Download Section 1.1.1 - All Things Media

Algebra I
Part 1
Algebra I Part 1
ISBN 978-1-935119-01-2
Quantum Scientific Publishing
Cover design by Scott Sheariss
Unit One
Section 1.1.1 – Real Numbers and the Number Line 2
Section 1.1.2 – Working with Fractions 9
Section 1.1.3 – Adding Real Numbers 25
Section 1.1.4 – Solve Problems Involving Addition and Subtraction of Real Numbers 30
Section 1.1.5 – Adding and Subtracting Real Numbers 36
Section 1.1.6 – Solve Problems that Involve Subtraction of Real Numbers 41
Section 1.1.7 – Multiplying and Dividing Real Numbers 46
Section 1.1.8 – Exponents and the Order of Operations 55
Section 1.1.9 – Translating Phrases and Sentences into Expressions and Equations 62
Section 1.1.10 – Evaluating Expressions and Confirming Solutions of Equations 67
Section 1.1.11 – Evaluating Expressions Using Real Numbers 73
Section 1.1.12 – Working with Irrational Numbers 78
Section 1.1.13 – Using the Commutative and Associative Properties 83
Section 1.1.14 – Using the Distributive Property 89
Section 1.1.15 – Using the Identity and Inverse Properties 92
Unit Two
Section 1.2.1 – Algebraic Terms 98
Section 1.2.2 – Algebraic Expressions and Equations 104
Section 1.2.3 − Solve Equations Using Addition 110
Section 1.2.4 − Solve Equations Using Subtraction 112
Section 1.2.5 − Solve Equations Using Multiplication 114
Section 1.2.6 − Solve Equations Using Division 117
Section 1.2.7 − Multi-Step Linear Equations 120
Section 1.2.8 − Linear Equations with Variable Terms on Both Sides 126
Table of Contents
Section 1.2.9 − Equations with Fractions and Decimals 131
Section 1.2.10 – Linear Identities and Equations with no Solution 138
Section 1.2.11 − Equations from Sentences 142
Section 1.2.12 – Formulas 148
Section 1.2.13 − Percent Problems 155
Section 1.2.14 − Distance Problems 164
Section 1.2.15 − Mixture Problems 168
Unit Three
Section 1.3.1 − Linear Inequalities 172
Section 1.3.2 − Linear Inequalities II 176
Section 1.3.3 − Solve Inequalities 183
Section 1.3.4 − Solve Inequalities 186
Section 1.3.5 − Solve Inequalities 189
Section 1.3.6 − Solve Inequalities 193
Section 1.3.7 − Solve Inequalities 197
Section 1.3.8 − Solve Inequalities 201
Section 1.3.9 − Compound Inequalities 205
Section 1.3.10 − Graph Solutions to Compound Inequalities 209
Section 1.3.11 − Compound Inequalities 213
Section 1.3.12 − Absolute Value Inequalities 216
Section 1.3.13 − Inequality Applications 220
Section 1.3.14 − Compound Inequality Applications 223
Section 1.3.15 − Inequality Applications 226
Unit One:
Section 1.1.1 – Real Numbers and the Number Line 8
Section 1.1.2 – Working with Fractions 15
Section 1.1.3 – Adding Real Numbers 31
Section 1.1.4 – Solve Problems Involving Addition and Subtraction
of Real Numbers 36
Section 1.1.5 – Adding and Subtracting Real Numbers 42
Section 1.1.6 – Solve Problems that Involve Subtraction of Real Numbers 47
Section 1.1.7 – Multiplying and Dividing Real Numbers 52
Section 1.1.8 – Exponents and the Order of Operations 61
Section 1.1.9 – Translating Phrases and Sentences into Expressions
and Equations 68
Section 1.1.10 – Evaluating Expressions and Confirming Solutions of Equations 73
Section 1.1.11 – Evaluating Expressions Using Real Numbers 79
Section 1.1.12 – Working with Irrational Numbers 84
Section 1.1.13 – Using the Commutative and Associative Properties 89
Section 1.1.14 – Using the Distributive Property 95
Section 1.1.15 – Using the Identity and Inverse Properties 98
Section 1.1.1 – Real Numbers and the Number
Line
Section Objectives:



