Transcript Terminology of Algebra

Elementary Algebra
Exam 1 Material
Familiar Sets of Numbers
• Natural numbers
– Numbers used in counting:
1, 2, 3, … (Does not include zero)
• Whole numbers
– Includes zero and all natural numbers:
0, 1, 2, 3, … (Does not include negative numbers)
• Fractions
– Ratios of whole numbers where bottom number can
not be zero:
Top number is called " numerator"
2 1 7
, , , etc
Bottom number is called " denominato r"
3 5 4
Prime Numbers
• Natural Numbers, not including 1, whose
only factors are themselves and 1
2, 3, 5, 7, 11, 13, 17, 19, 23, etc.
• What is the next biggest prime number?
29
Composite Numbers
• Natural Numbers, bigger than 1, that are
not prime
4, 6, 8, 9, 10, 12, 14, 15, 16, etc.
• Composite numbers can always be
“factored” as a product (multiplication) of
prime numbers
Factoring Numbers
• To factor a number is to write it as a product
of two or more other numbers, each of which
is called a factor
12 = (3)(4)
3 & 4 are factors
12 = (6)(2)
6 & 2 are factors
12 = (12)(1)
12 and 1 are factors
12 = (2)(2)(3)
2, 2, and 3 are factors
In the last case we say the 12 is “completely
factored” because all the factors are prime
numbers
Hints for Factoring Numbers
• To factor a number we can get two factors by writing any
multiplication problem that comes to mind that is equal to
the given number
• Any factor that is not prime can then be written as a
product of two other factors
• This process continues until all factors are prime
• Completely factor 28
28 = (4)(7)
28 = (2)(2)(7)
4 & 7 are factors, but 4 is not prime
4 is written as (2)(2), both prime
In the last case we say the 28 is “completely factored”
because all the factors are prime numbers
Other Hints for Factoring
• Some people prefer to begin factoring by thinking of the smallest
prime number that evenly divides the given number
• If the second factor is not prime, they again think of the smallest
prime number that evenly divides it
• This process continues until all factors are prime
• Completely factor 120
120 = (2)(60) 60 is not prime, and is divisible by 2
120 = (2)(2)(30)
30 is not prime, and is divisible by 2
120 = (2)(2)(2)(15) 30 is not prime, and is divisible by 3
120 = (2)(2)(2)(3)(5) all factors are prime
In the last case we say the 120 is “completely factored” because
all the factors are prime numbers
Fundamental Principle of
Fractions
• If the numerator and denominator of a
fraction contain a common factor, that
factor may be divided out to reduce the
fraction to lowest terms:
12
• Reduce to lowest terms by factoring:
18
1
1
12 2  2  3 2


18 21  3  31 3
When common factors are divided out, "1" is left in each place.
Summarizing the Process of
Reducing Fractions
• Completely factor both numerator and
denominator
• Apply the fundamental principle of
fractions: divide out common factors
that are found in both the numerator and
the denominator
When to Reduce Fractions to
Lowest Terms
• Unless there is a specific reason not to
reduce, fractions should always be
reduced to lowest terms
• A little later we will see that, when adding
or subtracting fractions, it may be more
important to have fractions with a common
denominator than to have fractions in
lowest terms
Multiplying Fractions
•
•
•
•
Factor each numerator and denominator
Divide out common factors
Write answer
Example:
1 1
1
5
2  2 35
4 15




21
33 2  2  7
9 28
1 1 1
Dividing Fractions
• Invert the divisor and change problem to
multiplication
• Example:
2 3
 
3 4
2 4
 
3 3
8
9
Adding Fractions Having a
Common Denominator
• Add the numerators and keep the common
denominator
• Example:
5
2 3
 
7
7 7
Adding Fractions Having a
Different Denominators
• Write equivalent fractions having a “least
common denominator”
• Add the numerators and keep the common
denominator
• Reduce the answer to lowest terms
Finding the Least Common
Denominator, LCD, of Fractions
• Completely factor each denominator
• Construct the LCD by writing down each
factor the maximum number of times it is
found in any denominator
Example of Finding the LCD
• Given two denominators, find the LCD:
18
,
24
• Factor each denominator:
18  2  3  3
24  2  2  2  3
• Construct LCD by writing each factor the maximum
number of times it’s found in any denominator:
What is the maximum number of times 2 is a factor? 3
What is the maximum number of times 3 is a factor? 2
LCD 
2  2  2  3  3  72
Writing Equivalent Fractions
• Given a fraction, an equivalent fraction is found by
multiplying the numerator and denominator by a
common factor
• Given the following fraction, write an equivalent fraction
having a denominator of 72:
5
18
How many times does 18 go into 72? 4
• Multiply numerator and denominator by 4:
5
5 4 20
  
