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College Algebra 12th

R. David Gustafson, Jeff Hughes

Chapter 0

A Review of Basic Algebra

Educators


Problem 1

Fill in the blanks.
A __________ is a collection of objects.

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Problem 2

Fill in the blanks.
If every member of one set $B$ is also a member of a second set $A$, then $B$ is called a __________ of $A$.

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Problem 3

Fill in the blanks.
If $A$ and $B$ are two sets, the set that contains all members that are in sets $A$ and $B$ or both is called the __________ of $A$ and $B$.

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Problem 4

Fill in the blanks.
If $A$ and $B$ are two sets, the set that contains all members that are in both sets is called the __________ of $A$ and $B$.

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Problem 5

Fill in the blanks.
A real number is any number that can be expressed as a __________ .

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Problem 6

Fill in the blanks.
A __________ is a letter that is used to represent a number.

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Problem 7

Fill in the blanks.
The smallest prime number is __________.

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Problem 8

Fill in the blanks.
All integers that are exactly divisible by 2 are called __________ integers.

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Problem 9

Fill in the blanks.
Natural numbers greater than 1 that are not prime are called __________ numbers.

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Problem 10

Fill in the blanks.
Fractions such as $\frac{2}{3}, \frac{8}{2},$ and $-\frac{7}{9}$ are called __________ numbers.

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Problem 11

Fill in the blanks.
Irrational numbers are __________ that don't terminate and don't repeat.

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Problem 12

Fill in the blanks.
The symbol __________ is read as "is less than or equal to."

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Problem 13

On a number line, the ___________ numbers are to the left of $0 .$

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Problem 14

Fill in the blanks.
The only integer that is neither positive nor negative is __________.

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Problem 15

Fill in the blanks.
The Associative Property of Addition states that $(x+y)+z=$ __________.

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Problem 16

Fill in the blanks.
The Commutative Property of Multiplication states that $x y=$ __________.

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Problem 17

Fill in the blanks.
Use the Distributive Property to complete the statement: $5(m+2)=$ __________.

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Problem 18

Fill in the blanks.
The statement $(m+n) p=p(m+n)$ illustrates the _____________ Property of __________.

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Problem 19

Fill in the blanks.
The graph of an __________ is a portion of a number line.

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Problem 20

Fill in the blanks.
The graph of an open interval has _________ endpoints.

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Problem 21

Fill in the blanks.
The graph of a closed interval has __________ endpoints.

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Problem 22

Fill in the blanks.
The graph of a __________ interval has one endpoint.

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Problem 23

Fill in the blanks.
Except for $0,$ the absolute value of every number is __________.

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Problem 24

Fill in the blanks.
The __________ between two distinct points on a number line is always positive.

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Problem 25

Let
$\mathbf{N}=$the set of natural numbers
$\mathbf{W}=$the set of whole numbers
$\mathbf{Z}=$the set of integers
$\mathbf{Q}=$the set of rational numbers
$\mathbf{R}=$the set of real numbers
Determine whether each statement is true or false. Read the symbol $c$ as "is a subset of."
$$\mathrm{N} \subset \mathrm{W}$$

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Problem 26

Let
$\mathbf{N}=$the set of natural numbers
$\mathbf{W}=$the set of whole numbers
$\mathbf{Z}=$the set of integers
$\mathbf{Q}=$the set of rational numbers
$\mathbf{R}=$the set of real numbers
Determine whether each statement is true or false. Read the symbol $c$ as "is a subset of."
$$Q \subset R$$

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Problem 27

Let
$\mathbf{N}=$the set of natural numbers
$\mathbf{W}=$the set of whole numbers
$\mathbf{Z}=$the set of integers
$\mathbf{Q}=$the set of rational numbers
$\mathbf{R}=$the set of real numbers
Determine whether each statement is true or false. Read the symbol $c$ as "is a subset of."
$$\mathrm{Q} \subset \mathrm{N}$$

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Problem 28

Let
$\mathbf{N}=$the set of natural numbers
$\mathbf{W}=$the set of whole numbers
$\mathbf{Z}=$the set of integers
$\mathbf{Q}=$the set of rational numbers
$\mathbf{R}=$the set of real numbers
Determine whether each statement is true or false. Read the symbol $c$ as "is a subset of."
$$\mathrm{Z} \subset \mathrm{Q}$$

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Problem 29

Let
$\mathbf{N}=$the set of natural numbers
$\mathbf{W}=$the set of whole numbers
$\mathbf{Z}=$the set of integers
$\mathbf{Q}=$the set of rational numbers
$\mathbf{R}=$the set of real numbers
Determine whether each statement is true or false. Read the symbol $c$ as "is a subset of."
$$\mathrm{W} \subset \mathrm{Z}$$

