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Finish the excel spreadsheet and read the chapter. Show all the work so I can get the full points. Show work and then add in answer. Must be good at algebra. Finish the excel spreadsheet and read the chapter. Show all the work so I can get the full points. Show work and then add in answer. Must be good at algebra. Finish the excel spreadsheet and read the chapter. Show all the work so I can get the full points. Show work and then add in answer. Must be good at algebra. Finish the excel spreadsheet and read the chapter. Show all the work so I can get the full points. Show work and then add in answer. Must be good at algebra.
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Algebra I for the Community College
Collection Editor:
Ann Simao
Algebra I for the Community College
Collection Editor:
Ann Simao
Authors:
Denny Burzynski
Wade Ellis
Online:
< http://legacy.cnx.org/content/col11598/1.3/ >
OpenStax-CNX
This selection and arrangement of content as a collection is copyrighted by Ann Simao. It is licensed under the
Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/).
Collection structure revised: December 19, 2014
PDF generated: December 19, 2014
For copyright and attribution information for the modules contained in this collection, see p. 327.
Table of Contents
1 Chapter 1: Introduction to Real Numbers and Algebraic Expressions
1.1
1.2
2 Chapter 2: Solving Linear Equations and Inequalities
2.1
2.2
3 Chapter 3: Graphing Linear Equations and Inequalities
3.1
3.2
4 Chapter 4: Solving Systems of Linear Equations
4.1
4.2
4.3
5 Chapter 5: Exponents and Polynomials
5.1
5.2
Index
Attributions
Chapter 1 Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Chapter 1 Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Chapter 2 Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Chapter 2 Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 3 Part A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Chapter 3 Part B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 156
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
Solving Systems of Linear Equations by Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Solving Systems of Linear Equations by Substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
Introduction to Systems of Linear Equations: Solving by Graphing . . . . . . . . . . . . . . . . . . . . . . . . 239
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
Chapter 5 Part X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Chapter 5 Part Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
iv
Available for free at Connexions
Chapter 1
Chapter 1: Introduction to Real
Numbers and Algebraic Expressions
1.1 Chapter 1 Part A
1
1.1.1 Real Numbers
1.1.1.1 Section Overview




Positive and Negative Numbers
Reading Signed Numbers
Opposites
The Double-Negative Property
1.1.1.2 Positive and Negative Numbers
Positive and Negative Numbers
sign
number
+ and − Notation
positive
negative
Each real number other than zero has a
negative
associated with it. A real number is said to be a
if it is to the right of 0 on the number line and
positive
if it is to the left of 0 on the number line.
note:
A number is denoted as
A number is denoted as
if it is directly preceded by a plus sign or no sign at all.
if it is directly preceded by a minus sign.
1.1.1.3 Reading Signed Numbers
The plus and minus signs now have two meanings :
The plus sign can denote the operation of addition or a positive number.
The minus sign can denote the operation of subtraction or a negative number.
To avoid any confusion between “sign” and “operation,” it is preferable to read the sign of a number as
“positive” or “negative.” When “+” is used as an operation sign, it is read as “plus.” When “−” is used as
an operation sign, it is read as “minus.”
1 This
content is available online at .
Available for free at Connexions
1
CHAPTER 1.
2
CHAPTER 1: INTRODUCTION TO REAL NUMBERS AND
ALGEBRAIC EXPRESSIONS
1.1.1.3.1 Sample Set A
Read each expression so as to avoid confusion between “operation” and “sign.”
Example 1.1
−8
Example 1.2
+ (−2)
Example 1.3
should be read as “negative eight” rather than “minus eight.”
4
should be read as “four plus negative two” rather than “four plus minus two.”
−6 + (−3)should
be read as “negative six plus negative three” rather than “minus six plus minus
three.”
Example 1.4
−15−(−6)should be read as “negative fteen minus negative six” rather than “minus fteen minus
minus six.”
Example 1.5
−5 +
Example 1.6
7 should be read as “negative ve plus seven” rather than “minus ve plus seven.”
0−2
should be read as “zero minus two.”
1.1.1.3.2 Practice Set A
Exercise 1.1.1.1
+
Exercise 1.1.1.2
2 + (−8)
Exercise 1.1.1.3
−7 + 5
Exercise 1.1.1.4
−10 − (+3)
Exercise 1.1.1.5
−1 − (−8)
Exercise 1.1.1.6
Write each expression in words.
6
0
(Solution on p. 60.)
1
(Solution on p. 60.)
(Solution on p. 60.)
(Solution on p. 60.)
(Solution on p. 60.)