18 18 4 72
Adding Fractions
• Find a least common denominator, LCD,
for the fractions
• Write each fraction as an equivalent
fraction having the LCD
• Write the answer by adding numerators
as indicated, and keeping the LCD
• If possible, reduce the answer to lowest
terms
Example
5
7

18 24
•
Find a least common denominator, LCD, for the rational expressions:
We have already found that the LCD is 72
•
Write each fraction as an equivalent fraction having the LCD:
5 7
5 4 7 3
20 21

    


18 24 18 4 24 3
72 72
•
Write the answer by adding or subtracting numerators as indicated, and keeping the
LCD:
20 21


72 72
•
41
72
If possible, reduce the answer to lowest terms
41
72
Won' t reduce because numerator and denominato r have no common factor
Subtracting Fractions
• Find a least common denominator, LCD,
for the fractions
• Write each fraction as an equivalent
fraction having the LCD
• Write the answer by subtracting
numerators as indicated, and keeping
the LCD
• If possible, reduce the answer to lowest
terms
Example
7 5

10 12
•
Find a least common denominator, LCD, for the rational expressions:
10  2  5
12  2  2  3
•
LCD  2  2  3  5  60
Write each fraction as an equivalent fraction having the LCD:
7 5
7 6 5 5
42 25
     


10 12 10 6 12 5
60 60
•
Write the answer by adding or subtracting numerators as indicated, and keeping the
LCD:
42 25


60 60
•
17
60
If possible, reduce the answer to lowest terms
17
60
Won' t reduce because numerator and denominato r have no common factor
Improper Fractions
& Mixed Numbers
• A fraction is called “improper” if the numerator is bigger
than the denominator
7
3
• There is nothing wrong with leaving an improper fraction
as an answer, but they can be changed to mixed
numbers by doing the indicated division to get a whole
number plus a fraction remainder
7
1
 2
3
3
• Likewise, mixed numbers can be changed to improper
fractions by multiplying denominator times whole
number, plus the numerator, all over the denominator
3
5  4  3 23
4 

5
5
5
Doing Math Involving Improper
Fractions & Mixed Numbers
• Convert all numbers to improper fractions
then proceed as previously discussed
3 7
23 7
23 3 7 5
34
69 35
4  
 
   


5 3
5 3
5 3 3 5
15
15 15
34
4
2
15
15
Either answer is okay
Homework Problems
• Section: 1.1
• Page: 11
• Problems: Odd: 7 – 29, 33 – 51, 55 – 69
• MyMathLab Homework 1.1 for practice
• MyMathLab Homework Quiz 1.1 is due for
a grade on the date of our next class
meeting
Exponential Expressions
34
“3” is called the base
“4” is called the exponent
• An exponent that is a natural number tells how many
times to multiply the base by itself
Example: What is the value of 34 ?
(3)(3)(3)(3) = 81
5  5  5  5  125
3
• An exponent applies only to the base (what it touches)
3  2  3  2  2  2  2  48
4
• Meanings of exponents that are not natural numbers will
be discussed later
Order of Operations
• Many math problems involve more than one
math operation
• Operations must be performed in the following
order:
–
–
–
–
Parentheses (and other grouping symbols)
Exponents
Multiplication and Division (left to right)
Addition and Subtraction (left to right)
• It might help to memorize:
– Please Excuse My Dear Aunt Sally
Order of Operations
•
•
•
•
•
Example:
P
E
MD
AS
5  24   3  8  2  3
2
5  24  3  6  3
2
5  24  9  6  3
5  8  9  2
6
Example of Order of Operations
(A fraction bar is a grouping symbol; top and bottom should be simplified separately )
• Evaluate the following expression:
7  315  3  4  6  23
2
38  5
7 9 68
27
7  315  12  6  23
2
33
16  6  8
27
7  33  6  23
2
33
22  8
27
7  33  6  8
39
14
27
Inequality Symbols
• An inequality symbol is used to compare numbers:
• Symbols include:
greater than:

greater than or equal to: 
less than:

less than or equal to:

not equal to:

