Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

Chapter 1

Algebra and Problem Solving - all with Video Answers

Educators

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Section 1

Some Basics of Algebra

Problem 1

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
A letter that can be any one of a set of numbers is called a(n)_______________________.

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Ankit Gupta

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Problem 2

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
A letter representing a specific number that never-changes is called a(n)_______.

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Ankit Gupta

Numerade Educator

Problem 3

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
When $x=10,$ the_______of the expression 4 x is 40.

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Ankit Gupta

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Problem 4

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
In $a^{b},$ the letter $a$ is called the____________ and the letter $b$ is called the_____________.

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Ankit Gupta

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Problem 5

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
When all variables in a variable expression are replaced by numbers and a result is calculated, we say that we are_______________the expression.

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Ankit Gupta

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Problem 6

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
To calculate $4+12 \div 3 \cdot 2,$ the first operation that we perform is_____________.

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Ankit Gupta

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Problem 7

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
A number that can be written in the form $a / b$ where $a \text { and } b \text { are integers (with } b \neq 0),$ is said to be a(n)__________________number.

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Ankit Gupta

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Problem 8

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
A real number that cannot be written as a quotient of two integers is an example of a(n)____________number.

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Ankit Gupta

Numerade Educator

Problem 9

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
Division can be used to show that $\frac{7}{40}$ can be written as a(n)______________decimal.

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Ankit Gupta

Numerade Educator

Problem 10

Choose from the following list the word or words that best complete each statement.
base,
constant,
division,
evaluating,
exponent,
irrational,
rational,
repeating,
terminating,
value,
variable.
Division can be used to show that $\frac{13}{7}$ can be written as a(n)___________decimal.

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Problem 11

Use mathematical symbols to translate each phrase.
Five less than some number

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Problem 12

Use mathematical symbols to translate each phrase.
Ten more than some number

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Problem 13

Use mathematical symbols to translate each phrase.
Twice a number

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Problem 14

Use mathematical symbols to translate each phrase.
Eight times a number

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Problem 15

Use mathematical symbols to translate each phrase.
Twenty-nine percent of some number

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Problem 16

Use mathematical symbols to translate each phrase.
Thirteen percent of some number

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Problem 17

Use mathematical symbols to translate each phrase.
Six less than half of a number

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Problem 18

Use mathematical symbols to translate each phrase.
Three more than twice a number

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Problem 19

Use mathematical symbols to translate each phrase.
Seven more than ten percent of some number

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Problem 20

Use mathematical symbols to translate each phrase.
Four less than six percent of some number

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Problem 21

Use mathematical symbols to translate each phrase.
One more than the difference of two numbers

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Problem 22

Use mathematical symbols to translate each phrase.
One more than the difference of two numbers

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Problem 23

Use mathematical symbols to translate each phrase.
Ninety miles per every four gallons of gas

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Problem 24

Use mathematical symbols to translate each phrase.
One hundred words per every sixty seconds

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Problem 25

garden with the given length of a side. Use$A=s^{2}$$$
\text { Side }=6 \mathrm{ft}
$$

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Problem 26

Find the area of a square flower garden with the given length of a side. Use$A=s^{2}$
$$
\text { Side }=12 \mathrm{ft}
$$

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Ankit Gupta

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Problem 27

Find the area of a square flower garden with the given length of a side. Use$A=s^{2}$
$$
\text { Side }=0.5 \mathrm{m}
$$

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Problem 28

Find the area of a square flower garden with the given length of a side. Use$A=s^{2}$
$$
\text { Side }=2.5 \mathrm{m}
$$

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Ankit Gupta

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Problem 29

Find the area of a triangular fireplace with the given base and height. Use $A=\frac{1}{2} b h$
(image cannot copy)
$$
\text { Base }=5 \mathrm{ft}, \text { height }=7 \mathrm{ft}
$$

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Ankit Gupta

Numerade Educator

Problem 30

Find the area of a triangular fireplace with the given base and height. Use $A=\frac{1}{2} b h$
(image cannot copy)
$$
\text { Base }=2.9 \mathrm{m}, \text { height }=2.1 \mathrm{m}
$$

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Ankit Gupta

Numerade Educator

Problem 31

Find the area of a triangular fireplace with the given base and height. Use $A=\frac{1}{2} b h$
(image cannot copy)
$$
\text { Base }=7 \mathrm{ft}, \text { height }=3.2 \mathrm{ft}
$$

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Ankit Gupta

Numerade Educator

Problem 32

Find the area of a triangular fireplace with the given base and height. Use $A=\frac{1}{2} b h$
(image cannot copy)
$$
\text { Base }=3.6 \mathrm{ft}, \text { height }=4 \mathrm{ft}
$$