(Solution on p. 60.)
+ (−11)
1.1.1.4 Opposites
Opposites
Opposites
On the number line, each real number, other than zero, has an image on the opposite side of 0. For this
reason, we say that each real number has an opposite.
opposite signs.
are the same distance from zero but have
The opposite of a real number is denoted by placing a negative sign directly in front of the number.
Thus, if
a
is any real number, then
−a
is its opposite.
note: The letter “a” is a variable. Thus, “a” need not be positive, and “−a” need not be negative.
Available for free at Connexions
3
If
a
is any real number,
−a
is opposite
a
on the number line.
1.1.1.5 The Double-Negative Property
The number
a
is opposite
−a
on the number line. Therefore,
− (−a)
is opposite
−a
on the number line.
This means that
− (−a) = a
Double-Negative Property: − (−a) = a
From this property of opposites, we can suggest the double-negative property for real numbers.
a is a real
− (−a) = a
If
number, then
1.1.1.5.1 Sample Set B
Find the opposite of each number.
Example 1.7
If
a=
2, then
Example 1.8
If
a = −4,
−a = −2.
then
Also,
− (−a) = − (−2) = 2.
−a = − (−4) = 4.
1.1.1.5.2 Practice Set B
Exercise 1.1.1.7
Exercise 1.1.1.8
Exercise 1.1.1.9
Exercise 1.1.1.10
Exercise 1.1.1.11
Exercise 1.1.1.12
− [− (−7)]
Exercise 1.1.1.13
Also,
− (−a) = a = − 4.
Find the opposite of each number.
(Solution on p. 60.)
8
(Solution on p. 60.)
17
(Solution on p. 60.)
-6
(Solution on p. 60.)
-15
(Solution on p. 60.)
-(-1)
Suppose
a
is a positive number. Is
(Solution on p. 60.)
(Solution on p. 60.)
−a
positive or negative?
Available for free at Connexions
CHAPTER 1.
4
Exercise 1.1.1.14
a
Exercise 1.1.1.15
Suppose
is a negative number. Is
CHAPTER 1: INTRODUCTION TO REAL NUMBERS AND
ALGEBRAIC EXPRESSIONS
(Solution on p. 60.)
−a
positive or negative?
Suppose we do not know the sign of the number
(Solution on p. 60.)
k.
Is
−k
positive, negative, or do we not know?
1.1.1.6 Exercises
Exercise 1.1.1.16
Exercise 1.1.1.17
A number is denoted as positive if it is directly preceded by
A number is denoted as negative if it is directly preceded by
Exercise 1.1.1.18
−7
Exercise 1.1.1.19
−5
Exercise 1.1.1.20
Exercise 1.1.1.21
Exercise 1.1.1.22
− (−1)
Exercise 1.1.1.23
(Solution on p. 60.)
.
.
How should the number in the following 6 problems be read? (Write in words.)
(Solution on p. 60.)
(Solution on p. 60.)
15
11
(Solution on p. 60.)
− (−5)
Exercise 1.1.1.24
5+3
Exercise 1.1.1.25
3+8
Exercise 1.1.1.26
+ (−3)
Exercise 1.1.1.27
1 + (−9)
Exercise 1.1.1.28
−7 − (−2)
Exercise 1.1.1.29
For the following 6 problems, write each expression in words.
(Solution on p. 60.)
(Solution on p. 60.)
15
(Solution on p. 60.)
0 − (−12)
Exercise 1.1.1.30
− (−2)
Exercise 1.1.1.31
− (− )
Exercise 1.1.1.32
For the following 6 problems, rewrite each number in simpler form.
(Solution on p. 60.)
16
(Solution on p. 60.)
− [− (−8)]
Available for free at Connexions
5
Exercise 1.1.1.33
− [− (− )]
Exercise 1.1.1.34
7 − (−3)
Exercise 1.1.1.35
20
(Solution on p. 60.)
6 − (−4)
1.1.1.6.1 Exercises for Review
Exercise 1.1.1.36

Exercise 1.1.1.37
Exercise 1.1.1.38
Exercise 1.1.1.39
Exercise 1.1.1.40
2
( here ) Find the quotient;
3
( here ) Solve the proportion:
(Solution on p. 60.)
27.
5
9
=
60
x
4
( here ) Use the method of rounding to estimate the sum: 5829
(Solution on p. 61.)
+ 8767
5
( here ) Use a unit fraction to convert 4 yd to feet.
(Solution on p. 61.)
6
( here ) Convert 25 cm to hm.
