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:
etc,4
7,
5
1,
3
2r"denominato" called isnumber Bottom
numerator"" called isnumber Top
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 factor12 = (3)(4) 3 & 4 are factors12 = (6)(2) 6 & 2 are factors12 = (12)(1) 12 and 1 are factors12 = (2)(2)(3) 2, 2, and 3 are factorsIn 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) 4 & 7 are factors, but 4 is not prime28 = (2)(2)(7)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 2120 = (2)(2)(30) 30 is not prime, and is divisible by 2120 = (2)(2)(2)(15) 30 is not prime, and is divisible by 3120 = (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:
• Reduce to lowest terms by factoring:
18
12
332
322
3
2
place.each in left is "1" out, divided are factorscommon When
1
1
1
1
18
12
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:
28
15
9
41 1
1
1
1121
5
722
53
33
22
Dividing Fractions
• Invert the divisor and change problem to multiplication
• Example:
4
3
3
2
3
4
3
2
9
8
Adding Fractions Having a Common Denominator
• Add the numerators and keep the common denominator
• Example:
7
3
7
27
5
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:,
• Factor each denominator:
• Construct LCD by writing each factor the maximum number of times it’s found in any denominator:
18
18
24
24
332 3222
factor? a is 2 timesofnumber maximum theisWhat
LCD 33222
factor? a is 3 timesofnumber maximum theisWhat 32
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:
• Multiply numerator and denominator by 4:
18
572? into go 18 does many times How 4
4
4
18
5
18
5
72
20
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
• Find a least common denominator, LCD, for the rational expressions:
• Write each fraction as an equivalent fraction having the LCD:
• Write the answer by adding or subtracting numerators as indicated, and keeping the LCD:
• If possible, reduce the answer to lowest terms
24
7
18
5
72 is LCD that thefoundalready have We
factorcommon no haver denominato andnumerator because reducet Won'
3
3
24
7
4
4
18
5
24
7
18
5
72
21
72
20
72
21
72
20
72
41
72
41
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
• Find a least common denominator, LCD, for the rational expressions:
• Write each fraction as an equivalent fraction having the LCD:
• Write the answer by adding or subtracting numerators as indicated, and keeping the LCD:
• If possible, reduce the answer to lowest terms
12
5
10
7
32212
5210
factorcommon no haver denominato andnumerator because reducet Won'
5
5
12
5
6
6
10
7
12
5
10
7
60
25
60
42
60
17
605322 LCD
60
25
60
42
60
17
Improper Fractions& Mixed Numbers
• A fraction is called “improper” if the numerator is bigger than the denominator
• 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
• Likewise, mixed numbers can be changed to improper fractions by multiplying denominator times whole number, plus the numerator, all over the denominator
3
7
3
7
3
12
5
34
5
345
5
23
Doing Math Involving Improper Fractions & Mixed Numbers
• Convert all numbers to improper fractions then proceed as previously discussed
3
7
5
34
3
7
5
23
5
5
3
7
3
3
5
23
15
35
15
6915
34
15
42
15
34
okay isanswer Either
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
“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 itselfExample: What is the value of 34 ?(3)(3)(3)(3) = 81
• An exponent applies only to the base (what it touches)
• Meanings of exponents that are not natural numbers will be discussed later
35 555 125
423 22223 48
43
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
3283425 2 363425 2 369425
2985 6
Example of Order of Operations
• Evaluate the following expression:
2
3
583
26431537
2
3
33
26121537
2
3
33
26337
93
86337
27
8697
27
8616
27
822
27
14
)separately simplified be should bottom and topsymbol; grouping a isbar fraction (A
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:
.
95
437 593
231
Expressions InvolvingInequality Symbols
• Expressions involving inequality symbols may be either true or false
• Determine whether each of the following is true or false:
523
45
3
2
11
629
437
LCD ith thefraction w equivalentan each toconvert fractions, comparingWhen
True
False
False
42
2
5
3
5
5
2
11 4
10
6
10
55 4
10
49 4
10
94 False
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
Nine is greater than eleven minus four
Three is not equal to eight minus six
1237
4119
683
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
symbol5 > 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
symbol3 < 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:
• Variable – A letter or other symbol used to represent a number whose value varies or is unknown
Examples of variables:
3 65
4
x n 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 rootsExamples of expressions:
• Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables
32 x5n
104 wy 92
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 :
• Consider the next similar, but slightly different, example
3n4n
4
81
43 3333
Example
• Evaluate the expression for :
• Notice the difference between this example and the previous one – it illustrates the importance of using a parenthesis in place of the variable
3n4n
4
81
43 3333
Example
• Evaluate the expression for : 2xx13
13
213
11
Example
• Evaluate the expression for : 4,3 yx
2212
13
2212 xy
234212
9812
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
5x
x3
4x
8x
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
4x
x7
6x
9x
x2
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
x7
x4
x2
xxx or 2
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
x
2
8
x
x
6
Phrases Translating to Expressions Involving Multiple Math Operations
English Phrase
• 4 less than 3 times a number
• the quotient of 5 and twice a number
• 6 times the difference between a number and 5
Algebra Expression
43 x
x2
5
56 x
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
x74
5
xx
43 x
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:
.
