• Home
  • Textbooks
  • Algebra and Trigonometry
  • Basic Concepts of Algebra

Algebra and Trigonometry

Judith A. Beecher, Judith A. Penna, Marvin L. Bittinger

Chapter 0

Basic Concepts of Algebra - all with Video Answers

Educators

AG

Section 1

The Real-Number System

View

Problem 1

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are rational numbers?

Daniel Carr
Daniel Carr
Numerade Educator
01:35

Problem 2

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are natural numbers?

AG
Ankit Gupta
Numerade Educator
02:07

Problem 3

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are irrational numbers?

AG
Ankit Gupta
Numerade Educator
00:37

Problem 4

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are integers?

AG
Ankit Gupta
Numerade Educator
01:38

Problem 5

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are whole numbers?

AG
Ankit Gupta
Numerade Educator
01:53

Problem 6

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are real numbers?

AG
Ankit Gupta
Numerade Educator
01:26

Problem 7

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are integers but not natural numbers?

AG
Ankit Gupta
Numerade Educator
00:31

Problem 8

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are integers but not whole numbers?

AG
Ankit Gupta
Numerade Educator
00:53

Problem 9

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are rational numbers but not integers?

AG
Ankit Gupta
Numerade Educator
01:08

Problem 10

Text Unavailable

Consider the numbers
$$\begin{array}{l}
\frac{2}{3}, 6, \sqrt{3},-2.45, \sqrt{26}, 18 . \overline{4},-11, \quad \sqrt[3]{27,}, 5 \frac{1}{6}, \\
7.151551555 \ldots,-\sqrt{35}, \quad \sqrt[5]{3},-\frac{8}{7}, 0, \sqrt{16}
\end{array}$$
Which are real numbers but not integers?

AG
Ankit Gupta
Numerade Educator
01:18

Problem 11

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x |-5 \leq x \leq 5\}$$

AG
Ankit Gupta
Numerade Educator
01:03

Problem 12

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x |-2<x<2\}$$

AG
Ankit Gupta
Numerade Educator
01:18

Problem 13

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x |-3<x \leq-1\}$$

AG
Ankit Gupta
Numerade Educator
00:57

Problem 14

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x | 4 \leq x<6\}$$

AG
Ankit Gupta
Numerade Educator
01:06

Problem 15

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x | x \leq-2\}$$

AG
Ankit Gupta
Numerade Educator
01:03

Problem 16

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x | x > -5\}$$

AG
Ankit Gupta
Numerade Educator
01:15

Problem 17

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x | x > 3.8\}$$

AG
Ankit Gupta
Numerade Educator
01:03

Problem 18

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x | x \geq \sqrt{3}\}$$

AG
Ankit Gupta
Numerade Educator
01:13

Problem 19

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x | 7 < x\}$$

AG
Ankit Gupta
Numerade Educator
01:02

Problem 20

Text Unavailable

Write interval notation. Then graph the interval.
$$\{x |-3>x\}$$

AG
Ankit Gupta
Numerade Educator
00:43

Problem 21

Text Unavailable

Write interval notation for the graph.
GRAPH CANNOT COPY.

AG
Ankit Gupta
Numerade Educator
00:38

Problem 22

Text Unavailable

Write interval notation for the graph.
GRAPH CANNOT COPY.

AG
Ankit Gupta
Numerade Educator
00:44

Problem 23

Text Unavailable

Write interval notation for the graph.
GRAPH CANNOT COPY.

AG
Ankit Gupta
Numerade Educator
00:49

Problem 24

Text Unavailable

Write interval notation for the graph.
GRAPH CANNOT COPY.

AG
Ankit Gupta
Numerade Educator
00:47

Problem 25

Text Unavailable

Write interval notation for the graph.
GRAPH CANNOT COPY.

AG
Ankit Gupta
Numerade Educator
01:07

Problem 26

Text Unavailable

Write interval notation for the graph.
GRAPH CANNOT COPY.

AG
Ankit Gupta
Numerade Educator
00:41

Problem 27

Text Unavailable

Write interval notation for the graph.
GRAPH CANNOT COPY.

AG
Ankit Gupta
Numerade Educator
00:44

Problem 28

Text Unavailable

Write interval notation for the graph.
GRAPH CANNOT COPY.

