Download Building the Higher Term (Creating Equivalent Fractions)

Index
Chapter 1: The Real Number System
Section
1.1 Fractions
1.2 Exponents, Order of Operations, and Inequaity
1.3 Variables, Expressions, and Equations
1.4 Real Numbers and the Number Line
1.5 Adding and Subtracting Real Numbers
1.6 Multiplying and Dividing Real Numbers
1.7 Properties of the Real Numbers
1.8 Simplifying Expression
Practice Test
Pages
2-21
22-26
27-33
34-39
40-46
47-52
53-56
57-59
60-66
1
§1.1 Fractions
Outline
Definitions
Natural Numbers
Numerator
Denominator
Proper Fractions
Improper Fractions
Mixed numbers
Factors
Factor Tree
Multiple
Prime
Composite
Least Common Multiple (Denominator)
Prime Factorization
Reciprocal
Lowest Terms (Reducing)
Product
Pie Chart (Circle Graph) Multiplicative Identity Element
Writing fractions in lowest terms
Prime Factorization
Greatest Common Factor
Finding all factors
Divisibility of a number
Method 1: Prime Factorization
Method 2: GCF Method
Multiplying & Dividing Fractions
Mixed NumbersImproper Fractions
Improper Fractions Mixed Numbers
Method for Multiplying
Reciprocals & Method for Dividing
Adding & Subtracting Fractions and Mixed Numbers
Adding/Subtracting with Like Denominators
Method 1: Adding/Subtracting Mixed #'s using Improper Fractions
Method 2: Adding/Subtracting Mixed #'s Vertically
Fundamental Principle of Fractions
Building Higher Terms
Finding Least Common Multiples(Denominator)
Adding/Subtracting with Unlike Denominators
Application Problems
Word Problems – Handy Pointers
Pie Charts
Approximation
Homework
p. 10-14 All except #64, 78, 79 & 80
Review of fractions is the most important arithmetic review that we need for algebra. We
will use the core concepts of fractions many times to come, especially in our study of
rational expressions. You should focus on mastering the concepts of prime factorization,
reducing using prime factorization, finding a least common denominator and building a
higher term in order to add fractions with unlike denominators.
First, we need to review some vocabulary for fractions. Recall that
2  Numerator
3  Denominator
Remember also that a fraction can represent a division problem!
Example: What is the numerator of 5/8 ?
2
Example: What is the denominator of
19
/97 ?
Fractions represent a part of something. The numerator represents how many pieces of
the whole are represented. The denominator tells us how many pieces that the whole has
been divided into.
We like to represent fractions in what we refer to as lowest terms, which means that the
numerator and denominator have no factors in common except one. There are two
technical ways of putting a fraction into lowest terms. The first way uses greatest
common factors and the other uses prime numbers. Prime factorization is not shown
extensively in your book but I will be giving examples using both. First, we must digress
and discuss some definitions and some methods of factoring.
There are 2 classifications of all natural numbers {1,2,3,4…} greater than 1. They are
either prime or composite. (Note that one is not considered either prime or composite!)
A prime number is a number that has only one and itself as factors. A factor is a number
used in a product. A product is the answer to a multiplication problem.
Example:
7 – 17
19 – 119
29 – 129
Only one times the number itself yields a
prime!
It is helpful to have some of the prime numbers memorized, I believe that it is most useful
to know that 2,3,5,7,11,13,17,19,23, and 29 are prime and the most important of those
are 2,3,5,7 and 11.
A composite number is a number that has more factors than one and itself. The
definition of composite in the English language is “something that is made up of many
things”. This holds true for math as well, it is a number made up of many factors.
To find all the factors of a number (not the prime factors), simply 1) start at one times the
number itself and write them down with a good amount of space between, 2) then go on
to the two and ask is the number at hand divisible by two (is two a factor?), 2a) if it is
write two next to one and the other factor that yields your product next to the number
itself (there should still be space between 2 and the other factor), 3) continue on with each
successive natural number until you've "met in the middle."
3
Example:
14 – 1,
,14
14  2 = 7, so 2 & 7 are factors of 14
1, 2,
7, 14
14 isn't divisible by 3
14 isn't divisible by 4
14 isn't divisible by 5
14 isn't divisible by 6
Now I've "met in the middle" since the next # is 7 which
is in my list already, so I've found all the factors.
Example:
Find the factors of 9 and 18
All the numbers in the examples above are composite numbers because they have factors
other than one and themselves. Said another way, each composite number contains 1 and
itself as factors as well as at least one other number.
In order to find all the prime factors of a composite number, we will use a method called
prime factorization. The method goes like this: 1) What is the smallest prime number
that our number is divisible by? 2) What times that prime gives our number? 3) Once
we have these two factors we circle the prime number and focus on the one that isn’t
prime. 4) If there is one that isn’t prime, we ask the same two questions again, until
we have found all the prime numbers that our number is divisible by. 5) Then we
rewrite our composite number as a product of all the circled primes. 6) Finally, we can
use exponential notation to write them in a simplified manner. When multiplied
together all the primes must yield the composite number or there is an error.
12
/ \
2
6
/ \
2 3
12 = 223 = 22  3
Whether you use a factor tree as I have here, or use one of the other methods is up to you,
but I find that the very visual factor tree works nicely.
4
Example:
Find the prime factorization of 15 and 24
Here are some important hints about whether a number is divisible by 2, 3, 5, and 10.
You should take some time to go over these, they certainly help with division problems
as well as factoring.
Hints for 2: Is the number even? Does it end in 0, 2, 4, 6, or 8?
Example:
248
Hints for 3: Add the digits. Is the sum evenly divisible by 3?
Example:
123
Hints for 5: Does the number end in 0 or 5?
Example:
205
Hints for 10: Does the number end in 0?
Example:
350
5
NOW, back to the task we began to discuss on page 2, putting fractions in
lowest terms. First, I will discuss the method that your book does not –
Prime Factorization.
Prime Factorization Method
Step 1: Factor numerator and denominator into prime factors
Step 2: Cancel all factors in common in both numerator and denominator.
Step 3:
Rewrite the fraction.
Example: Reduce
12
/24 to its lowest terms.
Step 1: 12 = 2  2  3 .
24 2  2  2  3
Step 2: Cancel the 2 of the 2's and the 3's
Step 3: Rewrite 1/2 .
Note: Canceling is division and when you divide any number by itself the result is
always one! This can cause problems if we do not realize that we are dividing out the
common factors leaving a one behind when all the factors in the numerator cancel! The
tendency then becomes to write a 0, but it is always 1!
Example: Reduce
27
/81 to its lowest terms.
Now the method discussed in your book, the greatest common factor (GCF) method.
First you must know that the GCF of two numbers is the largest number that is a factor of
both. So if you find all the factors of the numerator and denominator (see above for finding
all factors of a number) the GCF is the largest factor that is in both lists. We will now put the
same fractions as above (Prime Factorization Method) in lowest terms using the GCF method.
6
Greatest Common Factor Method
Step 1: Find the GCF of numerator and denominator
Step 2: Factor the numerator and denominator using GCF
Step 3: Cancel the GCF from the denominator and numerator
Step 4: Rewrite the fraction
Example: Reduce
12
/24 to its lowest terms.
Step 1: 12 – 1, 2, 3, 4, 6, 12
24 – 1, 2, 3, 4, 6, 8, 12, 24
GCF = 12
Step 2: 12 = 12  1
24 12  2
Step 3: Cancel the 12's
Step 4: Rewrite 1 .