• Examples:
59
3 9  5
7 3  4
1 3  2
.
Expressions Involving
Inequality Symbols
• Expressions involving inequality symbols may be
either true or false
• Determine whether each of the following is true
or false:
3 2  5
92  6
False
True
False
7 3  4
11 3
  4 When comparing fractions, convert each to an equivalent fraction w ith the LCD
2 5
11 5 3 2
55 6
49
9
   4
 4
4
4 4
False
2 5 5 2
10 10
10
10
Translating to Expressions
Involving Inequality Symbols
• English expressions may sometimes be
translated to math expressions involving
inequality symbols:
Seven plus three is less than or equal to twelve
7  3  12
Nine is greater than eleven minus four
9  11  4
Three is not equal to eight minus six
3  86
Equivalent Expressions
Involving Inequality Symbols
• A true expression involving a “greater than” symbol can be
converted to an equivalent statement involving a “less then” symbol
– Reverse the expressions and reverse the direction of the inequality
symbol
5 > 2 is equivalent to:
2<5
• Likewise, a true expression involving a “less than symbol can be
converted to an equivalent statement involving a “greater than”
symbol by the same process
– Reverse the expressions and reverse the direction of the inequality
symbol
3 < 7 is equivalent to:
7>3
Homework Problems
• Section: 1.2
• Page: 21
• Problems: Odd: 5 – 19, 23 – 49, 53 – 79,
83 – 85
• MyMathLab Homework 1.2 for practice
• MyMathLab Homework Quiz 1.2 is due for
a grade on the date of our next class
meeting
Terminology of Algebra
• Constant – A specific number
Examples of constants: 3 6
4
5
• Variable – A letter or other symbol used to
represent a number whose value varies or
is unknown
n
Examples of variables: x
A
Terminology of Algebra
• Expression – constants and/or variables
combined in a meaningful way with one or more
math operation symbols for addition, subtraction,
multiplication, division, exponents and roots
Examples of expressions:
23
5 x
10
4
n
y  9 w
2
• Only the first of these expressions can be
simplified, because we don’t know the numbers
represented by the variables
Terminology of Algebra
• If we know the number value of each variable in
an expression, we can “evaluate” the expression
• Given the value of each variable in an
expression, “evaluate the expression” means:
– Replace each variable with empty parentheses
– Put the given number inside the pair of parentheses
that has replaced the variable
– Do the math problem and simplify the answer
Example
• Evaluate the expression for
n
n  3:
4

4
3
3333
4
81
• Consider the next similar, but slightly different,
example
Example
• Evaluate the expression for
n  3:
n
4
 
4
 3
 3333
 81
4
• Notice the difference between this example and the
previous one – it illustrates the importance of using a
parenthesis in place of the variable
Example
• Evaluate the expression for x  2 :
13  x
13   
13  2
11
Example
• Evaluate the expression for x  3, y  4 :
12  2 y  x
12  2   
2