AG
Ankit Gupta
Numerade Educator
00:47

Problem 29

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$6 \in N$$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 30

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$0 \notin N$$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 31

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$3.2 \in \mathbb{Z}$$

AG
Ankit Gupta
Numerade Educator
01:31

Problem 32

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$-10 . \overline{1} \in \mathbb{R}$$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 33

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$-\frac{11}{5} \in \mathbf{Q}$$

AG
Ankit Gupta
Numerade Educator
00:58

Problem 34

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$-\sqrt{6} \in \mathbb{Q}$$

AG
Ankit Gupta
Numerade Educator
00:52

Problem 35

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$\sqrt{11} \notin \mathbb{R}$$

AG
Ankit Gupta
Numerade Educator
00:46

Problem 36

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$-1 \in W$$

AG
Ankit Gupta
Numerade Educator
01:14

Problem 37

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$24 \notin W$$

AG
Ankit Gupta
Numerade Educator
00:59

Problem 38

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$1 \in \mathbb{Z}$$

AG
Ankit Gupta
Numerade Educator
01:40

Problem 39

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$1.089 \notin \mathbb{D}$$

AG
Ankit Gupta
Numerade Educator
00:57

Problem 40

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$\mathbb{N} \subseteq \mathbb{W}$$

AG
Ankit Gupta
Numerade Educator
01:00

Problem 41

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$w \subseteq \mathbb{Z}$$

AG
Ankit Gupta
Numerade Educator
00:59

Problem 42

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$\mathbb{Z} \subseteq \mathbb{N}$$

AG
Ankit Gupta
Numerade Educator
01:03

Problem 43

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$\mathbf{Q} \subseteq \mathbf{R}$$

AG
Ankit Gupta
Numerade Educator
01:03

Problem 44

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$\mathbb{Z} \subseteq \mathbb{Q}$$

AG
Ankit Gupta
Numerade Educator
01:04

Problem 45

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$\mathrm{R} \subseteq \mathbb{Z}$$

AG
Ankit Gupta
Numerade Educator
01:10

Problem 46

Text Unavailable

In Exercises the following notation is used: $\mathbb{N}=$ the set of natural numbers, $\mathbb{W}=$ the set of whole numbers, $\mathbb{Z}=$ the set of integers, $\mathbb{Q}=$ the set of rational numbers, $1=$ the set of irrational numbers, and $\mathbb{R}=$ the set of real numbers. Classify the statement as true or false.
$$Q \subseteq 1$$