2
Example: Reduce
27
/81 to its lowest terms.
Of all the operations with fractions, multiplication is the easiest! It is extremely straight
forward.
7
Multiplying Fractions or Mixed Numbers
Step 1: Convert any mixed numbers to improper fractions.
Step 2: Multiply the numerators (tops)
Step 3: Multiply the denominators (bottoms)
Step 4: Simplify by writing as a mixed number and/or reducing
First, recall that a mixed number is a whole number added to a fraction, denoted simply
by a whole number with a fraction written next to it. These can be changed to an
improper fraction – a fraction in which the numerator is larger than the denominator.
(We should also note that a proper fraction also exists and it is a fraction in which the
numerator is smaller than the denominator.) Let's quickly review how to change a mixed
number to an improper fraction.
Mixed NumberImproper Fraction
Step 1: Multiply the whole number by the denominator of the fraction
Step 2: Add the numerator of the fraction to the product in step1
Step 3: Put sum over the original denominator
Example:
Convert
1½
to an improper fraction.
Let's quickly review writing an improper fraction as a mixed number. We just finished
our discussion of reducing (also called lowest term) and I believe that it is always easiest to
convert to a mixed number when the improper fraction is in its lowest terms, but it is not
necessary, therefore step one can become step three.
Improper Fraction Mixed Number
Step 1:
Reduce the improper fraction to its lowest terms
Step 2:
Divide the denominator into the numerator
Step 3:
Write the whole number and put the remainder over the
denominator.
Let's try one example before we resume our discussion of multiplication.
Example:
Change to a mixed number
27
/4
8
Now let's return to multiplication.
Example:
Multiply
/25  5/7
21
Note: You may also use principles of canceling and therefore eliminate the need for
reducing. Because the original numerators are the factors of the resulting numerator
and the same for the denominators the principles of converting to lowest terms by method
of GCF or prime factorization apply!
Example:
Multiply
3
/4  1 1/4
Note: The denominators multiply even though they are the same! This does not seem
difficult until after we cover addition, but try to keep it in mind!!!
Before we discuss dividing fractions we must define a reciprocal. A reciprocal can be
defined as flipping the fraction over, which means making the denominator the numerator
and the numerator the denominator. Another way that I frequently speak of taking a
reciprocal is saying to invert it. The true definition of a reciprocal, however, is the
number that when multiplied by the number at hand will yield the multiplicative identity
element, one.
9
Dividing Fractions
Step 1: Convert all mixed numbers to improper fractions
Step 2: Take the reciprocal the divisor (that is the second number that
you're dividing by)
Step 3: Multiply the inverted divisor by the dividend (the number that you
are dividing into, the first number)
Step 4: Simplify the answer if necessary by reducing and/or changing
to a mixed number.
5
Example:
Divide:
Example:
Divide:
5
Example:
Divide:
5
/8 
/8 
3
/4
/8 
3
/5
2
/3
Note: If you tend to do your canceling before multiplying then do not jump the gun!
Canceling can only be done when there is a multiplication problem!!! In this problem,
you can not cancel the 5's because it is division at this point, not multiplication!
10
Example:
Divide: 2 1/8 
3
/5
Note: Since division is multiplication by the reciprocal, dividing mixed numbers is no
different from multiplying them! We must always convert them to improper fractions
before performing any operation on them.
In addition (subtraction) we have two cases to consider. The first case is the easiest, when
the denominators are alike, and the second requires using the LCD to add fractions with
unlike denominators. We will discuss the LCD shortly.
Adding (Subtracting) Fractions with Common Denominators
Step 1: Add the numerators
Step 2: Bring along the common denominator
Step 3: Simplify if necessary by reducing and/or changing to a mixed
number
Example:
4
/5 +
3
/5
11
Example:
3
/8 +
1
/8
Note: The truly unfortunate thing about addition of fraction is that you can only simplify
after the problem is complete, so you always have to keep your eyes open for an answer
that is not in simplest form!
Along with adding/subtracting fractions we need to discuss adding/subtracting mixed
numbers. There are two approaches for doing this. The first is to change both to
improper fractions and follow the steps for adding fractions. This method is fine for
adding/subtracting fractions that have small whole numbers and/or small denominators,
but it can get messy when those numbers get large. The second method handles large
numbers well, but can get tricky when subtracting a larger fraction from a smaller one. (If
you want to know which fraction is the larger of two find their cross products and the
larger indicates the larger fraction. Cross products are the product of the denominator of
one fraction and the numerator of the other, and are written next to the numerator factor's
fraction.)
Adding/Subtracting Mixed Numbers using Improper Fractions
Step 1:
Step 2:
Step 3:
Step 4:
Step 5:
Convert to improper fractions
Find a least common denominator if necessary
Build the higher term if necessary (step 2 was taken)
Add/subtract numerators and place over LCD
Simplify by reducing and/or changing to a mixed number
Example:
Subtract
5 2/3  3 1/3
12
Adding/Subtracting Mixed Numbers using Mixed Numbers
Step 1:
Step 2:
Step 3:
Step 4:
Put in a vertical fashion. First on & top second on bottom.
Find a least common denominator if necessary
Build the higher term if necessary (step 2 was taken)
Add/subtract numerators of fraction portion and place over LCD
Note: If the numerator of the top fraction is less than that of the bottom, you can't reverse the order of
subtraction, you must borrow.
Step 4a:
To borrow subtract one from the whole number (like when
subtracting large numbers) and add that whole to your fraction by
making a fraction of the LCD over the LCD.
Step 5: Add whole numbers
Step 6: Simplify by reducing and/or changing the fractional portion to
a mixed number.
Note: If you change the fractional portion to a mixed number you must further simplify the answer by
adding the whole number and this mixed number.
Step 6a:
Add the fractional part's mixed number and the whole number by
adding the wholes and rewrite as a mixed number.
Example:
Add
1¾ + 2¾
Example:
Subtract
306 15/100  125 58/100
Note: I think that you could see how much more difficult this could become if we used
the first approach with these large whole numbers and large denominators!
13
Now let's proceed to the discussion of necessary elements required to add/subtract two
fractions with unlike denominators. First we need to know how to build a higher term
and then we need to know how to find a least common multiple (known as a least common
denominator when applied to fractions).
To build a higher term you must know the Fundamental Principle of Fractions.
Essentially this principle says that as long as you do the same thing (multiply or divide by the
same number) to both the numerator and denominator you will get an equivalent fraction.
Here it is in symbols:
Fundamental Principle of Fractions
ac = a
or
ac = a
bc
b
bc
b
This is used to build an equivalent fraction – a fraction that represents the same
quantity.
Example:
1
/4
and
2
/8
are equivalent fractions.
Building the Higher Term (Creating Equivalent Fractions)
Step 1:
Step 2:
Step 3:
Step 4:
Decide or know what the new denominator is to be.
Use division to decide what the "c" will be (as in the fundamental principle of
fractions.)
Multiply both the numerator and denominator by the "c"
Rewrite the fraction.
Example:
Write 1/8 as an equivalent fraction with a
denominator of 24
Next we must discuss how to find a least common multiple (LCM). The least common
multiple is the smallest number for which 2 or more numbers are factors. Another way
of saying the same thing is to use the word multiple and say that the LCM is the smallest
number which is a multiple of 2 or more numbers. A multiple is a number created when
you multiply a certain number by the natural numbers. For instance, the multiples of 4
are 4, 8, 12, 16, … We will use LCM’s to find the least common denominator (LCD)
so that we can add fractions with unlike denominators. A least common denominator is
the smallest number which is a multiple of all the denominators.