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Problem 30

Let
$\mathbf{N}=$the set of natural numbers
$\mathbf{W}=$the set of whole numbers
$\mathbf{Z}=$the set of integers
$\mathbf{Q}=$the set of rational numbers
$\mathbf{R}=$the set of real numbers
Determine whether each statement is true or false. Read the symbol $c$ as "is a subset of."
$$\mathbf{R} \subset \mathbf{Z}$$

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Problem 31

Let $A=\{a, b, c, d, e\}, B=(d, e, f, g\},$ and $C=\{a, c, e, f\} .$ Find each set.
$$A \cup B$$

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Problem 32

Let $A=\{a, b, c, d, e\}, B=(d, e, f, g\},$ and $C=\{a, c, e, f\} .$ Find each set.
$$A \cap B$$

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Problem 33

Let $A=\{a, b, c, d, e\}, B=(d, e, f, g\},$ and $C=\{a, c, e, f\} .$ Find each set.
$$A \cap C$$

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Problem 34

Let $A=\{a, b, c, d, e\}, B=(d, e, f, g\},$ and $C=\{a, c, e, f\} .$ Find each set.
$$B \cup C$$

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Problem 35

Determine whether the decimal form of each fraction terminates or repeats.
$$\frac{9}{16}$$

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Problem 36

Determine whether the decimal form of each fraction terminates or repeats.
$$\frac{3}{8}$$

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Problem 37

Determine whether the decimal form of each fraction terminates or repeats.
$$\frac{3}{11}$$

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Problem 38

Determine whether the decimal form of each fraction terminates or repeats.
$$\frac{5}{12}$$

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Problem 39

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are natural numbers?

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Problem 40

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are whole numbers?

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Problem 41

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are integers?

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Problem 42

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are rational numbers?

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Problem 43

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are irrational numbers?

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Problem 44

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are prime numbers?

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Problem 45

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are composite numbers?

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Problem 46

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are even integers?

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Problem 47

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are odd integers?

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Problem 48

Consider the following set:
$$\left\{-5,-4,-\frac{2}{3}, 0,1, \sqrt{2}, 2,2.75,6,7\right\}$$
Which numbers are negative numbers?

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Problem 49

Graph each subset of the real numbers on a number line.
The natural numbers between 1 and 5

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Problem 50

Graph each subset of the real numbers on a number line.
The composite numbers less than 10

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Problem 51

Graph each subset of the real numbers on a number line.
The prime numbers between 10 and 20

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Problem 52

Graph each subset of the real numbers on a number line.
The integers from $-2$ to $4$

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Problem 53

Graph each subset of the real numbers on a number line.
The integers between $-5$ and $0$

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Problem 54

Graph each subset of the real numbers on a number line.
The even integers between $-9$ and $-1$

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Problem 55

Graph each subset of the real numbers on a number line.
The odd integers between $-6$ and $4$

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Problem 56

Graph each subset of the real numbers on a number line.
$-0.7,1.75,$ and $3 \frac{7}{8}$

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Problem 57

Write each inequality in interval notation and graph the interval.
$$x>2$$

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Problem 58

Write each inequality in interval notation and graph the interval.
$$x<4$$

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Problem 59

Write each inequality in interval notation and graph the interval.
$$0<x<5$$

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Problem 60

Write each inequality in interval notation and graph the interval.
$$-2<x<3$$

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Problem 61

Write each inequality in interval notation and graph the interval.
$$x>-4$$

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Problem 62

Write each inequality in interval notation and graph the interval.
$$x<3$$

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Problem 63

Write each inequality in interval notation and graph the interval.
$$-2 \leq x<2$$

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Problem 64

Write each inequality in interval notation and graph the interval.
$$-4<x \leq 1$$

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Problem 65

Write each inequality in interval notation and graph the interval.
$$x \leq 5$$

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Problem 66

Write each inequality in interval notation and graph the interval.
$$x \geq-1$$

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Problem 67

Write each inequality in interval notation and graph the interval.
$$-5<x \leq 0$$

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Problem 68

Write each inequality in interval notation and graph the interval.
$$-3 \leq x<4$$

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Problem 69

Write each inequality in interval notation and graph the interval.
$$-2 \leq x \leq 3$$

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Problem 70

Write each inequality in interval notation and graph the interval.
$$-4 \leq x \leq 4$$

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Problem 71

Write each inequality in interval notation and graph the interval.
$$6 \geq x \geq 2$$

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Problem 72

Write each inequality in interval notation and graph the interval.
$$3 \geq x \geq-2$$

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Problem 73

Write each pair of inequalities as the intersection of two intervals and graph the result.
$$x>-5 \text { and } x<4$$