2
12  24   3
12  8  9
13
2
Translating English Phrases Into
Algebraic Expressions
• Many English phrases can be translated
into algebraic expressions:
– Use a variable to indicate an unspecified
number
– Identify key words that imply:
•
•
•
•
Add
Subtract
Multiply
Divide
Phrases that Translate to Addition
English Phrase
• A number plus 5
• The sum of 3 and a
number
• 4 more than a number
• A number increased
by 8
Algebra Expression
x5
3 x
x4
x 8
Phrases that Translate to
Subtraction
English Phrase
• 4 less than a number
• A number subtracted
from 7
• 6 subtracted from a
number
• a number decreased
by 9
• 2 minus a number
Algebra Expression
x4
7 x
x6
x 9
2 x
Phrases that Translate to
Multiplication
English Phrase
• 7 times a number
• the product of 4 and a
number
• double a number
• the square of a
number
Algebra Expression
7x
4x
2x
x 2 or x  x
Phrases that Translate to Division
English Phrase
• the quotient of 2 and
a number
• a number divided by 8
• 6 divided by a number
Algebra Expression
2
x
x
8
6
x
Phrases Translating to Expressions
Involving Multiple Math Operations
English Phrase
Algebra Expression
• 4 less than 3 times a
number
3x  4
• the quotient of 5 and
twice a number
5
2x
• 6 times the difference
between a number and 5
6x  5
Phrases Translating to Expressions
Involving Multiple Math Operations
English Phrase
• the difference between 4
and 7 times a number
• the quotient of a number
and 5, subtracted from
the number
• the product of 3, and a
number increased by 4
Algebra Expression
4  7x
x
x
5
3x  4
Equations
• Equation – a statement that two expressions
are equal
– Equations always contain an equal sign, but an
expression does not have an equal sign
• Like a statement in English, an equation may be
true or false
• Examples:
5  9  14
47  5
T or F?
T or F?
True
False
.
Equations
• Most equations contain one or more
variables and the truthfulness of the
equation depends on the numbers that
replace the variables
• Example: x  4  9
• What value of x makes this true? x  5
• A number that can replace a variable to
make an equation true is called a solution
5 is a solution t o the equation
Distinguishing Between
Expressions & Equations
• Expressions contain constants, variables
and math operations, but NO EQUAL
SIGN
x  49
• Equations always CONTAIN AN EQUAL
SIGN that indicates that two expressions
have the same value
x49
Solutions to Equations
• Earlier we said that any numbers that can
replace variables in an equation to make a true
statement are called solutions to the equation
• Soon we will learn procedures for finding
solutions to an equation
• For now, if we have a set of possible solutions,
we can find solutions by replacing the variables
with possible solutions to see if doing so makes
a true statement
Finding Solutions to Equations
from a Given Set of Numbers
• From the following set of numbers, 3, 4, 5
find a solution for the equation:
2x  3  11
• Check x = 3
23  3  11?
9  11
• Check x = 4
24  3  11?
11  11
• Check x = 5
25  3  11?
13  11
From the given set of numbers, 4 is the only solution
Writing Equations
from Word Statements
• The same procedure is used as in
translating English expressions to
algebraic expressions, except that any
statement of equality in the English
statement is replaced by an equal sign
• Change the following English statement to
an equation, then find a solution from the
set of numbers 3, 4, 5
• Four more than twice a number is ten
2x  4  10 The solution is : 3
Homework Problems
• Section: 1.3
• Page: 29
• Problems: Odd: 13 – 55, 59 – 81
• MyMathLab Homework 1.3 for practice
• MyMathLab Homework Quiz 1.3 is due for
a grade on the date of our next class
meeting
Sets of Numbers
• Natural numbers
– Numbers used in counting:
1, 2, 3, … (Does not include zero)
• Whole numbers
– Includes zero and all natural numbers:
0, 1, 2, 3, … (Does not include negative numbers)
• Integers
– Includes all whole numbers and their opposites
(negatives):
…, -3, -2, -1, 0, 1, 2, 3, …
Number Line
• Draw a line, choose a point on the line, and label it as 0
• Choose some unit of length and place a series of points,
spaced by that length, left and right of the 0 point
• Points to the right of zero are labeled in order 1, 2, 3, …
• Points to the left of zero starting at the point closest to
zero and moving left are labeled in order, -1, -2, -3, …
5
0
5
• Notice that for any integer on the number line, there is
another integer the same distance on the other side of
zero that is the opposite of the first
• A number line is used for graphing integers and other
numbers
Graphing Integers
on a Number Line
• To graph an integer on a number line we
place a dot at the point that corresponds to
the given number and we label the point
with the number
• The number label is called the “coordinate”
of the point
• Graph -2:
2
5
0
5
Rational Numbers
• The next set of numbers to be considered will fill in some
of the gaps between the integers on a number line
• Rational numbers
– Numbers that can be written as the ratio of two integers
– This includes all integers since they can be written as
themselves over 1
– This includes all fractions and their opposites (- ½ , ½, etc.)
– It also includes all decimals that either terminate ( .57 ) or have a
a sequence of digits that form an infinitely repeating pattern at
the end (.666…, written as .6, etc.)
Graphing Rational Numbers
•
•
•
Positive rational numbers will correspond to a point right of zero and
negative rational numbers will correspond to a point left of zero
To find the location of the point, consider the mixed number equivalent of
the given number
If the number is positive:
– go to the right to the whole number
– divide the next interval into the number of divisions indicated by the denominator
of the fraction
– continue to the right from the whole number to the division indicated by the
numerator
– Place a dot at that point and label it with the coordinate
•
If the number is negative:
– go to the left to the whole number
– divide the next interval into the number of divisions indicated by the denominator
of the fraction
– continue to the left from the whole number to the division indicated by the
numerator
– Place a dot at that point and label it with the coordinate
Examples of Graphing
Rational Numbers
• Graph
3
5
3
0
5
3
5
1
7
• Graph 
3
0
1
2
3