AG
Ankit Gupta
Numerade Educator
00:44

Problem 47

Text Unavailable

Name the property illustrated by the sentence.
$$3+y=y+3$$

AG
Ankit Gupta
Numerade Educator
00:54

Problem 48

Text Unavailable

Name the property illustrated by the sentence.
$$6(x z)=(6 x) z$$

AG
Ankit Gupta
Numerade Educator
00:43

Problem 49

Text Unavailable

Name the property illustrated by the sentence.
$$-3 \cdot 1=-3$$

AG
Ankit Gupta
Numerade Educator
00:41

Problem 50

Text Unavailable

Name the property illustrated by the sentence.
$$4(y-z)=4 y-4 z$$

AG
Ankit Gupta
Numerade Educator
00:31

Problem 51

Text Unavailable

Name the property illustrated by the sentence.
$$5 \cdot x=x \cdot 5$$

AG
Ankit Gupta
Numerade Educator
01:02

Problem 52

Text Unavailable

Name the property illustrated by the sentence.
$$7+(x+y)=(7+x)+y$$

AG
Ankit Gupta
Numerade Educator
00:44

Problem 53

Text Unavailable

Name the property illustrated by the sentence.
$$2(a+b)=(a+b) 2$$

AG
Ankit Gupta
Numerade Educator
00:43

Problem 54

Text Unavailable

Name the property illustrated by the sentence.
$$-11+11=0$$

AG
Ankit Gupta
Numerade Educator
00:55

Problem 55

Text Unavailable

Name the property illustrated by the sentence.
$$-6(m+n)=-6(n+m)$$

AG
Ankit Gupta
Numerade Educator
00:33

Problem 56

Text Unavailable

Name the property illustrated by the sentence.
$$t+0=t$$

AG
Ankit Gupta
Numerade Educator
00:32

Problem 57

Text Unavailable

Name the property illustrated by the sentence.
$$8 \cdot \frac{1}{8}=1$$

AG
Ankit Gupta
Numerade Educator
00:42

Problem 58

Text Unavailable

Name the property illustrated by the sentence.
$$9 x+9 y=9(x+y)$$

AG
Ankit Gupta
Numerade Educator
00:21

Problem 59

Text Unavailable

Simplify.
$$|-8.15|$$

AG
Ankit Gupta
Numerade Educator
00:17

Problem 60

Text Unavailable

Simplify.
$$|-14.7|$$

AG
Ankit Gupta
Numerade Educator
00:20

Problem 61

Text Unavailable

Simplify.
$$|295|$$

AG
Ankit Gupta
Numerade Educator
00:14

Problem 62

Text Unavailable

Simplify.
$$|-93|$$

AG
Ankit Gupta
Numerade Educator
00:18

Problem 63

Text Unavailable

Simplify.
$$|-\sqrt{97}|$$

AG
Ankit Gupta
Numerade Educator
00:15

Problem 64

Text Unavailable

Simplify.Simplify.
$$\left|\frac{12}{19}\right|$$

AG
Ankit Gupta
Numerade Educator
00:14

Problem 65

Text Unavailable

Simplify.Simplify.
$$|0|$$

AG
Ankit Gupta
Numerade Educator
00:13

Problem 66

Text Unavailable

Simplify..
$$|15|$$

AG
Ankit Gupta
Numerade Educator
00:17

Problem 67

Text Unavailable

Simplify.
$$\left|\frac{5}{4}\right|$$

AG
Ankit Gupta
Numerade Educator
00:16

Problem 68

Text Unavailable

Simplify.
$$|-\sqrt{3}|$$

AG
Ankit Gupta
Numerade Educator
01:42

Problem 69

Text Unavailable

Find the distance between the given pair of points on the number line.
$$-8,14$$

AG
Ankit Gupta
Numerade Educator
01:43

Problem 70

Text Unavailable

Find the distance between the given pair of points on the number line.
$$-5.2,0$$

AG
Ankit Gupta
Numerade Educator
01:07

Problem 71

Text Unavailable

Find the distance between the given pair of points on the number line.
$$-9,-3$$

AG
Ankit Gupta
Numerade Educator
02:25

Problem 72

Text Unavailable

Find the distance between the given pair of points on the number line.
$$\frac{15}{8}, \frac{23}{12}$$

AG
Ankit Gupta
Numerade Educator
02:15

Problem 73

Text Unavailable

Find the distance between the given pair of points on the number line.
$$6.7,12.1$$

AG
Ankit Gupta
Numerade Educator
01:21

Problem 74

Text Unavailable

Find the distance between the given pair of points on the number line.
$$-15,-6$$

AG
Ankit Gupta
Numerade Educator
02:02

Problem 75

Text Unavailable

Find the distance between the given pair of points on the number line.
$$-\frac{3}{4}, \frac{15}{8}$$

AG
Ankit Gupta
Numerade Educator
01:53

Problem 76

Text Unavailable

Find the distance between the given pair of points on the number line.
$$-3.4,10.2$$

AG
Ankit Gupta
Numerade Educator
01:17

Problem 77

Text Unavailable

Find the distance between the given pair of points on the number line.
$$-7,0$$

AG
Ankit Gupta
Numerade Educator
01:01

Problem 78

Text Unavailable

Find the distance between the given pair of points on the number line.
$$3,19$$

AG
Ankit Gupta
Numerade Educator
00:46

Problem 79

Text Unavailable

To the student and the instructor: The Synthesis exercises found at the end of every exercise set challenge students to combine concepts or skills studied in that section or in preceding parts of the text.
Between any two (different) real numbers there are many other real numbers. Find each of the following.
Answers may vary.
An irrational number between 0.124 and 0.125

AG
Ankit Gupta
Numerade Educator
00:53

Problem 80

Text Unavailable

To the student and the instructor: The Synthesis exercises found at the end of every exercise set challenge students to combine concepts or skills studied in that section or in preceding parts of the text.
Between any two (different) real numbers there are many other real numbers. Find each of the following.
Answers may vary.
A rational number between $-\sqrt{2.01}$ and $-\sqrt{2}$

AG
Ankit Gupta
Numerade Educator
00:42

Problem 81

Text Unavailable

To the student and the instructor: The Synthesis exercises found at the end of every exercise set challenge students to combine concepts or skills studied in that section or in preceding parts of the text.
Between any two (different) real numbers there are many other real numbers. Find each of the following.
Answers may vary.
A rational number between $-\frac{1}{101}$ and $-\frac{1}{100}$

AG
Ankit Gupta
Numerade Educator
00:50

Problem 82

Text Unavailable

To the student and the instructor: The Synthesis exercises found at the end of every exercise set challenge students to combine concepts or skills studied in that section or in preceding parts of the text. Between any two (different) real numbers there are many other real numbers. Find each of the following. Answers may vary.
An irrational number between $\sqrt{5.99}$ and $\sqrt{6}$

AG
Ankit Gupta
Numerade Educator
01:06

Problem 83

Text Unavailable

The hypotenuse of an isosceles right triangle with legs of length 1 unit can be used to "measure" a value for $\sqrt{2}$ by using the Pythagorean theorem, as shown.
$$\begin{array}{l}
c^{2}=1^{2}+1^{2} \\
c^{2}=2 \\
c=\sqrt{2}
\end{array}$$
Draw a right triangle that could be used to "measure" $\sqrt{10}$ units.
FIGURE CANNOT COPY.

AG
Ankit Gupta
Numerade Educator