1495 F?or T
574
True
F?or T False
Equations
• Most equations contain one or more variables and the truthfulness of the equation depends on the numbers that replace the variables
• Example:
• What value of x makes this true?
• A number that can replace a variable to make an equation true is called a solution
94 x5x
equation theosolution t a is 5
Distinguishing Between Expressions & Equations
• Expressions contain constants, variables and math operations, but NO EQUAL SIGN
• Equations always CONTAIN AN EQUAL SIGN that indicates that two expressions have the same value
94 x
94 x
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, find a solution for the equation:
• Check x = 3
• Check x = 4
• Check x = 5
1132 x
5,4,3
?11342
?11352
?11332 119
1111
1113 solutiononly theis 4 numbers, ofset given theFrom
Writing Equationsfrom 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
• Four more than twice a number is ten
5,4,3
1042 x :issolution The 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, …
• 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
0 55
Graphing Integerson 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:
0 55
2
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 GraphingRational Numbers
• Graph
• Graph
5
3
3
7
5
30
3
12
0
0 11
5
3
2
3
7
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: 3,5,
Notes on Square Roots
• The square root of is written as and represents a number that multiplies by itself to give
• We know that the number that multiplies by itself to give is , so we write
• is a terminating decimal, so is a rational number• The square root of is written as and represents a
number that multiplies by itself to give• We know of no number that multiplies by itself to give ,
but a calculator gives a decimal approximation that fills the screen without showing a repeating pattern at the end. is an irrational number
• Square roots may be rational, irrational, or neither
4 44
4 224 4
5 55
4
5
5
More Notes on Square Roots
• The square root of is written as , but it does not exist in the real number system (no real number multiplies by itself to give a negative
• 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
9 9
9
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 GraphingIrrational Numbers
• Graph 3
3
7320508.1 7.1
02 1
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: . These type of numbers (complex numbers) will be discussed in college algebra.
7
Ordering Real Numbers
• Given two real numbers, represented by the variables a and b, one of the following order relationships is true:
a = ba equals b if they graph at the same locationa < ba is less than b, if a is left of b on a number linea > ba is greater than b, if a is right of b on a number line
2?- 7- isWhy 2- ofleft is 7-
Additive Inversesof 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 oppositeThe additive inverse of 5 is:The additive inverse of -3 is:
• 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:
5- 3
7
Absolute Valueof 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 numberThe absolute value of 5 is shown by :and is equal to:The absolute value of -3 is shown by:and is equal to:
5 5
3 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:• Reasoning:
– Negatives scored:– Positives scored:
• _________ won by ____, so
1826
1826
Negatives 8 1826 8
Second Example of Addition
• Example:
• Reasoning: – Negatives scored:– Positives scored:
• _________ won by ____, so:Positives 2
2
7)2()9(835
16295 18783
7)2()9(835
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:
• Reasoning: – Negatives scored:– Positives scored:
• _________ won by ________, so:
6
5
4
3
1212 9 10
twelfths9 twelfths10
Positives twelfth1 6
5
4
3
12
1
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
• Example:
• Reasoning: – Negatives scored:– Positives scored:
• _________ won by ____, so:
18.23.5
Negatives
3.518.2 12.3
18.2
3.5
much? howby won Negatives
12.3
18.23.5 12.3
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”
.
46 4 negative add 6 meansfour positivesubtract 6 46 2
35 3 positive add 5 means 3 negativesubtract 5 35 8
27 2 positive add 7 negative means 2 negativesubtract 7 negative 27 5
Problems Involving BothAddition and Subtraction
• Example:
• Identify subtraction:
• Add opposite:
• Reasoning: – Negatives scored:– Positives scored:
• _________ won by ____, so:
7410653
7410653 7410653
4653710 17
18
Negatives 1
17410653
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 DividingReal 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:
. 56
72
30
14
3
124
8
6
4
3
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:
25
18
12
5
25
18
12
5
2
3
5
1
10
3
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:
6
5
3
2
5
6
3
22
15
4
Division Involving Zero
• People are often confused when division involves zero – the rule must be memorized!– Division by zero is always undefined
– Otherwise, division into zero is always zero
• Explanation comes from checking answer:
.
undefined is 0
12
0 12
0
34
12
1234
0 12
0
0012
?0
12
12?0 !!Impossible
Undefined
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:
.