7
3
2 0
1
Irrational Numbers
• It may seem that rational numbers would fill up all the
gaps between integers on a number line, but they don’t
• The next set of numbers to be considered will fill in the
rest of the gaps between the integers and rational
numbers on a number line
• Irrational numbers
– Numbers that can not be written as the ratio of two integers
– This includes all decimals that do not terminate and do not have
a sequence of digits that form an infinitely repeating pattern at
the end
– Included in this set of numbers are any square roots of positive
numbers that will not simplify to get rid of square root sign
– Examples:  , 5 ,  3
Notes on Square Roots
• The square root of 4 is written as 4 and represents a
number that multiplies by itself to give 4
• We know that the number that multiplies by itself to give 4
is 2 , so we write 4  2
•
4 is a terminating decimal, so 4 is a rational number
• The square root of 5 is written as 5 and represents a
number that multiplies by itself to give 5
• We know of no number that multiplies by itself to give 5 ,
but a calculator gives a decimal approximation that fills
the screen without showing a repeating pattern at the
end. 5 is an irrational number
• Square roots may be rational, irrational, or neither
More Notes on Square Roots
• The square root of  9 is written as  9 ,
but it does not exist in the real number
system (no real number multiplies by itself
to give a negative
•  9 is not rational or irrational. It’s not
real, but is a type of number called an
imaginary number, that will be studied in
college algebra
Graphing Irrational Numbers
•
•
•
Positive irrational numbers will correspond to a point right of zero and
negative irrational numbers will correspond to a point left of zero
To find the approximate location of the point, consider the decimal
approximation
If the number is positive:
– go to the right to the whole number
– divide the next interval into the number of divisions of accuracy desired (tenths,
hundredths, etc.)
– continue to the right from the whole number to the division indicated by the digits
right of the decimal point
– Place a dot at that point and label it with the coordinate
•
If the number is negative:
– go to the left to the whole number
– divide the next interval into the number of divisions of accuracy desired (tenths,
hundredths, etc.)
– continue to the left from the whole number to the division indicated by the digits
right of the decimal point
– Place a dot at that point and label it with the coordinate
Example of Graphing
Irrational Numbers
• Graph
 3
 1.7320508
 3
2
1
0
 1.7
Real Numbers
• The set of rational numbers and the set of irrational
numbers have no numbers in common
• When the two sets of numbers are put together they
make up a new set of numbers called “real numbers”
• Every real number is either rational or irrational
• There is a one-to-one correspondence between points
on a number line and the set of real numbers
• There are some numbers that are not real numbers, an
example is:  7 . These type of numbers (complex
numbers) will be discussed in college algebra.
Ordering Real Numbers
• Given two real numbers, represented by the
variables a and b, one of the following order
relationships is true:
a=b
a equals b if they graph at the same location
a<b
a is less than b, if a is left of b on a number line
a>b
a is greater than b, if a is right of b on a number
line
Why is - 7  - 2?
- 7 is left of - 2
Additive Inverses
of Real Numbers
• Every real number has an additive inverse
• The additive inverse of a real number is the number
located on a number line the same distance from zero,
but in the opposite direction
• The additive inverse of a number is the same as its
opposite
The additive inverse of 5 is: - 5
The additive inverse of -3 is: 3
• Placing in negative sign in front of a number is a way of
indicating the additive inverse of the number
• If we want to indicate the additive inverse of -7, we can
place a negative sign in front of -7:
- (-7) is the same as: 7
Absolute Value
of Real Numbers
• Every real number has an absolute value
• The absolute value of a real number is its
“distance” from zero
• Distance is never negative, so absolute value is
never negative
• Absolute value of a number is indicated by
placing vertical bars around the number
The absolute value of 5 is shown by : 5 and
is equal to: 5
The absolute value of -3 is shown by:  3 and
is equal to: 3
8  8
7 7
0 0
Homework Problems
• Section: 1.4
• Page: 39
• Problems: All: 9 – 20
Odd: 23 – 27, 35 – 63
• MyMathLab Homework 1.4 for practice
• MyMathLab Homework Quiz 1.4 is due for
a grade on the date of our next class
meeting
Addition of Real Numbers
• Addition – like a game between two teams,
“Positive” and “Negative,” the answer to the
problem is the answer to the question, “Who
won the game, and by how much?”
• Example:  26  18
• Reasoning:
– Negatives scored: 26
– Positives scored: 18
Negatives won by ____,
• _________
8 so  26 18   8
Second Example of Addition
• Example:  5  3  8  (9)  (2)  7
• Reasoning:
– Negatives scored: 5  9  2  16
– Positives scored:
3  8  7  18
Positives won by ____,
2 so:
• _________
 5  3  8  (9)  (2)  7  2
Addition of Signed Fractions
• Addition rule is the same for all signed numbers,
but you must first write each fraction as an
equivalent fraction where all fractions have a
common denominator
• Example:  3  5   9  10
4
6
12 12
• Reasoning:
– Negatives scored:
– Positives scored:
9 twelfths
10 twelfths
3 5
1
Positives won by ________,
• _________
1 twelfth so:   
4
6
12
Addition of Signed Decimals
• Addition rule is the same for all signed
numbers, but be sure to line up decimal
points before adding or subtracting
Negatives won by how much?
• Example:  5.3  2.18
5 .3
• Reasoning:
 2.18
– Negatives scored: 5.3
3.12
2.18
– Positives scored:
Negatives won by ____,
3.12 so:
• _________
 5.3  2.18   3.12
Subtraction of Real Numbers
• Subtract means “add the opposite”
• All subtractions are changed to “add the
opposite” and then the problem is done
according to addition rules already discussed
• In identifying a subtraction problem remember
that the same symbol, - , is used between
numbers to mean “subtract” and in front of a
number to mean “negative number”
64
6 subtract positive four means 6 add negative 4
5   3
5 subtract negative 3 means 5 add positive 3
 7   2
6   4  2
5   3. 8
negative 7 subtract negative 2 means negative 7 add positive 2
 7   2   5
Problems Involving Both
Addition and Subtraction
 3   5  6  10  4   7
• Example:
• Identify subtraction:  3   5  6  10  4   7
• Add opposite:  3   5   6  10   4   7
• Reasoning:
– Negatives scored: 3  5  6  4  18
– Positives scored:
10  7  17
Negatives won by ____,
1 so:
• _________
 3   5  6  10  4   7  1
Homework Problems
• Section: 1.5
• Page: 49
• Problems: Odd: 7 – 97
• MyMathLab Homework 1.5 for practice
• MyMathLab Homework Quiz 1.5 is due for
a grade on the date of our next class
meeting
Multiplying and Dividing
Real Numbers
• Multiplication and Division of signed
numbers follows the rule:
– Do problem as if both were positive
– Answer is positive if signs were the same
– Answer is negative if signs were opposite
• Examples:
 65   30
 2 7  14
12
 4
3
3
6