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Problem 74

Write each pair of inequalities as the intersection of two intervals and graph the result.
$$x \geq-3 \text { and } x<6$$

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Problem 75

Write each pair of inequalities as the intersection of two intervals and graph the result.
$$x \geq-8 \text { and } x \leq-3$$

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Problem 76

Write each pair of inequalities as the intersection of two intervals and graph the result.
$$x>1 \text { and } x \leq 7$$

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Problem 77

Write each inequality as the union of two intervals and graph the result.
$$x<-2 \text { or } x>2$$

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Problem 78

Write each inequality as the union of two intervals and graph the result.
$$x \leq-5 \text { or } x>0$$

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Problem 79

Write each inequality as the union of two intervals and graph the result.
$$x \leq-1 \text { or } x \geq 3$$

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Problem 80

Write each inequality as the union of two intervals and graph the result.
$$x<-3 \text { or } x \geq 2$$

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Problem 81

Write each expression without using absolute value symbols.
$$|13|$$

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Problem 82

Write each expression without using absolute value symbols.
$$|-17|$$

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Problem 83

Write each expression without using absolute value symbols.
$$|0|$$

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Problem 84

Write each expression without using absolute value symbols.
$$-|63|$$

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Problem 85

Write each expression without using absolute value symbols.
$$-|-8|$$

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Problem 86

Write each expression without using absolute value symbols.
$$|-25|$$

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Problem 87

Write each expression without using absolute value symbols.
$$-|32|$$

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Problem 88

Write each expression without using absolute value symbols.
$$-|-6|$$

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Problem 89

Write each expression without using absolute value symbols.
$$|\pi-5|$$

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Problem 90

Write each expression without using absolute value symbols.
$$|8-\pi|$$

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Problem 91

Write each expression without using absolute value symbols.
$$|\pi-\pi|$$

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Problem 92

Write each expression without using absolute value symbols.
$$|2 \pi|$$

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Problem 93

Write each expression without using absolute value symbols.
$$|x+1| \text { and } x \geq 2$$

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Problem 94

Write each expression without using absolute value symbols.
$$|x+1| \text { and } x \leq-2$$

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Problem 95

Write each expression without using absolute value symbols.
$$|x-4| \text { and } x<0$$

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Problem 96

Write each expression without using absolute value symbols.
$$|x-7| \text { and } x>10$$

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Problem 97

Find the distance between each pair of points on the number line.
3 and 8

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Problem 98

Find the distance between each pair of points on the number line.
-5 and 12

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Problem 99

Find the distance between each pair of points on the number line.
-8 and -3

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Problem 100

Find the distance between each pair of points on the number line.
6 and -20

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Problem 101

What subset of the real numbers would you use to describe the populations of Memphis and Miami?

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Problem 102

What subset of the real numbers would you use to describe the subdivisions of an inch on a ruler?

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Problem 103

Find the distance between each pair of points on the number line.
What subset of the real numbers would you use to report temperatures in London and Lisbon?

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Problem 104

What subset of the real numbers would you use to describe the prices of hoodies at Aéropostale?

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Problem 105

The average low temperature in International Falls, Minnesota, in January is $-7^{\circ} \mathrm{F}$. The average high temperature is $15^{\circ} \mathrm{F}$. Determine the degrees difference between the average high and the average low.

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Problem 106

Harbin, China, is one of the world's coldest cities and known for its ice and snow festivals. In February, the average nightly low temperature is $-20^{\circ} \mathrm{C}$ and the average daily high temperature is $-7^{\circ} \mathrm{C}$. What is the temperature drop from day to night?

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Problem 107

Explain why $-x$ could be positive.

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Problem 108

Explain why every integer is a rational number.

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Problem 109

Is the statement $|a b|=|a| \cdot|b|$ always true? Explain.

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Problem 110

Is the statement $\left|\frac{a}{b}\right|=\frac{|a|}{|b|}(b \neq 0)$ always true? Explain.

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Problem 111

Is the statement $|a+b|=|a|+|b|$ always true? Explain.

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Problem 112

Explain why it is incorrect to write $a<b>c$ if $a<b$ and $b>c$.

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Problem 113

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
There are six integers between $-3$ and $3$.

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Problem 114

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
$\frac{725}{0}$ is a rational number because $725$ and $0$ are integers.

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Problem 115

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
$\infty$ is a real number.

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Problem 116

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
$$|a-b|=|b-a|$$

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Problem 117

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
$$\varnothing \subset\left\{5, \pi, \sqrt{3}, \frac{13}{4}\right\}$$

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Problem 118

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
$$\varnothing \subset \varnothing$$

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Problem 119

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
There are six subsets of $\{11,22,33\}$.

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Problem 120

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
A set is always a subset of itself.

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