14
Prime Factorization Method to Find LCM (LCD)
Step 1:
Step 2:
Step 3:
Factor the numbers (denominators for LCD) using prime factorization
Take note of the unique factors in the factorizations
Create a LCM (LCD) by using each unique prime factor the
number of times that it appears the most (largest exponent, variables
included), for any one number (not the total number of times that it
appears!)
Example:
Find the LCD of 1/12 & 2/15
Note: A negative is not considered to be a part of an LCD, they are only positive.
Example:
Find the LCD of 2/3, 3/5 and 5/7
Note: If they are all primes then the LCM is their product!
15
Example:
Find the LCD of 2/5, 7/25, 13/50
Note: If the largest is a multiple of the smaller one(s) then the largest is the LCM
Now that we know how to build a higher term and find an LCD we can put these two
steps together in order to find an LCD, build the higher term using the LCD and finally
add or subtract the fractions at hand. Let's go over the steps for putting it all together.
Adding/Subtract Fractions with Unlike Denominators
Step 1:
Step 2:
Step 3:
Step 4:
Find the LCD
Build the higher term using LCD
Add/Subtract the new fractions with common denominators.
Simplify the fraction if possible by reducing and/or converting to a
mixed number.
Example:
2
/5 
1
/3
16
Example: 1 1/4 + 3 2/3
Example:
9 2/15  2 13/25
We can also apply our working knowledge of fractions to the real world and to
interpretation of a special kind of graph called a Pie Chart. First, we should discuss some
common sense approaches to application problems. What I mean is that we should not
read an application problem (Yes, I am talking about the dreaded word problem) and
automatically say I can't. You don't know until you've tried with an open mind! The first
key is translation from English to math. Just with anything in life you must ask yourself
what is being expected of you; mastering a few simple phrasings and using common
sense in relation to wording can accomplish this. Here is a simple process for working
word problems:
17
Solving Word Problems
Step 1:
*Step 2:
*Step 3:
*Step 4:
*Step 5:
*Step 6:
Step 7:
Read the problem carefully before doing anything. Understand
what the question is that needs to be answered.
Write down in shorthand all information given and needed.
(*This is a step that I must see on paper or points are
deducted.)
Decide how the information given, can be used to obtain the
information needed (this is actually part of what you need to do in step 1 to
understand the problem, but it comes in written form here) . Write down an
equation using shorthand notation and symbols that will be used
to find the information needed.
Plug given information (usually numbers at this time) into the equation
in step 3.
Solve the equation (at this time it will be a simple computation).
Find the answer to the problem (most times this will be the same as step 5).
Give the answer using correct labels.
Check your answer, making sure that your answer makes sense.
Some helpful hints in deciding the operation being done to arrive at the answer (the
operation in the formula) are as follows:
Addition – More than, altogether, somehow you are combining things to see a
total, perimeter (the distance around an object)
Subtraction – Less or smaller than, taking away from, how many are left, a total
prior to (before) something happening that increased the amount,
decreasing, difference
Multiplication – Repeated addition, number of times some amount, number of a
single serving or single type (key word being of)
Division –
Parts of, Subdividing into sections, pieces or parts, Equal parts
Example:
If I have a recipe that calls for 2 ½ cups of flour and I
want to double the recipe (make 2 times the number of servings),
how many cups of flour should I use?
18
Example:
The three sides of a triangle measure as follows. Side 1
is 2 1/8 feet, side 2 is 3 5/8 feet, and side 3 is 7 5/12 feet.
Find the perimeter of the triangle.
Example:
If I have a string that is 25 ¾ inches long and I need it to
be divided into 4 equal pieces, how long will each piece
be?
19
Example:
At the end of the day I have more garbage than I started
with. My can is marked off in fractional increments to
one. It is at ¾ at the end of the day and it has increased
by 3/8. At what fractional increment did it begin the day?
Example:
At the end of the day my garbage can is at the ¾ mark
and at the beginning of the day it was at 1/8, what was the
diference between these two measurements?
20
Pie charts are used to represent parts of a whole. This is what fractions do for us so a pie
chart is a nice visual representation of fractional parts of things. We use pie charts to
show many statistics, and most commonly you will see them as percentages, but since a
percentage is a fraction, it is the same. The whole can be one or one hundred thousand or
any other whole number. This opens pie charts up for applications of fractions using
addition, subtraction and multiplication.
Example:
a)
Look at the chart on page 13 of your book. It shows the
fractional portion of the 8 million immigrants in the US
between 1990 and 1997 broken down by the major
"cultural" groups.
Asian – 3/10
European – 13/100
Latin – 13/25
Other – Remaining
How much larger was the fractional portion of Latin
immigration than European?
b)
Together what fraction did the Asian, European and Latin
immigrants make up?
c)
How many of the 8 million people were Asian?
21
§1.2 Exponents, Order of Operations, and Inequality
Outline
Definitions
Exponent
Base
Operators
Braces
Brackets
Parentheses
Translation
Mathematical Statements
Inequality Symbols
Exponents
Base
Repeated Multiplication
Order of Operations
PEMDAS
Inequality Symbols
<,>, , , 
Translation
Mathematical Statement
Mathematical Expression
Bar Graph
Practical Application of Inequality Symbols
Homework
Order of Operations
Grouping Symbols
Mathematical Expression
p. 20-22 #1-4,6,10,18-22,24-33mult.of3,38,40,41,42-54even,
55-61odd,71,72,75,76,78,81
What is an exponent? An exponent represents repeated multiplication. The exponent,
the little number that is written above and to the right of the base, tells us how many
times to use the base as a factor. The base, the main number, can be any real number.
Using a number as a factor means multiplying it by itself.
Base  5 2 Exponent
Example:
What is the base of 23 ?
Example:
What is the exponent of 23 ?
Example:
What is 23 ?
22
There are also some special names for exponents 2 and 3. The exponent 2 is called the
square and 3 is called the cube. There are also several ways to refer to an exponent – we
can say "to the (exponent's) power" or "to the power of (exponent)"
Not only can the base be a whole number but it can also be a fraction, an expression, etc.
Let's try an example using fractions.
Example:
(3/5)2
I would highly recommend that you memorize the perfect squares for quickly squaring
simple numbers and for future reference.
12=1, 22=4, 32=9, 42=16, 52=25, 62=36, 72=49, 82=64, 92=81, 102=100, 112=121, 122=144, 132=169,
142=196, 152=225, 252=625.
The following perfect cubes may also be helpful:
13=1, 23=8, 33=27, 43=64, & 53=125
When many operators (addition, subtraction, multiplication & division) are used in a
single mathematical expression it can become confusing as to which one to do first. We
solve this by using order of operations and grouping symbols. Grouping symbols such
as brackets [ ], braces {}, and parentheses ( ) help us to tell others what we wish them to
do first, which leads directly into the order of operations – the order in which operations
are done when in an expression together.
Order of Operations
Parentheses, Brackets or Braces
Exponents
Multiplication and Division in order from left to right
Addition and Subtraction in order from left to right
A trick for reminding yourself which operations in what order is the
following: Please Elect My Dear Aunt Sally or simply PEMDAS.
Example:
7 + 22  2
23
Example:
5  (1 + 9)  21
Example:
32[3 + 3(22)]  5
Example:
4  7/3 + 1
2
/3  1/2 + (1/3)2
Next on our agenda is a small discussion of the inequality symbols, which will lead us
directly into a discussion of translation problems.