4
8
.
Multiplying Signed Fractions
• Basic rule has already been discussed
• Otherwise, remember to:
– Divide out factors common to top & bottom
– Multiply top factors to get top
– Multiply bottom factors to get bottom
• Example: 1
3
2
5
3
 5  18   5  18 
          
10
 12  25   12  25 
Dividing Signed Fractions
• Basic rule has already been discussed
• Otherwise, remember to:
– Invert the second fraction and change
problem to multiplication
– Complete using rules for multiplication
• Example:
2
 2   5   2  6 
4
           
 3   6   3  5 
5
1
Division Involving Zero
• People are often confused when division
involves zero – the rule must be memorized!
– Division by zero is always undefined
 12
is undefined
0
– Otherwise, division into zero is always zero
0
0
 12
• Explanation comes from checking answer:
 12
 3
4
4 3  12
0
0
 12
120  0
 12
 ? Undefined
0
.
0?  12 Impossible !!
Order of Operations
• Many math problems involve more than one
math operation
• Operations must be performed in the following
order:
–
–
–
–
Parentheses (and other grouping symbols)
Exponents
Multiplication and Division (left to right)
Addition and Subtraction (left to right)
• It might help to memorize:
– Please Excuse My Dear Aunt Sally
Homework Problems
• Section: 1.6
• Page: 63
• Problems: Odd: 11 – 73, 77 – 113
• MyMathLab Homework 1.6 for practice
• MyMathLab Homework Quiz 1.6 is due for
a grade on the date of our next class
meeting
Averaging Real Numbers
• To average a set of real numbers we add all the
numbers and then divide by the number of
numbers in the set
• Find the average of the following set of numbers:
 3, 8, 5,  7, 2
 3  8  5   7   2
5
Average 