2,7,5,8,3
5
27583 Average
5
5 1
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
257
1,348
35,750
Yes
No
Yes
Yes
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
357
1,348
45,750
Yes
No
Yes
Yes
:digits of Sum
:digits of Sum
:digits of Sum
:digits of Sum
33
15
16
21
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?51,976,1043571,34845,750
Yes
Yes
No
No
:digits Last two:digits Last two
:digits Last two:digits Last two
457
4850
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
357
1,348
45,750
No
No
No
Yes
:digitLast
:digitLast
:digitLast
:digitLast
47
80
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?51,976,1043571,34845,750
Yes
NoNo
Yes
:digits of Sum
:digits of Sum
:digits of Sum
33
16
21
Even
OddEven
Even
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?51,976,1043571,34845,750
Yes
NoNo
:digits Last three:digits Last three:digits Last three:digits Last three
104357348750
No
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
357
1,348
45,750
No
NoNo
No
:digits of Sum:digits of Sum
:digits of Sum
:digits of Sum
3315
1621
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:
• Associative Property – the way real numbers are grouped during addition or multiplication does not effect the result:
baabandabba
bcacabandcbacba
cba abc
Properties of Real Numbers
• Commutative Property Examples:
• Associative Property Examples:
x3
32 x
2x
3x
x2
yx2
32 x
xy2
Properties of Real Numbers
• Identity Property for Addition – when zero is added to a number, the result is still the number:
• Identity Property for Multiplication – when one is multiplied by a number, the result is still the number:
aaandaa 00
aaandaa 11
Properties of Real Numbers
• Identity Property for Addition Example:
• Identity Property for Multiplication Examples:
03
13
2
3
3
2
5
7
3
3
5
7
x30 x3
Properties of Real Numbers
• Inverse Property for Addition – when the opposite (negative) of a number is added to the number, the result is zero:
• Inverse Property for Multiplication – when the reciprocal of a number is multiplied by the number, the result is one
00 aaandaa
11
11
aa
anda
a
Reciprocals of Real Numbers
• Zero has no reciprocal
• Reciprocals of other integers are formed by putting 1 over the number
• Reciprocals of fraction are formed by switching the numerator and denominator
undefined is 0
1
:is 3- of reciprocal The 3
1
:is 5
3 of reciprocal The
3
5
Properties of Real Numbers
• Inverse Property for Addition Examples:
• Inverse Property for Multiplication Examples:
xx 22
5
15
3
2
3
200
11 3
4
4
3
Properties of Real Numbers
• Distributive Property – multiplication can be distributed over addition or subtraction without changing the result
acabcbaandacabcba
Illustration of Distributive Property
33523 5323523
15673 2121
4434 x 34434 xx 12434 xx
Illustration of Distributive Property
• Distributive Property works both directions:
• If two terms contain a common factor, that factor can be written outside parentheses with the remaining factors remaining as terms inside parentheses
cbaacabandcbaacab
222 yx yx 3 nmnm __33
Illustration of Distributive Property
• Use the Distributive Property “backwards” to write each of the following in a different way:
_155x
35155 xx
_810x
452810 xx
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:
• Variable – A letter or other symbol used to represent a number whose value varies or is unknown
Examples of variables:
3 65
4
x n 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 wayExamples of expressions:
• Only the first of these expressions can be simplified, because we don’t know the numbers represented by the variables
32 x5n
104 wy 92
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:
• 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 25 x BA y3
2
2 m 25x AB y3
2
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
m
25x
ABy
3
22
1
5
1 3
2
Like Terms
• Recall that a term is a _________ , a ________, or a _______ of a ________ 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:
constantvariable productconstant
variables
yxandyx 22 73
Determine Like Terms
• Given the term:
• Which of the following are like terms?
34xy
yx32
3
5
2xy
335 yx
354. 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:
• The reason for this can be shown by the distributive property:
xx 72 x9
xxxx 97272
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:
• Reasoning:
xx 72 x5
xxxx 57272
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: xyxyx 26194yyxxx 21964
yx 219164 yx 179
head!your in done be
can steps twoMiddle
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:
• Associative Property justifies answer! !!
yx3 yx 33
xy3 xy3 yx33
+ 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:
y32 y312 y32 y 1
423 xx 41213 xx
423 xx 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
• Example: 232 xxy
232 xxy yxx 232 yx36head!your in done becan step Middle
simplify!t Won'
Simplifying an Expression
• Get rid of parentheses by multiplying or distributing
• Combine like terms
• Example:
xxxx 422253
xxxx 441053
143 x
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