24
Inequality Symbols

Does Not Equal
<
Strictly Less Than
>
Strictly Greater Than

Less Than or Equal To

Greater Than or Equal To
Some helpful hints in reading the greater than and less than symbols:
The arrow points to the smaller number.
Put teeth in the mouth and the smaller eats the larger (you may have learned this as a
child, and therefore I pose this here to jog a memory!)
We have both strictly less than and less than or equal to, and both can be used to refer to
the same case, but the less than or equal to can also indicate equality. The less than or
equal to refers to a case in which either the strict inequality or the equality holds true.
Example:
a)
State whether each statement is true or false.
7<9
b)
9 > 11
c)
15  7 + 8
Example:
a)
b)
Write the statement using symbols (translate).
12 is less than 21
15 is greater than or equal to 13
Now we need to discuss the most important concept in this section. Without total control
of translation word problems will become unmanageable. Translation means taking a
sentence written in English and putting it into mathematical symbols. We will be
translating mathematical sentences like those above that contain the words and phrases
that refer to the inequality symbols. These should not be confused with translation of
mathematical expressions that contain operators. Two phrasing for the operators addition
and subtraction can appear similar to those for mathematical sentences [10 greater than 4 (4 +
10) versus 10 is greater than 4 (10 < 4)] . Mathematical expressions always contain operators,
whereas mathematical sentences contain inequality symbols and not operators.
25
The authors now choose to discuss translation of bar graph in their relation to inequality
symbols and translation of mathematical sentences. I will not present a formal discussion
here. I will remind you that more is the same as greater, at most means less than or equal
to, and not less than means greater than or equal to.
26
§1.3 Variables, Expressions and Equations
Outline
Definitions
Equation
expression
evaluation
solution
Variable
elements
set
ellipsis notation
root
sequence
set-builder notation
define the variable
all phrases in charts
Algebraic Expression
Evaluation
Translation
Differentiate from Equation
Algebraic Equation
Is a number or a member of a set a solution?
Translation
Differentiate from Expression
Translation
Phrases for Operators
Key for order in subtraction
Phrases for Equality
Phrases for Variables
Phrasing for Parentheses
Sets
Homework p. 27-28 #1-9all, 12-42mult.of3, 43-58all, 60, 63, 66, #68-76all
In this section we will be discussing algebraic expressions and algebraic equations. The
difference between an algebraic expression and an algebraic equation is an equal sign.
An algebraic expression does not contain an equal sign and an algebraic equation
does! An algebraic expression is a collection of numbers, variables (a letter of the alphabet
used to denote an unknown number), operators and grouping symbols.
Example: All the following are algebraic EXPRESSIONS
a)
3x
b)
3x2  5
c)
6(y4) + 2
d)
5z  16
5
(Recall that fraction bars are grouping symbols)
An algebraic expression can be evaluated. This means that given specific values for the
variables, we can give a numeric answer to the expression.
27
Example: Evaluate 3x2 + 5y  z given x = 2 y = 1 and z = 3
Example: Evaluate 2c + 15 given c = 3 and k = 7
k
Example: Evaluate 2 ( z + 5 ) 
4
/z + k2 given z = 2 and k = 5
An algebraic equation sets up equivalence between an algebraic expression and a
numeric expression (a number or numbers with operators) or between two algebraic
expressions. Since an equation sets up an equivalence relationship it can be solved. This
means that a solution or root can be found which will make the equation true. A solution
or root is a value that can be put in place of the variable to make the equation true. We
are not developing a method for solving equations yet, but we do need to practice
deciding if the solution is correct. This is done in the same manner as evaluating an
expression, except we then must decide if the resulting statement is true, if it is then we
have found a solution and if it is false then we don't have a solution.
28
Example:
a)
b)
Example:
The following are examples of algebraic EQUATIONS
5 + x = 9
9  z = 5  2z
Is 2 a solution to the following equation?
x2 + 2 = 8; 2)
(Your book will pose this question in this manner:
x2 + 2 = 8
This section also helps you to practice your skills for writing algebraic expressions or
equations by translating the following words and phrases into operators. Following is a
helpful chart, after which we will practice translating some expression. Remember that
translation is the key to all word problems.
Addition
Word
sum
more than
added to
greater than
increased by
years older
than
total of
plus
Subtraction
Word
difference of
*years
younger than
diminished
by
*less than
Phrasing
The sum of 7 and 2
5 more than 10
6 added to 10
7 greater than 9
4 increased by 20
15 years older than John. John is 20.
7 + 2
10 + 5
10 + 6
9 + 7
4 + 20
20 + 15
The total of 6 and 28
8 plus 281
6 + 28
8 + 281
Phrasing
The difference of 5 and 2
The difference of 2 and 1
Sam's age if he is 3 years younger
than John. John is 7.
15 diminished by 9
21 diminished by 15
17 less than 49
7 less than 17
Symbols
Symbols
5  2
2  1
7  3
15
21
49
17




9
15
17
7
29
decreased by
*subtract(ed)
from
take away
subtract
less
29 decreased by 15
15 decreased by 7
Subtract 13 from 51
Subtract 51 from 103
79 take away 61
54 subtract 2
16 less 4
29  15
15  7
51  13
103  51
79  61
54  2
16  4
* - Means that the numbers come in opposite order than they appear in the sentence.
Multiplication
Word
product
times
twice
multiplied by
at
"fractional
part" of
"Amount" of
"$" or "¢"
percent of
Phrasing
The product of 6 and 5
24 times 7
Twice 24
8 multiplied by 15
9 items at $5 a piece
A quarter of 8
Amount of money in 25 dimes
(nickels, quarters, pennies, etc.)
3 percent of 15
Symbols
65
24(7)
2(24)
8*15
($5)9
(¼)(8) or
8
/4 .
($0.1)(25) or (10)(25) ¢
0.03(15)
Division
Word
divide
quotient
Phrasing
Symbols
Divide 81 by 9
81  9
The quotient of 6 and 3
6  3
The quotient of 24 and 6
24  6
divided by
100 divided by 20
100  20
20 divided by 5
20  5
ratio of
The ratio of 16 to 8
16  8
The ratio of 8 to 2
8  2
shared equally 65 apples shared equally among 5
65  5
among
people
Note: Division can also be written in the following equivalent ways, i.e. x  6 = x/6 =
6x = x
6
Exponents
Words
Phrasing
Algebraic Expression
squared
Some number squared
x2
square of
The square of some number
x2
cubed
Some number cubed
x3
cube of
The cube of some number
x3
(raised) to
Some number (raised) to the power of x6
the power of 6
30
Equality
Words
yields
equals
is
will be
was
Phrasing
A number and 7 yields 17. Let x = #.
7 and 9 equals 16
The sum of 5 and 4 is 9.
12 decreased by 4 will be 8.
The quotient of 12 and 6 was 2.
Algebraic Equation
x + 7 = 17
7 + 9 = 16
5 + 4 = 9
12  4 = 8
12  6 = 2
Note: Any form of the word is can be used to mean equal.
Parentheses
Parentheses are indicated in four ways. The first is the use of a comma, such as: The
product of 5, and 16 less than a number. The second is the use of two operators'
phrases next to one another. An example of this would be 17 decreased by the sum of
9 and 2. Notice how decreased by is followed by the sum of and not a number, this
indicates that we will be doing the sum first; hence a set of parentheses will be needed.