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Ankit Gupta

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Problem 33

Evaluate each expression using the values provided.
$$
3(x-7)+2, \text { for } x=10
$$

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Problem 34

Evaluate each expression using the values provided.
$$
5+(2 x-3), \text { for } x=8
$$

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Problem 35

Evaluate each expression using the values provided.
$$
12+3(n+2)^{2}, \text { for } n=1
$$

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Problem 36

Evaluate each expression using the values provided.
$$
(n-10)^{2}-8, \text { for } n=15
$$

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Problem 37

Evaluate each expression using the values provided.
$4 x+y,$ for $x=2$ and $y=3$

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Problem 38

Evaluate each expression using the values provided.
$8 a-b,$ for $a=5$ and $b=7$

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Problem 39

Evaluate each expression using the values provided.
$20+r^{2}-s,$ for $r=5$ and $s=10$

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Problem 40

Evaluate each expression using the values provided.
$m^{3}+7-n,$ for $m=2$ and $n=8$

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Problem 41

Evaluate each expression using the values provided.
$2 c \div 3 b,$ for $b=2$ and $c=6$

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Problem 42

Evaluate each expression using the values provided.
$3 z \div 2 y,$ for $y=1$ and $z=6$

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Problem 43

Evaluate each expression using the values provided.
$3 n^{2} p-3 p n^{2},$ for $n=5$ and $p=9$

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Problem 44

Evaluate each expression using the values provided.
$2 a^{3} b-2 b^{2},$ for $a=3$ and $b=7$

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Problem 45

Evaluate each expression using the values provided.
$5 x \div(2+x-y),$ for $x=6$ and $y=2$

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Problem 46

Evaluate each expression using the values provided.
$3(m+2 n) \div m,$ for $m=7$ and $n=0$

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Problem 47

Evaluate each expression using the values provided.
$[10-(a-b)]^{2},$ for $a=7$ and $b=2$

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Problem 48

Evaluate each expression using the values provided.
$[17-(x+y)]^{2},$ for $x=4$ and $y=1$

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Problem 49

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically.
This may involve looking at an earlier exercise or example,
or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem.
Evaluate each expression using the values provided.
$[5(r+s)]^{2},$ for $r=1$ and $s=2$

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Ankit Gupta

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Problem 50

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically.
This may involve looking at an earlier exercise or example,
or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem.
Evaluate each expression using the values provided.
$$
[3(a-b)]^{2}, \text { for } a=7 \text { and } b=5
$$

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Ankit Gupta

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Problem 51

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically.
This may involve looking at an earlier exercise or example,
or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem.
Evaluate each expression using the values provided.
$$
x^{2}-[3(x-y)]^{2}, \text { for } x=6 \text { and } y=4
$$

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Problem 52

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically.
This may involve looking at an earlier exercise or example,
or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem.
Evaluate each expression using the values provided.
$$
m^{2}-[2(m-n)]^{2}, \text { for } m=7 \text { and } n=5
$$

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Ankit Gupta

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Problem 53

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically.
This may involve looking at an earlier exercise or example,
or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem.
Evaluate each expression using the values provided.
$$
(m-2 n)^{2}-2(m+n), \text { for } m=8 \text { and } n=1
$$

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Ankit Gupta

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Problem 54

To the student and the instructor: Throughout this text selected exercises are marked with the icon Ahal. Students who pause to inspect an Ahal exercise should find the answer more readily than those who proceed mechanically.
This may involve looking at an earlier exercise or example,
or performing calculations in a more efficient manner. Some Ahal exercises are left unmarked to encourage students to always pause before working a problem.
Evaluate each expression using the values provided.
$$
(r-s)^{2}-3(2 r-s), \text { for } r=11 \text { and } s=3
$$

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Problem 55

Use roster notation to write each set.
The set of letters in the word "algebra"

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Problem 56

Use roster notation to write each set.
The set of all days of the week

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Problem 57

Use roster notation to write each set.
The set of all odd natural numbers

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Problem 58

Use roster notation to write each set.
The set of all even natural numbers

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Problem 59

Use roster notation to write each set.
The set of all natural numbers that are multiples of 10

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Problem 60

Use roster notation to write each set.
The set of all natural numbers that are multiples of 5

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Problem 61

Use set-builder notation to write each set.
The set of all even numbers between 9 and 99

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Problem 62

Use set-builder notation to write each set.
The set of all multiples of 5 between 7 and 79