7
1.1.2 Real Number Line
1.1.2.1 Overview



The Real Number Line
The Real Numbers
Ordering the Real Numbers
1.1.2.2 The Real Number Line
Real Number Line
visually
In our study of algebra, we will use several collections of numbers.
display the numbers in which we are interested.
The
real number line
allows us to
A line is composed of in nitely many points. To each point we can associate a unique number, and with
Coordinate
Graph
each number we can associate a particular point.
The number associated with a point on the number line is called the
coordinate
graph
The point on a line that is associated with a particular number is called the
Construction of the Real Number Line
We construct the real number line as follows:
of the point.
of that number.
2 “Decimals: Division of Decimals”
3 “Ratios and Rates: Proportions”
4 “Techniques of Estimation: Estimation by Rounding”
5 “Measurement and Geometry: Measurement and the United States System”

6 “Measurement and Geometry: The Metric System of Measurement”
7 This content is available online at .
Available for free at Connexions
CHAPTER 1.
6
CHAPTER 1: INTRODUCTION TO REAL NUMBERS AND
ALGEBRAIC EXPRESSIONS
1. Draw a horizontal line.
2. Choose any point on the line and label it 0. This point is called the
origin
.
3. Choose a convenient length. This length is called “1 unit.” Starting at 0, mark this length o in both
directions, being careful to have the lengths look like they are about the same.
We now de ne a real number.
Real Number
real number
Positive and Negative Real Numbers
A
is any number that is the coordinate of a point on the real number line.
collection of real numbers
positive real numbers
negative real numbers
The collection of these in nitely many numbers is called the
. The real numbers
whose graphs are to the right of 0 are called the
appear to the left of 0 are called the
. The real numbers whose graphs
.
The number 0 is neither positive nor negative.
1.1.2.3 The Real Numbers
The collection of real numbers has many subcollections. The subcollections that are of most interest to us
Natural Numbers
natural numbers (N ): {1, 2, 3, . . . }
are listed below along with their notations and graphs.
The
Whole Numbers
whole numbers (W ): {0, 1, 2, 3, . . . }
The
Integers
integers (Z): {. . . , − 3, − 2, −1, 0, 1, 2, 3, . . . }
Notice that every natural number is a whole number.
The
Rational Numbers
rational numbers (Q):
a
b
b 6= 0
Fractions
Notice that every whole number is an integer.
The
and
Rational numbers are real numbers that can be written in the form
are integers, and
.
Rational numbers are commonly called
fractions.
Available for free at Connexions
a/b, where
7
Division by 1
b
Division by 0
Since
can equal 1, every integer is a rational number:
Recall that
10/2 = 5
since
2 · 5 = 10.
However, if
a
1
= a.
10/0 = x,
then
0 · x = 10.
But
0 · x = 0,
not 10. This
suggests that no quotient exists.
0/0 = x. If 0/0 = x, then 0 · x = 0. But this means that x could be any number, that
0 · 4 = 0, or 0/0 = 28 since 0 · 28 = 0. This suggests that the quotient is indeterminant.
Now consider
Is Unde ned or Indeterminant
0/0 = 4
x/0
since
is,
Division by 0 is unde ned or indeterminant.
Do not divide by 0.
Rational numbers have decimal representations that either terminate or do not terminate but contain a
repeating block of digits. Some examples are:
3
= 0.75
|4 {z }
Terminating
15
= 1.36363636 . . .
|11
{z
}
Nonterminating, but repeating
Some rational numbers are graphed below.
Irrational Numbers
irrational numbers (Ir):
The
Irrational numbers are numbers that cannot be written as the quotient of
two integers. They are numbers whose decimal representations are nonterminating and nonrepeating. Some
examples are
4.01001000100001 . . .
π = 3.1415927 . . .
Notice that the collections of rational numbers and irrational numbers have no numbers in common.
When graphed on the number line, the rational and irrational numbers account for every point on the
number line. Thus each point on the number line has a coordinate that is either a rational or an irrational
number.
In summary, we have
1.1.2.4 Sample Set A
The summaray chart illustrates that
Example 1.9
Every natural number is a real number.
Available for free at Connexions
CHAPTER 1.
8
CHAPTER 1: INTRODUCTION TO REAL NUMBERS AND
ALGEBRAIC EXPRESSIONS
Example 1.10
Example 1.11
Every whole number is a real number.
No integer is an irrational number.
1.1.2.5 Practice Set A
Exercise 1.1.2.1
Exercise 1.1.2.2
Exercise 1.1.2.3
Exercise 1.1.2.4
Exercise 1.1.2.5
Exercise 1.1.2.6
(Solution on p. 61.)