Next, you may notice that the expected 'and' between the two numbers being operated
upon is after a prepositional phrase [A phrase that consists of a preposition (usually of in our
case) and the noun it governs (usually number in our case) and acts like an adjective or adverb]. For
example: The sum of 9 times a number and the number. Usually we would see the 'and'
just after the number 9, but it does not appear until after the prepositional phrase 'of 9
times a number'. If you think of this in a logical manner, what you should see is that you
have to have two numbers to operate upon before you can complete the operation, which
would require the use of parentheses to tell you to find a number first! Finally, you may
notice a phrase containing another operator after the 'and' where you would expect a
number. An example here might be: The difference of 51 and the product of 9 and a
number. The note about thinking in a logical manner applies here too! You must have
two numbers to operate on!
Variables
Variables are an undisclosed number. If you are not told what variable to use by the
author, you should always define the variable as part of the problem. We define the
variable by saying "Let x = #" or whatever letter you choose to be your unknown, the
most commonly used variable is x, but you will find that sometimes it makes more sense
to use another letter.
Example:
a)
b)
Write an algebraic expression for the following. Let x = #
The difference of five times a number and two
The sum of five and a number increased by two
31
c)
The product of a number and two, decreased by seven
d)
The quotient of seven and two times a number
Example:
a)
Write an algebraic equation for the following:
Two times a number yields ten
b)
The ratio of a number and two is the same as eight
c)
The total of eighteen and five times a number equals
eleven times the number
d)
The product of five and the sum of two and a number is
equivalent to the quotient of forty and the number
A set of numbers is a group of numbers that meet some predefined criteria. A set is
denoted by braces and can list all elements (in math usually numbers), describe the elements
using an algebraic expression or words (called set-builder notation), or list a sequence (a
pattern by which numbers are related using addition or multiplication) using ellipsis notation [the use
of three periods (…) to denote continuation of a pattern] . Your authors will ask you to determine
which numbers from a is a solution to an equation that your have translated.
32
Example:
Which of the number from {2, 5, 9, 12} is a solution to
the following algebraic equations?
a)
The product of 2 and the sum of x and 3 is 30.
b)
The square of x less 9 yields 16.
33
§1.4 Real Numbers and the Number Line
Outline
Definitions
Whole Numbers
Natural Numbers
Integers
Positive Numbers
Negative Numbers
Signed Numbers
Rational Numbers
Irrational Numbers
Number Line
Real Numbers
Negative
Opposite
Additive Inverse
Absolute Value
Order Property of Real Numbers
Coordinate
Double Negative
Identity Element of Additon
Real Numbers
Whole Numbers {0, 1, 2, 3, …}
Natural Numbers {1, 2, 3, …}
Integers {…, -3, -2, -1,0, 1, 2, 3, …}
Positive Numbers (see natural numbers)
Negative Numbers {x | x  integers < 0}
Signed Numbers {x | x  integers, x0}
Usefulness in everyday life
Rational Numbers {x/y | x  integers, y  integers, y0}
Irrational Numbers {x | x can't be represented by a rational number}
Number Line
Visual representation of Real Numbers
Graphing on
Whole numbers
Rational Numbers
Opposites
Definition
Negative (shorthand notation)
Additive Inverse
Double Negative is opposite of a negative number
Absolute Values
Definition
Notation
Opposite of
Comparing Opposites and Absolute Values
Simplify 1st then compare
Decimals
Appendix B, p. 649
Homework
p. 36-38 #9-18all, 21, 22, 29, 30, #35-41all, 44, 47, 50, 53, 56, 59, 62, #65-68all
p. 652 #2-18even, #22-38even
Real Numbers are all numbers that can be represented on a number line.
A number line is a continuing line that represents the real numbers. At the center is
zero, moving to the left from zero are the negative numbers and moving to the right from
zero are the positive numbers. The fact that moving left on the number line encounters
increasingly smaller numbers, while moving right encounters increasingly larger numbers
is known as the order property of real numbers.
34
Graphing on a Number Line
In order to graph a number on a number line all we must do is locate the number and then
put a dot on the point and then label it with the coordinate. A coordinate is the number
associated with the point being labeled.
Example:
Label 0, -1 and 2 on the number line below.
We have now discussed 3 set of numbers thus far – the natural numbers {1, 2, 3, …}, the
whole numbers {0, 1, 2, 3, …}, and the real numbers {x | x is any number on the number
line}. The number line is going to help us with our next set of number, the integers. The
integers are the natural numbers, their opposites (negatives) and 0 {…, -3, -2, -1, 0, 1, 2,
3, …}. Opposites are numbers that are the same distance from zero but on the opposite
side of zero when consulting a number line. Said another way, place a negative sign in
front of a number and you have created its opposite (although it may need to be simplified when
considered in this manner, but we'll discuss that a little later) .
The integers are made up of positive numbers, negative numbers and zero. The positive
and negative numbers (the integers to the right of zero are called the positive numbers and those to the
left are negative) make up a set of numbers called the signed numbers {x | x  integers, x 
0}. The signed numbers are useful to us in describing everyday circumstances.
Temperatures above zero degrees are positive and those below zero are negative. If we
have money in our checking account we have positive number balances and if we are
overdrawn we have negative number balances. If we are above sea level we have heights
measured in positive numbers, but if we go below sea level we have heights (depths)
measured in negative numbers.
On the number line there is always one unit between each whole number that is marked.
Between these whole numbers are the fractions. Fractions are part of the irrational
numbers – numbers that can be represented as a quotient of two real numbers, where the
bottom (divisor) can’t be zero {x/y | x  , y  , y  0}. The other part of the irrational
numbers is the integers, because every integer can be written as a fraction by placing
it over 1.
35
Graphing Fractions
Graphing fractions is the same as graphing any other number; it just requires that we have
marks between whole numbers. There will be one less mark than the number in the
denominator since one whole is represented by the denominator over itself. For instance,
if we wish to represent thirds, you will have two marks between 0 and 1, 1 and 2, etc.
because the marks break the line into 3 parts between 0 and 1, 1 and 2, etc. Let’s graph
some fractions.
Example:
Graph the following fractions.
a)
½
b)
¾
Note: Both ¾ and ½ can be graphed on the same line if you realize that 2/4 is equivalent
to ½.
Graphing a mixed number is the same as graphing a fraction. You must recall that it is
the whole number plus the fraction, meaning that it is the fractional portion greater than
the whole portion of the mixed number. Therefore, in order to graph a mixed number
you would label your number line to at least 1 number larger than the whole portion and
then divide the line between the whole portion and the next largest number into the
fractional parts represented by the fraction.
Example:
Draw your own number line and graph 12/5.
Note: If you were asked to graph an improper fraction it is easiest to divide the
numerator by the denominator to change it to a mixed number and graphing as described
above.
36
The last set of numbers that we must discuss is the irrational numbers. The irrational
numbers are numbers that can't be written as a quotient of integers. Any number that is
a non-repeating, non-terminating decimal is an irrational number. Some examples of
such numbers are as follows: , - 7 , 0.325079325774…, etc.
The real numbers can now be described as the set of rational and irrational numbers {x |
xrational numbers or x irrational numbers}. The only note left is that although our
discussions in this class will be restricted to the real numbers, there are more sets outside
the reals. One such set we have a chance to encounter in Chapter 9. They are the
imaginary numbers and they represent square roots of negative numbers such as  -3.