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Problem 63

Use set-builder notation to write each set.
$\{0,1,2,3,4\}$

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Problem 64

Use set-builder notation to write each set.
$\{-3,-2,-1,0,1,2\}$

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Problem 65

Use set-builder notation to write each set.
$\{11,13,15,17,19\}$

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Problem 66

Use set-builder notation to write each set.
$\{24,26,28,30,32\}$

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Problem 67

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?.
$$
-8.7,-3,0, \frac{2}{3}, \sqrt{7}, 6
$$

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Ankit Gupta

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Problem 68

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?.
$$
-\frac{9}{2},-4,-1.2,0, \sqrt{5}, 3
$$

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Ankit Gupta

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Problem 69

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?.
$$
-17,-0.01,0, \frac{5}{4}, 8, \sqrt{77}
$$

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Ankit Gupta

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Problem 70

Which numbers in the list provided are (a) whole numbers? (b) integers? (c) rational numbers? (d) irrational numbers? (e) real numbers?.
$$
-6.08,-5,0,1, \sqrt{17}, \frac{99}{2}
$$

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Ankit Gupta

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Problem 71

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
196 \in \mathbb{N}
$$

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Problem 72

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\mathrm{N} \subseteq W
$$

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Problem 73

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\mathrm{W} \subseteq Z
$$

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Problem 74

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\sqrt{8} \in \mathbb{Q}
$$

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Problem 75

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\frac{2}{3} \in \mathbb{Z}
$$

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Problem 76

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\mathrm{H} \subseteq R
$$

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Problem 77

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\sqrt{10} \in \mathbb{R}
$$

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Problem 78

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
4.3 \notin \mathbb{Z}
$$

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Problem 79

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\mathbb{Z} \nsubseteq N
$$

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Problem 80

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\mathbb{Q} \subseteq R
$$

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Problem 81

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\mathbb{Q} \subseteq Z
$$

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Ankit Gupta

Numerade Educator

Problem 82

Classify each statement as either true or false. The following sets are used:
$\mathrm{N}=$ the set of natural numbers; $W=$ the set of whole numbers; $\mathbb{Z}=$ the set of integers; $\mathbb{C}=$ the set of rational numbers; $\mathrm{H}=$ the set of irrational numbers; $\mathrm{R}=$ the set of real numbers.
$$
\frac{8}{15} \in H
$$

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Problem 83

To the student and the instructor: Writing exercises, denoted by $[\%,$ are meant to be answered using sentences. Because answers to many writing exercises will
vary, solutions are not listed at the back of the book.
What is the difference between rational numbers and integers?

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Problem 84

To the student and the instructor: Writing exercises, denoted by $[\%,$ are meant to be answered using sentences. Because answers to many writing exercises will
vary, solutions are not listed at the back of the book.
Charlie insists that $15-4+1 \div 2 \cdot 3$ is $2 .$ What error is he making?

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Problem 85

To the student and the instructor: Synthesis exercises are designed to challenge students to extend the concepts
or skills studied in each section. Many synthesis exercises require the assimilation of skills and concepts from several sections.
5. Is the following true or false, and why?
$$
\{2,4,6\} \subseteq\{2,4,6\}
$$

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Problem 86

To the student and the instructor: Synthesis exercises are designed to challenge students to extend the concepts
or skills studied in each section. Many synthesis exercises require the assimilation of skills and concepts from several sections.
On a quiz, Mia answers $6 \in \mathbb{Z}$ while Giovanni writes $\{6\} \in \mathbb{Z} .$ Giovanni's answer does not receive full credit while Mia's does. Why?

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Problem 87

Translate to an algebraic expression.
The quotient of the sum of two numbers and their difference

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Problem 88

Translate to an algebraic expression.
Three times the sum of the cubes of two numbers

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Problem 89

Translate to an algebraic expression.
Half of the difference of the squares of two numbers

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Problem 90

Translate to an algebraic expression.
The product of the difference of two numbers and their sum

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Problem 91

Translate to an algebraic expression.
The set of all whole numbers that are not natural numbers

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Problem 92

Translate to an algebraic expression.
The set of all integers that are not whole numbers

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Problem 93

Translate to an algebraic expression.
$\{x | x=5 n, n \text { is a natural number }\}$

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Problem 94

Translate to an algebraic expression.
$\{x | x=3 n, n \text { is a natural number }\}$

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Problem 95

Translate to an algebraic expression.
$\{x | x=2 n+1, n \text { is a whole number }\}$

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Problem 96

Translate to an algebraic expression.
$$
\{x | x=2 n, n \text { is an integer }\}
$$

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Problem 97

Translate to an algebraic expression.
Draw a right triangle that could be used to measure $\sqrt{13}$ units.

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