If so, give an example of an actual
situation in which it makes sense for the value of a variable to be 0.
Concept Check
Find the smallest possible value for the expression x 
4
1
(when x = 2)
2
5
when x is a positive integer.
x
Optional Student Activity
Consider the following variable expressions:
a  b
2
, a2  b2 ,  a  b  , a3  b3 ,  a  b  , a4  b4 .
3
4
1. By trying different values of a and b, is
a  b
2
 a 2  b 2 always true?
2. By trying different values of a and b, is
a  b
3
 a 3  b3 always true?
3. By trying different values of a and b, is
a  b
4
 a 4  b 4 always true?
4. On the basis of your answers to 1–3, is
a  b
n
No
No
No
 a n  b n always true when n is a natural number?
No
Objective 1.3C
New Vocabulary
like terms
combining like terms
Concept Check
Name the property that justifies each lettered step used in simplifying the expression.
1. 3  x  y   2 x
a.
3x  3 y   2x
b.
 3 y  3x   2 x
c. 3 y   3x  2 x 
Distributive Prop.
Comm. Prop. of Add.
Ass. Prop. of Add.
d. 3 y   3  2  x
3 y  5x
Distributive Prop.
2. y   3  y 
a.
y   y  3
Comm. Prop. of Add.
b.
 y  y  3
Ass. Prop. of Add.
c.
1y  1y   3
d.
1  1 y  3
2y  3
Mult. Prop. of One
Distributive Prop.
3. 5  3a  1
a. 5  3a   5 1
Distributive Prop.
b. (5  3)a  5(1)
15a  5 1
c. 15a  5
Ass. Prop. of Mult.
Mult. Prop. of One
Optional Student Activity
Simplify:
 x  a  x  b  x  c  . . . x  y  x  z 
0
Objective 1.4A
New Vocabulary
See the list of verbal phrases that translate into mathematical operations.
Concept Check
1. Write five phrases that would translate into the expression y + 6.
For example: the sum of y and 6; the total of y and 6; 6 more than y; 6 added to y; y increased by 6
2. Write four phrases that would translate into the expression c – 18.
For example: 18 less than c; the difference between c and 18; c minus 18; c decreased by 18
3. Write three phrases that would translate into the expression 5b.
For example: 5 times b; the product of 5 and b; 5 multiplied by b
4. Write three phrases that would translate into the expression
x
.
4
For example: x divided by 4; the quotient of x and 4; the ratio of x to 4
Optional Student Activity
Complete each statement with the word even or odd.
1. If k is an odd natural number, then k + 1 is an ______ natural number.
2. If n is a natural number, then twice n is an _____ natural number.
Even
Even
3. If m and n are even natural numbers, then the product of m and n is an ______ natural number.
Even
4. If m and n are odd natural numbers, then the sum of m and n is an _____ natural number.
Even
5. If m and n are odd natural numbers, then the product of m and n is an _______ natural number.
Odd
6. If m is an even natural number and n is an odd natural number, then the sum of m and n is an _____
natural number.
Odd
Objective 1.4B
Concept Check
1. The sum of two numbers is 14. Express both numbers in terms of the same variable. Then show that the
sum of the two expressions is 14.
2. A wire for a guitar is 12 ft long and is cut into two pieces. Express the lengths of the two pieces in terms
of the same variable. Then show that the sum of the two lengths is 12.
Optional Student Activity
Translate each of the following formulas into a variable expression, replacing the word is with an equals
sign. Use the letter P for perimeter and the letter A for area. The symbol for pi is .
1. The perimeter of a rectangle is the sum of twice the length and twice the width.
P = 2L + 2W
2. The area of a rectangle is the product of the length and the width.
3. The area of a triangle is one-half of the base times the height.
4. The circumference of a circle
A = LW
1
A  bh
2
is the product of pi and the diameter.
5. The area of a circle is the product of pi and the square of the radius.
C = d
A = r2
Answers to Focus on Problem Solving: Polya’s Four-Step Process
1. (Exercise 1 is on page 44.)
Understand the problem.
We must determine the number of ounces of water the cup will hold. To do so, we need the volume of
the cup. The dimensions of the cup are given in inches, so the volume will be in cubic inches. We will
need to convert cubic inches to fluid ounces.
Devise a plan.
Consult a reference book to find the formula for the volume of a cone (V 
1
 r 2 h) . The
3
conversion rate for cubic inches to fluid ounces is given on page 44 of the text: 1 in3  0.55 fl oz. The
plan is to find the volume of the cup in cubic inches and then convert the volume to fluid ounces.
Carry out the plan.
Find the volume of the cone.
r  1.5 , h  4
V

1
 r 2h
3
1
 (1.5)2 (4)
3
 9.424778 in3
3
Convert 9.424778 in to fluid ounces.
V  9.424778  0.55
 5.183628 fl oz
The cup will hold about 5.18 fl oz.
Review the solution.
It seems reasonable that a conical cup 4 in. tall would hold about 5 fl oz.
2.
Understand the problem.
We are asked to determine the dimensions of a 12-ounce soft drink can. We need to approximate the
length of a hand. We also need to know the formula for the volume of a right circular cylinder, and we
need to convert 12 fl oz to cubic inches.
Devise a plan.

2
From a reference book, find the formula for the volume of a right circular cylinder V   r h
 . We will
approximate the length of a hand to be 8 in. We can use this approximation and the formula C  2 r to
find the radius of the can. The conversion rate for cubic inches to fluid ounces is 0.55 fl oz  1 in . After
finding the radius of the can and converting 12 fl oz to cubic inches, we will find the height of the can.
3
Carry out the plan.
The length of the hand is 75% of the circumference.
0.75C  8
C  10.67 in.
Use the formula C  2 r to find the radius.
2 r  C
2 r  10.67
r  1.70 in.
Convert 12 fl oz to cubic inches.
12  0.55  21.82 in
3
Use the formula for the volume of a right circular cylinder to find the height of the can.
 r2h  V
 (1.70)2 h  21.82
h  2.40 in.
The radius of the can is approximately 1.70 in. The height is approximately 2.40 in.
Review the solution.
The radius of a conventional soda can is smaller than 1.70 in. (only about 1.25 in.). The height is greater
than 2.40 in. (almost 5 in.). But we can see that the dimensions are “in the right ball park.”
Answers to Projects and Group Activities: Water Displacement
1. The volume of the cylinder is V   r 2 h , where r  2 and h  10 .
V   r2h
V   (2) 2 (10)
V  40
The volume of water displaced is V = LWH, where L  30 , W  20 , and H  x .
V  LWH
40  (30)(20)x
0.21  x
The water will rise approximately 0.21 cm.
2. The volume of
2
2 4

of the sphere is V    r 3  where r  6 .
3
3 3

2 4

V    r3 
3 3

8
V   (6)3  192 
9
The volume of the water displaced is V  LWH
V  LWH
192  (20)(16)x
1.88  x
, where L  20 , W  16 , and H  x .
The water will rise approximately 1.88 in.
3. Find the volume of the statue by finding the volume of the water displaced by the statue.
V  LWH , where L  12 ,
W  12 , and H  0.42 .
V  1212 0.42  60.48
3
The volume of the statue is 60.48 in .
Density  weight ÷ volume
Density  15  60.48  0.25
3
The density of the statue is approximately 0.25 lb/ in .