You will have exercises to do that ask you to choose numbers from a set that meet the
criteria of the various sets. Realize that the sets overlap and a number may be a member
of more than one set of numbers! I never take valuable class time to go over examples
like this, but feel free to ask questions after attempting homework, if you feel it
necessary.
----------------------When talking about the number line, there are two properties that come about, the
absolute value and the opposite. These two properties are very similar and can become
confused, but if you learn their definitions you should never have a problem!!
The absolute value of a number is defined as the number of units a coordinate is located
away from zero on the number line. Absolute value does not take into account whether a
number is positive or negative – it is strictly the number without its sign! We indicate
absolute value in the following way
| number |
Example:
The absolute value of number
Simplify
a)
|7|
b)
| -3 |
37
The opposite is defined as the coordinate located the same distance on the opposite side
of zero. If a number is positive, then its opposite is negative, and likewise if a number is
negative then its opposite is positive. A shorthand way of writing the opposite of a is –a
and hence your authors' reference to the opposite as the negative. There is one more
name for the opposite and that is the additive inverse. The additive inverse is defined as
a number that when added to the number at hand will yield the identity element of
addition (zero). The number that does this is a number's opposite; hence the additive
inverse and the opposite (negative) are the same!
a + -a = 0
a's Additive Inverse/Opposite is -a
To possibly head off some complications, let's define something called a double negative.
A double negative it the opposite of a negative number. And when considered in this
respect while taking into account the definition of an opposite, it should be quite clear
that the opposite of a negative number is a positive number, but when it comes at you in
shorthand it can be confusing (some of you may also have some additional amo to use in
this regards in that you already know that a negative times a negative yields a positive).
- ( -a) = a
Example:
For all a  integers
Simplify
a)
- (-5)
b)
- (2)
c)
- (- (-7))
d)
-|-3|
Note: This is the most confusing of all opposite problems because it is read the opposite of the absolute
value of negative 3. Since the absolute value is defined as the number without its sign, we can’t determine
the opposite of the absolute value until that portion is sorted through. Thus, unlike its counterpart –(-3),
the answer is –3 for this problem because the absolute value of –3 is 3 and the opposite of 3 is –3. You
must always remember that absolute value signs are like parentheses, but they have a special meaning that
must be taken into account before anything to their left.
38
In order to compare absolute values and/or opposites you must first simplify their
meaning. I must see any simplification for full credit on problems! Then you must
use the order property of the real numbers (recall that this says that any number to the left of a
number is less than the number and any number to the right of a number is greater than the number).
Example:
Use < , > or = to compare each of the following (show any
simplification of the numbers that you are comparing) .
a)
-4
1
b)
-(-6)
13
c)
-|5|
- | -5|
d)
¼
¾
e)
-¼
-¾
f)
-32
(-3)2
/2
Note: This is a very important distinction! These are not equal because in the first case
the base is 3, not –3, because –3 = -1  3, and the exponent only applies to the number to
which it is written directly to the right of. It is the same as saying 3x2, in which case it is
only the x that is being squared not the 3! On the other hand in (-3)2 the parentheses
indicate that the base is indeed –3, which means that –3 is to be used as a factor twice.
As a final note in this section – you should review working with decimals in Appendix B
on page 649 of your book. You will be responsible for operations with decimals without
the use of a calculator and therefore you may need to review the basics. If you
experience difficulty, please come to me for help during office hours or ask questions in
class. Some homework has been assigned from this appendix and it is expected that you
have done it without the aid of a calculator.
39
§1.5 Adding and Subtracting Real Numbers
Outline
Definitions
Addend
Subtrahend
Minuend
Adding Real Numbers
Using the Number Line
Using Rules
Same signs
Opposite signs
Order of Operations
Still PEMDAS
Subtracting Real Numbers
Redefine subtraction as addition of the opposite
Use rules of addition
Order of Operations
Still PEMDAS
Translation Problems w/ Integers
Review §1.3 words and phrases
Applications w/ Integers
Decreases can be negative integers
Any number below zero is negative
Homework p. 46-50 #1-6all, #9-33mult.of3, #32, #42-96mult.of3, #98-108even
One way that you can add real numbers is using a number line. This is a very helpful
visual tool for learning how positive and negative numbers are added. Recall the order
property of the number line – recall that if we have a negative number we are traveling in
the negative direction on the number line (left) and we have a positive number we are
traveling in a positive direction on the number line.
-5
0
5
Adding on the Number Line
Step 1: Start at the first addend
Step 2: Move the appropriate number of units in the positive or
negative direction as indicated by the 2nd addend
Example: Add each of the following using a number line
a)
7 + (-3)
40
b)
-3 + (-4)
c)
-10 + 4
The other alternative has some rules to remember:
Adding Integers
If the signs are the same
Step 1: Add the absolute values
Step 2: Keep the common sign
If the signs are different
Step 1: Subtract the absolute value of the smaller from the larger
Step 2: Keep the sign of the larger
Example:
a)
Use the rules for adding integers to add the following
15 + 88
41
b)
9 + (-5)
Note: If you need to know which fraction is the larger of the two you may put them side
by side find the cross products and the one with the larger cross product is the larger
fraction. Here 1/5 is larger as 91= 9 is larger than 15=5. Since it is the absolute
values that we are dealing with, just disregard the sign until the end.
c)
-15 + (-42 )
d)
-80 + 15
Now, let’s do some that are a little more complex. Let’s throw in real numbers and add
more than 2 integers. Remember when adding more than 2 integers that you can use the
addition properties of numbers, they also apply to integers. Recall that 2 important
properties say that we can regroup numbers and move them around. I will use these
properties to add all numbers with the same signs and then add numbers with opposite
signs last when necessary.
42
Example:
a)
Add the following
1
/9 + (-1/5)
Note: If you need to know which fraction is the larger of the two you may put them side
by side find the cross products and the one with the larger cross product is the larger
fraction. Here 1/5 is larger as 91= 9 is larger than 15=5. Since it is the absolute
values that we are dealing with, just disregard the sign until the end.
b)
-15 ½ + (-42 ¼ )
c)
-80.5 + 15
d)
-9 + 10 + (-7)
43
e)
[-7 + (-8)] + [7 + (-3)]
Note: Given this problem without the brackets I would choose to add all the negative
numbers first, because that is a simple addition problem that yields a negative number
and then I would add the positive number to that result. Try it and see if you get the same
answer as we just got!
The first key to subtracting two real numbers is seeing subtraction problems as addition
problems. This is crucial in subtracting integers and does not conflict with anything that
you have already learned. Please do not fight this process as it is crucial in
understanding. If you are beyond this, please do not voice your objections, just bear with
me. In order to subtract two integers we need to rewrite all subtraction problems as
adding the opposite of the subtrahend (that’s the second number)!!
Example:
Rewrite each subtraction problem as an addition problem
and solve.
a)
39
b)
7  (-4)
c)
-9  6
d)
-12 (-6)
44
Once we see a subtraction problem as an addition problem there are no more rules to
learn! We already know how to add integers in two different ways – using the number
line and using the rules. Now all we must do is practice our new skill.
Let's practice some more with some problems that include real numbers, and absolute
values.
Example:
a)
Simplify each of the following.
| -5  7|  8
b)
| 19  7/8 |  | -11 + - 2/3 |
c)
| -19.07  (-2.125) | + (-205 + -3.1)
45
Your authors also include some translation problems in this section. They are the same
as those included in section 3, and although I think that they are very important I will skip
more examples in the interest of time. Some other problems that I would rather skip in
interest of time are the application problems involving signed numbers. I would like to
remind you that negative numbers represent decreases or going below zero, but I am
going to leave the actual problems up to you. I will give you an example of a word
problem and remind you that word problems are all about translation (watch order).