Use a number line to order numbers
Identify natural numbers, whole numbers, integers, rational numbers, and
irrational numbers
Find the absolute value of a real number
Use a Number Line to Order Numbers
The arrows on
the ends of the
number line
indicate that you
can never stop
counting, In
other words,
there is no
“last” number.
In mathematics, a number line is frequently used to visually represent the ordering of
numbers. A number line is drawn as a straight line with arrows on both ends. On the
number line, the number zero (0) is placed in the center of the line and the positive
numbers are placed sequentially and evenly spaced to the right of zero, and the negative
numbers are laid out the same way to the left of zero. Here is what the number line looks
like:
–5 –4 –3 –2 –1 0
3 4 5
2
1
In representing numbers on a number line in this manner, you can see, for instance, that
the number 5, which is greater than the number 2, is to the right of the number 2.
–5
–4
–3 –2 –1
3
2
1
0
5
4
Also the number 1, which is smaller than the number 3, is to the left of the number 3.
–5
–4
–3 –2 –1
3
2
1
0
4
5
This relationship holds true for any two numbers on the number line. If a number is to the
left of another number, then it is smaller than that other number, and if a number is to the
right of another number, then it is larger than that other number.
The symbols <
and > are
referred to as
inequality
symbols.
–1 > –5 is
simply referred
to as an
inequality or an
inequality
statement.
You can also clearly see the order relationships involving negative numbers on a number
line. Any negative number is to the left of any positive number, so any negative number
is always less than any positive number. Furthermore, for example, –5 is to the left of –3,
so –5 is less than –3.
–5
–4
–3 –2 –1
3
2
1
0
5
4
Also –1 is to the right of –5, so –1 is greater than –5.
–5
8
–4
–3 –2 –1
0
1
2
3
4
5
Notice that the number zero (0) is in between the negative and the positive numbers.
Therefore 0 is less than any positive number since it is to the left of any positive number
and 0 is greater than any negative number since it is to the right of any negative number.
Finally, there are symbols used to represent the words “is less than” and the words “is
greater than.” Specifically, the symbol < can be used in place of the words “is less than”
and the symbol > can be used in place of the words “is greater than.” For example, we
could write 5  2 instead of “5 is greater than 2” and we could write –6 < –3 instead of
“–6 is less than –3.”
Example 1
Draw a basic number line and place dots at the locations of the following numbers: 4, –2,
1, 0, –3, and 5. Then indicate whether each of the following statements is true or false.
a)
b)
c)
d)
e)
4 is less than 1
5 < –3
0 is greater than –2
–2 < –3
–3 > 1
Solution
–5
–4
–3 –2 –1
0
1
2
3
4
5
a) False. 4 is greater than 1 since it is to the right of 1.
b) False. 5 is greater than –3 since it is to the right of –3. The correct statement
would be 5 > –3.
c) True. 0 is greater than –2 since it is to the right of –2.
d) False. –2 is greater than –3 since it is to the right of –3. The correct statement
would be –2 > –3.
e) False. –3 is less than 1 since it is to the left of 1. The correct statement would be
–3 < 1.
Identify Natural Numbers, Whole Numbers, Integers, Rational
Numbers, and Irrational Numbers
The numbers that most of us encounter in our lives are called real numbers. There are
several types of real numbers. All types of real numbers can be represented on the
number line, which is often referred to as the real number line. The numbers we have
studied so far on the number line include natural numbers, whole numbers and integers.
The natural numbers, sometimes referred to as the counting numbers, begin with the
number 1 and include all of the numbers as if we were counting up from the number 1.
9
So the natural numbers include 1, 2, 3, 4, 5, 6, 7, 8, 9, and so on. The natural numbers
never end. There is no “last” natural number.
–5 –4 –3 –2 –1 0
1
Number line showing the natural numbers
3
2
5
4
The whole numbers include the number zero (0) and all of the natural numbers.
–5
–4
–3 –2 –1
3
2
1
0
5
4
Number line showing the whole numbers
Next, if you placed a negative sign on all of the natural numbers and combined all of
those with all of the whole numbers, you would have the entire collection of numbers
called the integers. The integers are the numbers we typically label on the number line.
–5
–4
–3 –2 –1
0
1
2
3
4
5
Number line showing the integers
5
–
can be
1
5
written as
1
5
or as
1
2.333333… may
also be written
as 2.3 . The
“bar” symbol
above the 3
indicates that the
3 repeats
indefinitely.
Finally, all of the real numbers fall into one of two types, either rational or irrational.
Rational numbers are numbers that can be written like a fraction or numbers that can be
1 7 10
expressed as a ratio. Some examples of rational numbers are , , –
, –5, 2, and 0.
2 3
9
The last three examples, –5, 2, and 0, don’t look like fractions, but we can make them
5 2
0
look like fractions by writing them as – , , and . Remember, any number divided by
1 1
1
the number 1 is equal to itself. Notice that the numerators and denominators of rational
numbers are integers. Let’s observe one last thing about rational numbers. If we were to
use our calculators to divide each rational number’s numerator by its denominator we
would obtain the following decimal numbers:
–
1
 0.5
2
7
 2.33333...
3
5
= –5
1
2
2
1
–
10
= –1.11111…
9
0
0
1
Notice that all of these rational numbers and in fact, every rational number, can be
written either as a decimal number that ends or as a decimal number that repeats. This is
the essential characteristic of any rational number.
10
So what happens if we have a number that cannot be written as a decimal number that
ends or as a decimal number that repeats? Here is an example. You might have seen the
number represented by the Greek letter  before. If you use your calculator to write  as
a decimal number, you discover that  is approximately equal to 3.141592654… .
Notice that this decimal number never ends and is non-repeating. This type of number is
an example of an irrational number. Another way to think of irrational numbers, such as
 , is that they cannot be written as a fraction or ratio,which has an integer in both the
numerator and the denominator. Other examples of irrational numbers include many
square roots such as 2 , 3 , and 5 . The decimal representations of these square roots
are 1.41421…, 1.73205…, and 2.23607… respectively. All are never-ending, nonrepeating decimal numbers. None of them can be written as a fraction with an integer
numerator and an integer denominator.
Any real number is either a rational or an irrational number. Either it can be written as an
ending or repeating decimal (rational number), or it cannot (irrational number).
Rational and irrational numbers may be represented on a number line, just like we
represent natural numbers, whole numbers and integers. The number line below shows
how we would represent some of the rational and irrational numbers we discussed above.
–5
–4
–3 –2 –1
–
10
9
2
1
0
1
2
2
3
4
5