 .1
5
5
Divisibility
• A real number is divisible by another if the
division has no remainder
• On the following slides are tests for
divisibility by all the numbers between 2
and 9, except for 7 (there is no test for
divisibility by 7)
• Memorize these tests
Test for Divisibility by 2
• A real number is divisible by 2 only if its
last digit is even
• Which of the following numbers are
divisible by 2?
31,976,104 Yes
No
257
Yes
1,348
Yes
35,750
Test for Divisibility by 3
• A real number is divisible by 3 only if the
sum of its digits is divisible by 3
• Which of the following numbers are
divisible by 3?
51,976,104 Sum of digits : 33 Yes
Sum of digits : 15
357
Yes
Sum of digits : 16
1,348
No
Sum of digits : 21
45,750
Yes
Test for Divisibility by 4
• A real number is divisible by 4 only if the
last two digits form a number that is
divisible by 4
• Which of the following numbers are
divisible by 4?
51,976,104 Last two digits : 4
Yes
Last two digits : 57
357
No
Last two digits : 48
1,348
Yes
Last two digits : 50
45,750
No
Test for Divisibility by 5
• A real number is divisible by 5 only if the
last digit is 5 or 0
• Which of the following numbers are
divisible by 5?
51,976,104
Last digit :
No
4
Last digit :
357
No
7
Last digit :
1,348
8
No
0
Last digit :
Yes
45,750
Test for Divisibility by 6
• A real number is divisible by 6 only if it
passes both the test for divisibility by 2
and divisibility by 3
• Which of the following numbers are
divisible by 6?
51,976,104 Even Sum of digits : 33 Yes
No
Odd
357
No
Even Sum of digits : 16
1,348
Yes
Even Sum of digits : 21
45,750
Test for Divisibility by 8
• A real number is divisible by 8 only if its
last three digits form a number divisible by
8
• Which of the following numbers are
divisible by 8?
51,976,104 Last three digits : 104 Yes
Last three digits : 357 No
357
Last three digits : 348 No
1,348
Last three digits : 750 No
45,750
Test for Divisibility by 9
• A real number is divisible by 9 only if the
sum of its digits is divisible by 9
• Which of the following numbers are
divisible by 9?
51,976,104 Sum of digits : 33
No
Sum of digits : 15
357
No
Sum of digits : 16
1,348
No
No
Sum of digits : 21
45,750
Homework Problems
• Section: 1.6
• Page: 63
• Problems: All: 115 – 119, 121 – 127
• MyMathLab Homework 1.6a for practice
• MyMathLab Homework Quiz 1.6a is due
for a grade on the date of our next class
meeting
Properties of Real Numbers
• Commutative Property – the order in
which real numbers are added or
multiplied does not effect the result:
a  b  b  a and
ab  ba
• Associative Property – the way real
numbers are grouped during addition or
multiplication does not effect the result:
a  b  c  a  b  c
a bc
and
abc  abc
abc
Properties of Real Numbers
• Commutative Property Examples:
3 x  x  3
x  2  2 x
• Associative Property Examples:
2  x  3  2  x  3
2xy  2xy
Properties of Real Numbers
• Identity Property for Addition – when
zero is added to a number, the result is still
the number:
a  0  a and
0a  a
• Identity Property for Multiplication –
when one is multiplied by a number, the
result is still the number:
a 1  a and 1 a  a
Properties of Real Numbers
• Identity Property for Addition Example:
3 0   3
0  3x  3 x
• Identity Property for Multiplication
Examples:
2
2
1 
3
3
3 7
7
 
3 5
5
Properties of Real Numbers
• Inverse Property for Addition – when the
opposite (negative) of a number is added
to the number, the result is zero:
a   a   0 and
aa  0
• Inverse Property for Multiplication –
when the reciprocal of a number is
multiplied by the number, the result is one
1
a   1 and
a
1
a 1
a
Reciprocals of Real Numbers
• Zero has no reciprocal
1
is undefined
0
• Reciprocals of other integers are
formed by putting 1 over the number
1
The reciprocal of - 3 is : 
3
• Reciprocals of fraction are formed by
switching the numerator and
denominator
3
The reciprocal of is :
5
5
3
Properties of Real Numbers
• Inverse Property for Addition
Examples:
 2x  2x  0
2  2
    0
3  3
• Inverse Property for Multiplication
Examples:
1
5  1
5
3 4
  1
4 3
Properties of Real Numbers
• Distributive Property – multiplication can
be distributed over addition or subtraction
without changing the result
ab  c  ab  ac and
ab  c  ab  ac
Illustration of Distributive Property
32  5  3   3 
32  5  32  35
37  6  15
21  21
4x  3  4   4 
4x  3  4x  4 3
4x  3  4 x 12
Illustration of Distributive Property
• Distributive Property works both directions:
ab  ac  ab  c and
ab  ac  ab  c
• If two terms contain a common factor, that
factor can be written outside parentheses
with the remaining factors remaining as
terms inside parentheses
2x  2 y  2 x  y 
3m  3n  __
3 m  n
Illustration of Distributive Property
• Use the Distributive Property “backwards”
to write each of the following in a different
way:
5x  15  _ 
5x  15  5x  3