Example:
In Death Valley the average winter temperature is 82
and several hundred miles away at the top of Mt.
Whitney, the average winter temperature is -23. What is
the difference in average temperatures between Death
Valley and Mt. Whitney in the winter?
46
§1.6 Multiplying and Dividing Real Numbers
Outline
Definitons
Quotient
Dividend
Divisor
Multiplicative Inverse
Reciprocal
Division
Average
Sigma Notation
Multiplication & Divison
Redefine Division – Multiplication by the reciprocal (see fractions)
Basic Truths
Multiplication by Zero – Zero
Division by Zero – Undifined
Zero divided by Anything – Zero
Multiplication & Division w/ Reals
++ = +
-  - = +
+  - = 
Combining All Concepts Thus Far
Order of Op Problems
Averages
Divisibility Tests
Only Covered in Book on p. 62
Homework p.60-62 #1-6all, #43, #46-70even, #83, #88-102even, #110, #116-119all,
#121&122
Let's first talk about some of the ways that multiplication and division can be written.
All of the following mean multiplication:
3x2
32
3(2)
(3)(2)
(3)2
In each of these the 3 and the 2 are called factors and the answer
would be called the product
Note: We will try to phase out the 1st in this list as it will become confusing when variables are involved.
All of the following mean division:
10  2
10
2
10
/2
210
In each of these the 10 is called the dividend, the 2 is called the
divisor and the answer would be called the quotient
The multiplicative inverse, also called the reciprocal, is any number which when
multiplied by the number itself yields 1. We are going to be redefining division just as
we did subtraction. Division is multiplication by the reciprocal (multiplicative inverse) of
47
the divisor (second number, outside number or denominator) . When we write a/b we are
actually saying “a  1/b”, so we see that multiplying by the inverse of a number is the
same as division by that number. Recall that division by zero is undefined, therefore
there is no reciprocal of zero!
Example:
a)
Rewrite each as a multiplication problem
56  8
b)
7/8
c)
8 24
NOW SOME IMPORTANT TRUTHS:
At this time I would like to remind you: the product of zero and anything is zero,
Multiplication by Zero – ZERO
a(0) = 0
division by zero is undefined and
Division by Zero – UNDEFINED
a = undefined
0
since no number, when multiplied by zero yields a
zero divided by anything is zero.
Zero Divided by Anything – ZERO
0
a
= 0
since a(0) = 0 [mult. prop. of zero]
48
Multiplying & Dividing Reals
Rule 1: Multiplication or Division of two positive numbers yields a positive number
(+  + = +)
Rule 2: Multiplication or Division of two negative numbers yields a positive number
(-  - = +)
Rule 3: Multiplication or Division of a negative and positive number yields a negative
number
(-  + = - or +  - = -)
Example: -5 x 3
Example: -15  -3
Example: -3  -5
Example: -21/3
Note: A negative fraction is negative whether the negative is associated with the
numerator, the denominator or just the entire fraction! -21/3 = 21/-3 = -(21/3)
Example: 3(5)
Example: 7-56
Example: 28
-7
49
Really this section is quite simple in the new rules that it brings up. There are only the
four and this business about division being multiplication really should be old hat since
we have already seen this in the discussion on fractions. Now all there is left to do is
practice all the more difficult things with the use of integer multiplication and division.
We will do some order of operation problems that involve integers, exponents,
parentheses, absolute values, and both multiplication/division and addition/subtraction. I
will leave all translation problems up to you without review, but I will stress that you
should ask for review if you are having difficulty. Try to review section 3 before trying
any translations.
Example:
a)
Simplify each of the following.
-2/5  1 7/8
b)
- 5.1/0.3
c)
(5  9)  | -8  2/3 |
50
d)
9  5  (3)(-7)  2 ½
7  9
e)
7  -8 + (-7)2 + 7
| -8  9 |
f)
| (9)(-1) + -11 |
-5 + -22 + | -9 |
51
The final thing that we will do before closing out the section is to cover averages. An
average is computed by finding the sum of elements and dividing it by the number of
elements. An average is the point that would "balance" all the elements if they were
weighted. I will define an average using sigma notation below (capital sigma is a Greek letter
that mathematicians use to denote addition of elements).
Average = x1 + x2 + x3 + x4 + … + xn
n
x  , n = # of elements
Note: The little numbers below and to the right of the variable are called subscripts and are used to
denote different elements without using a different variable name for each. By naming the last x n it
indicates that all the numbers in the set are added together, no matter how many elements there are, since
n is used to denote the number of elements.
Example:
Find the average of the following numbers.
7, 8, -9, -1, 12, -53
If I feel that we have time I may assign board problems from p.63 for each
person to do in class, to count as a homework assignment.
52
§1.7 Properties of Real Numbers
Outline
Properties of Real Numbers
Commutative Property of Addition
a + b = b + a
Allowed us to learn our addition facts without twice the work
Commutative Property of Multiplication
a  b = b  a
Allowed us to learn our multiplication facts without twice the work
Associative Property of Addition
(a + b) + c = a + (b + c)
Useful in adding columns or numbers
Associative Property of Multiplication
( a  b)  c = a  ( b  c )
Useful in multiplying multiple factors
Distributive Property
a(b + c) = ab + ac
Multiplication distributes over addition
Allows simplification of algebraic expressions(equations)
Identity Property of Addition
a + 0 = a
Allows us to solve algebraic equations (addition property of equality)
Identity Property of Multiplication
a  1 = a
Allows us to solve algebraic equations (multiplication property of equality)
Helps in building higher terms and reducing
Inverse Property of Addition
a + -a = 0
Further allows us to solve algebraic equations (addition property of equality)
Inverse Property of Multiplication
a  1/a = 1
Further allows us to solve algebraic equations (multiplication property of equality)
Homework p. 70-73 #11-30all, #36-50even, #51-54all, #64-78even, #87 & 88
Commutative Properties
This special property states that no matter in which order you add or multiply two
numbers the sum or product is still the same.
a + b = b + a
a  b = b  a
Example:
Example:
Example:
Example:
Example:
Commutative Property of Addition
Commutative Property of Multiplication
0 + 8 = 8 + 0
15 + 7 = 7 + 15
x + 7 = 7 + x
7(8) = 8(7)
8x = x8
53
Associative Properties
This property tells us that we can group numbers together in any way and add or multiply
them and still get the same answer. You learned this property when you learned to add
columns of numbers and found that it was easier to group numbers together and then add
the groups' sums. Or when you learned that the multiplication table was symmetric.
( a  b)  c = a  ( b  c )
(a + b) + c = a + (b + c)
Associative Prop. of Mult.
Associative Prop. of Add.
Example: 5 + 4 + 7 + 3 = ( 5 + 4 ) + ( 7 + 3 )
Example: 6 + ( 1 + 8 ) = ( 6 + 1 ) + 8
Example: (x + 7) + 3 = x + (7 + 3)
One very nice thing about the associative property of addition is that we can use it to add
terms that are alike, like terms. A term is any number, variable(s) or a variable(s)
multiplied by a number (referred to as a numeric coefficient). Like terms are terms that have
the exact same variable or variables. We can also use the associative property of
multiplication to group numeric coefficients together to create a single product. Note: You
might recall that when a number is written next to a variable it indicates multiplication.