Example 2
Standardized
Test Prep
State whether each of the following numbers is a natural number, a whole number, an
integer, a rational number or an irrational number. Some of the numbers may be more
than one type.
Which is an irrational
number?
b)
17
c)
17
d)
–17
e)
0
A. 
B.
1
3
3
C. 0.12
D.
25
Answer: B
a)
3
–
4
11
Solution
a)
Remember, any
real number
must be either
rational or
irrational, but it
cannot be both.
3
is a rational number. It is written as a fraction with integers in the numerator
4
and denominator and its decimal equivalent is –0.75, which is a decimal number
which ends.
–
b)
17 is a natural number, a whole number, an integer and a rational number.
17
Remember 17  .
1
c)
17 is an irrational number. Its decimal representation is 4.12311…, which is a
never-ending, non-repeating decimal.
d)
–17 is an integer and a rational number. It is not a natural number or a whole
number like the number 17, because it is negative.
e)
0 is a whole number, an integer, and a rational number. Remember the natural
numbers begin at 1.
Find the Absolute Value of a Real Number
The absolute value of a real number is simply the number made positive. If the original
number is already positive, its absolute value is itself. If the original number is negative,
its absolute value is the same number without the negative sign. So the absolute value of
4 is 4. The absolute value of –4 is 4. The symbol | | is used to indicate absolute value. So
symbolically, |4| = 4 and |–4| = 4.
An alternative way of thinking about absolute value is that absolute value represents the
distance of any number on the number line from the number 0. Remember the absolute
value of a number is always positive and distance is typically thought of as a positive
amount. So how far is the number 4 away from the number 0 on the number line? The
answer is 4, and the absolute value of 4 is 4. And how far is the number –4 from the
number 0 on the number line? Again, the answer is 4, and the absolute value of –4 is 4.
4 units from 0
–5
–4
–3 –2 –1
3
4
5
–3 –2 –1 0
1
3
2
–4 is a distance of 4 units away from 0, so |–4| =4.
4
5
0
1
2
4 is a distance of 4 units away from 0, so |4| = 4.
4 units from 0
–5
12
–4
Example 3
Determine the value of the following:
a)
|100|
b)
|–10|
1
2
c)

d)
|0|
e)
|–  |
Solution
Always simply give a positive number answer when finding the absolute value of any
number.
a)
|100| = 100
b)
|–10| = 10
c)

1 1

2 2
d)
|0| = 0
e)
|–  | = 
EXERCISE SET 1.1.1
For each pair of numbers below in Exercises 1 through 4, place dots on a number line
indicating their locations and then place either the < symbol or the > symbol between the
numbers, whichever is correct.
For example, given the pair of numbers,
draw a number line that looks like this:
–5
–4
–3 –2 –1
0
1
–3
2
2
3
4
5
13
and then place the > symbol between the numbers. (2 > –3)
1.
1
5
2.
–1
–5
3.
0
– 2
4.
1
2
5
In Exercises 5 through 10 below, state whether each real number is a natural number, a
whole number, an integer, a rational number or an irrational number. If it is more than
one type of number, indicate all the correct types.
For example, the number –8 is an integer and a rational number.
5.
8
6.
0
7.
–
8.
–
2
3
10
9.
–10
10.
In Exercises 11 through 14, determine the absolute value as requested.
For example, |20| = 20
11.
|1|
12.
|–1.1|
13.
|– 3 |
14.
14
5
6