Is every natural number a whole number?
(Solution on p. 61.)
Is every whole number an integer?
(Solution on p. 61.)
Is every integer a rational number?
(Solution on p. 61.)
Is every rational number a real number?
(Solution on p. 61.)
Is every integer a natural number?
(Solution on p. 61.)
Is there an integer that is a natural number?
1.1.2.6 Ordering the Real Numbers
Ordering the Real Numbers
A real number
of the graph of
b is said to be greater
a on the number line.
than a real number
a,
denoted
b > a,
if the graph of
b
is to the right
1.1.2.7 Sample Set B
As we would expect,
the right of
−5
5>2
since 5 is to the right of 2 on the number line. Also,
−2 > − 5
since
−2
on the number line.
1.1.2.8 Practice Set B
Exercise 1.1.2.7
Exercise 1.1.2.8
Exercise 1.1.2.9
Exercise 1.1.2.10
Exercise 1.1.2.11
(Solution on p. 61.)
Are all positive numbers greater than 0?
(Solution on p. 61.)
Are all positive numbers greater than all negative numbers?
(Solution on p. 61.)
Is 0 greater than all negative numbers?
(Solution on p. 61.)
Is there a largest positive number? Is there a smallest negative number?
(Solution on p. 61.)
How many real numbers are there? How many real numbers are there between 0 and 1?
Available for free at Connexions
is to
9
1.1.2.9 Sample Set C
Example 1.12
What integers can replace
x
so that the following statement is true?
−4 ≤ x < 2 This statement indicates that the number represented by −4 is less than or equal to x, and at the same time, x x is between −4 and 2. Speci cally, is strictly less than 2. This statement is an example of a compound inequality. The integers are Example 1.13 −4, − 3, − 2, − 1, 0, 1. Draw a number line that extends from including −2 −3 to 7. Place points at all whole numbers between and and 6. Example 1.14 Draw a number line that extends from −4 to 6 and place points at all real numbers greater than or equal to 3 but strictly less than 5. open circle It is customary to use a closed circle to indicate that a point is included in the graph and an to indicate that a point is not included. 1.1.2.10 Practice Set C Exercise 1.1.2.12 What whole numbers can replace (Solution on p. 61.) x so that the following statement is true? −3 ≤ x < 3 Exercise 1.1.2.13 Draw a number line that extends from equal to −4 (Solution on p. 61.) −5 to 3 and place points at all numbers greater than or but strictly less than 2. Available for free at Connexions CHAPTER 1. 10 CHAPTER 1: INTRODUCTION TO REAL NUMBERS AND ALGEBRAIC EXPRESSIONS 1.1.2.11 Exercises For the following problems, next to each real number, note all collections to which it belongs by writing for natural numbers, and R W for whole numbers, Z for integers, Q for rational numbers, Ir Exercise 1.1.2.14 Exercise 1.1.2.15 −12 Exercise 1.1.2.16 Exercise 1.1.2.17 −24 Exercise 1.1.2.18 86.3333 . . . Exercise 1.1.2.19 49.125125125 . . . Exercise 1.1.2.20 N for irrational numbers, for real numbers. Some numbers may require more than one letter. (Solution on p. 61.) 1 2 (Solution on p. 61.) 0 7 8 (Solution on p. 61.) (Solution on p. 61.) −15.07 For the following problems, draw a number line that extends from −3 Exercise 1.1.2.21 1 Exercise 1.1.2.22 −2 Exercise 1.1.2.23 − Exercise 1.1.2.24 Exercise 1.1.2.25 to 3. Locate each real number on the number line by placing a point (closed circle) at its approximate location. 1 2 (Solution on p. 61.) 1 8 (Solution on p. 61.) Is 0 a positive number, negative number, neither, or both? An integer is an even integer if it can be divided by 2 without a remainder; otherwise the number is odd. Draw a number line that extends from Exercise 1.1.2.26 −5 to 5 and place points at all negative even integers and at all positive odd integers. Draw a number line that extends from −3 (Solution on p. 61.) −5 to 5. Place points at all integers strictly greater than but strictly less than 4. For the following problems, draw a number line that extends from Exercise 1.1.2.27 −5 −2 Exercise 1.1.2.28 −3 Exercise 1.1.2.29 −4 Exercise 1.1.2.30 −5 to 5. Place points at all real numbers between and including each pair of numbers. and (Solution on p. 61.) and 4 and 0 Draw a number line that extends ... Purchase answer to see full attachment