Example:
a)
Simplify the following by using the associative property
to add like terms
8x + 5  2 + x
b)
8  (7n)
c)
2  3  6n
Distributive Property
Unlike addition, multiplication has another property called the distributive property. The
distributive property only works with multiplication and goes as follows, as it distributes
multiplication over addition:
a(b+c)=a(b) + a(c)
54
Example:
a)
Simplify each of the following using the distributive
property
4(3+2)
Note: Do not become confused by this simple example. It is meant to give you a warm
fuzzy about using the distributive property. The distributive property should not replace
order of operations if you have all numbers!
b)
x(3+5)
c)
2 ( 2x + 3 )
d)
5 ( x + y + 5z )
Identity Elements (Properties)
The identity element is the thing that gives the number itself back. They should not be
confused with the Inverse properties, which yield the Identity Elements.
a  1 = a
a + 0 = a
Multiplication's Identity Element is 1
Addition's Identity Element is 0
These properties are used in fractions most often. They are the properties that allow us to
reduce and build higher terms. For more information please see your book's explanation
on page 66.
55
Inverse Properties
The inverse properties are very useful properties that allow us to solve equations. Neither
you book nor I will cover this feature directly at this time, but when we cover solving
equations if you return to page 67 and review the examples that your book gives you will
see the similarities.
a + (-a) = 0
a  1/a = 1
Inverse Property of Addition
Inverse Property of Multiplication
The basic properties discussed in this section rarely if ever come up by themselves,
although you can expect to see one problem of the following type on any first test in my
classes. (Like problems 9 & 11-30 on p. 70-71 in your book)The importance of these properties is
how they help us to solve algebra problems.
Example:
_____
_____
_____
_____
_____
_____
_____
_____
Match each example with the property that it best
represents by writing the corresponding letter next to it.
5(t + x) = 5t + 5x
-6 + 6 = 0
5 + 8 = 8 + 5
(2  3)  5 = 2  (3  5)
1
/7  7 = 1
1
- /5  3/3 = -3/15
(-9 + 5) + 1 = -9 + (5 + 1)
3  (4  2) = (4  2)  3
a.
b.
c.
d.
e.
f.
g.
h.
Identity Element of Mult.
Associative Property of Mult.
Commutative Prop. of Mult.
Additive Inverse
Multiplicative Inverse
Distributive Property
Commutative Prop. of Addition
Associative Prop. of Addition
56
§1.8 Simplifying Expressions
Outline
Definition Review
Term
Like Term
Numeric Coefficient
New Definition
Simplifying (Combining Like Terms)
Combining Like Terms
Basics
Translation
Homework p. 76-78 #27-66mult.of3, #67-72all, #81-84all
Let’s take a few moments to review some important vocabulary that we will need for this
section.
Term – Number, variable, product of a number and a variable or a variable raised to a
power.
Example:
a)
b)
c)
d)
5
5x
xy
x2
Numeric Coefficient – The numeric portion of a term with a variable.
Example:
a)
What is the numeric coefficient?
3x2
b)
x
c)
- 5x/2
d)
–z
/2
57
Like Term – Terms that have a variable, or combination of variables, that are raised to the
exact same power.
Example:
a)
Are the following like terms?
7x
10x2
b)
- 15z
23z
c)
t
15t3
d)
5
5w
e)
xy
6xy
f)
x2y
- 2x2y2
Simplifying an algebraic expression by combining like terms means adding/subtracting
terms in an algebraic expression that are alike. Remember that terms in an algebraic
expression are separated by an addition sign (recall also that subtraction is addition of the
opposite, so once you change all subtraction to addition, you may simplify) and if multiplication is
involved the distributive property must first be applied.
Simplifying Algebraic Expressions
Step 1: Change all subtraction to addition
Step 2: Use the distributive property wherever necessary
Step 3: Group like terms
Step 4: Add numeric coefficients of like terms
Step 5: Don’t forget to separate each term in your simplified expression
by an addition symbol!!
Example:
a)
b)
Simplify
3x  12x
5y + 2 + 7  2y
58
c)
9  3(2y + 2)
Just for some much needed review and for some more experience in simplifying let's do
some translation.
Example:
Translate each of the following using x for the variable
and then simplify.
a)
The product of 5 and a number, added to the difference
of 7 and twice the number.
b)
The sum of a number and 14, subtracted from nine times
the number.
c)
Triple the difference of a number and 5, subtracted from
twice the number.
59
Chapter 1 Practice Test
Instructions: Put your name on the top before beginning. You will have approximately
one hour to complete this exam. You may not use a calculator. Please attach you note
card to the back of the test when finished. If you can’t complete a problem in the space
allotted, please direct me to the place where you have completed the work and label it
clearly. A problem without work will not receive full credit under any circumstances.
Word problems must show the appropriate set up, equation, substitution and work to
receive full credit. Show all work and box your final answer. Good luck!
1.
Simplify each of the following using your knowledge of fractions.
1
a)
/5  2/5
b)
1 2/3  3/8
c)
2
/5 + 4/5
d)
2
/27  1/18
e)
12 1/5 + 1 1/3
f)
12 ¼  5 ¾
60
2.
g)
0.115 + 1.5
h)
(2/5)2
Simplify the following using your knowledge of decimals.
a)
25  0.529
b)
75  0.03
c)
1.25  5
d)
(1.2)(0.05)
e)
(0.02)3
61
3.
Simplify using your knowledge of integers. Change all subtraction to
addition before simplifying.
a)
9 + -7
b)
-8 + -9
c)
-9 + 6
d)
-9  8
e)
-6  (-9)
f)
7  9
g)
-9  5
62
h)
-54  -9
i)
-22
j)
(-2)2
Note: There is a difference in the answers between i & j and you should be able to
explain this difference. If it is not clear ask in class!
4.
Simplify by using order of operations. Show all steps.
a)
9 + 15  3  8
b)
c)
d)
8 + -8  8
(8 + -8)  8
2(5  3) + 25  5 + 20  8  4
8  1/8 + -1
5  15 + 1/3
63
5.
6.
Translate using x as the variable, simplify and then evaluate
when x = 2.
a)
The sum of seven and twice the number, subtracted from the
difference of the number and five.
b)
The quotient of a number and 9, less a third of the number.
Compare each of the following using <, > or =. Show work in
simplifying the numbers that you are comparing.
a)
-(-5)
| -5 |
b)
- | -5 |
-(-5)
64
c)
0
/5
-|5|
2x  9 = 11, is x = 9 a solution. Explain.
7.
For the equation
8.
Using integers and only addition, write down all important
information in shorthand, write an expression and solve the following
word problem.
John had $51 in his account. He wrote a check for $27, then
another one for $28, finally he made a deposit of $30. What
was John's final balance.
9.
Match each example with the property that it best represents by
writing the corresponding letter next to it.
_____ 2(84) = (28)4
a) Inverse Property of Mult.
_____ 0 + 4 = 4
b) Commutative Prop. of Add.
_____ 1/4  4 = 1
c) Associative Prop. of Mult.
65
10.
_____ 2 + (5 + 3) = (5 + 3) + 2
d) Identity Element of Add.
_____ 4z + 4v = 4(z + v)
e) Distributive Property
Combine like terms to simplify.
a)
-5c + -5  3c
b)
-(a + b)  (5a  5b)
c)
5ax + 9  3(5ax + 5a)
d)
7x2 + 9x  5x2 + 9
66