10x  8  _ 

10x  8  25x  4
Homework Problems
• Section: 1.7
• Page: 74
• Problems: All: 1 – 30, 35 – 50, 55 – 80
• MyMathLab Homework 1.7 for practice
• MyMathLab Homework Quiz 1.7 is due for
a grade on the date of our next class
meeting
Terminology of Algebra
• Constant – A specific number
Examples of constants: 3  6
4
5
• Variable – A letter or other symbol used to
represent a number whose value varies or
is unknown
n
Examples of variables: x
A
Terminology of Algebra
• Expression – constants and/or variables combined with
one or more math operation symbols for addition,
subtraction, multiplication, division, exponents and roots
in a meaningful way
Examples of expressions:
23
5 x
10
4
n
y  9 w
2
• Only the first of these expressions can be simplified,
because we don’t know the numbers represented by the
variables
Terminology of Algebra
• Term – an expression that involves only a single
constant, a single variable, or a product
(multiplication) of a constant and variables
Examples of terms:
2
m
 5 x
2
A B
2
y
3
• Note: When constants and variables are
multiplied, or when two variables are multiplied,
it is common to omit the multiplication symbol
Previous example is commonly written:
2
m
 5x
2
AB
2
y
3
Terminology of Algebra
• Every term has a “coefficient”
• Coefficient – the constant factor of a term
– (If no constant is seen, it is assumed to be 1)
• What is the coefficient of each of the
following terms?
2
2
 5x
m
1
AB
2
5
1
2
y
3
2
3
Like Terms
• Recall that a term is a _________
constant , a
product of a ________
________,
constant
variable or a _______
variables
and _________
• Like Terms: terms are called “like terms”
if they have exactly the same variables
with exactly the same exponents, but may
have different coefficients
• Example of Like Terms:
2
3x y and
 7x y
2
Determine Like Terms
• Given the term:  4xy
• Which of the following are like terms?
3
3
5x y
3
2 3
xy
5
 2x y
3
3
.54 xy
Adding Like Terms
• When “like terms” are added, the result is
a like term and its coefficient is the sum of
the coefficients of the other terms
• Example:
2x  7 x  9x
• The reason for this can be shown by the
distributive property:
2x  7 x  2  7x  9x
Subtracting Like Terms
• When like terms are subtracted, the result
is a like term with coefficient equal to the
difference of the coefficients of the other
terms
• Example:
2 x  7 x   5x
• Reasoning:
2x  7 x  2  7x  5x
Simplifying Expressions by
Combining Like Terms
• Any expression containing more than one
term may contain like terms, if it does, all
like terms can be combined into a single
like term by adding or subtracting as
indicated by the sign in front of each term
• Example: Simplify: 4 x  19 y  6 x  2 y  x
Middle two steps can
be done in your head!
4 x  6 x  x  19 y  2 y
4  6 1x  19  2y
9 x  17 y
Review of Distributive Property
• Distributive Property – multiplication can
be distributed over addition or subtraction
• Some people make the mistake of trying to
distribute multiplication over multiplication
• Example: 3x  y   3 x  3 y
3xy  3 xy  3x3 y
• Associative Property justifies answer! !!
+ or – in Front of Parentheses
• When a + or – is found in front of a
parentheses, we assume that it means
“positive one” or “negative one”
• Examples:
2  3  y   2 13  y   2  3  y  1  y
3  x  2  x  4  3  1x  2 1x  4 
3 x  2  x  4  5
Multiplying Terms
• Terms can be combined into a single term
by addition or subtraction only if they are
like terms
• Terms can always be multiplied to form a
single term by using commutative and
associative properties of multiplication
2
• Example: 2 xy  3x Won' t simplify!
2 xy3x
2
  23xx y 
2
3
6x y
Middle step can be done in your head!
Simplifying an Expression
• Get rid of parentheses by multiplying or
distributing
• Combine like terms
• Example:
 3 x  5x  2  2 2 x  4  x
3x  5x 10  4x  4  x
3x 14
Homework Problems
• Section: 1.8
• Page: 80
• Problems: All: 5 – 30
Odd: 33 – 75
• MyMathLab Homework 1.8 for practice
• MyMathLab Homework Quiz 1.8 is due for
a grade on the date of our next class
meeting