Transcript
Page 1: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Elementary Algebra

Exam 1 Material

Page 2: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 3: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 4: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 5: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole 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

Page 6: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole 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

Page 7: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole 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

Page 8: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole 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

Page 9: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 10: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 11: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 12: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Dividing Fractions

• Invert the divisor and change problem to multiplication

• Example:

4

3

3

2

3

4

3

2

9

8

Page 13: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Adding Fractions Having a Common Denominator

• Add the numerators and keep the common denominator

• Example:

7

3

7

27

5

Page 14: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 15: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 16: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 17: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 18: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 19: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 20: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 21: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 22: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 23: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 24: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 25: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 26: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 27: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Order of Operations

• Example:

• P

• E

• MD

• AS

3283425 2 363425 2 369425

2985 6

Page 28: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 29: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 30: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 31: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 32: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 33: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 34: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 35: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 36: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 37: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Example

• Evaluate the expression for :

• Consider the next similar, but slightly different, example

3n4n

4

81

43 3333

Page 38: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 39: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Example

• Evaluate the expression for : 2xx13

13

213

11

Page 40: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Example

• Evaluate the expression for : 4,3 yx

2212

13

2212 xy

234212

9812

Page 41: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 42: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 43: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 44: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 45: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 46: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 47: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 48: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 49: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 50: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 51: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 52: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 53: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 54: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 55: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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, …

Page 56: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 57: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 58: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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.)

Page 59: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 60: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Examples of GraphingRational Numbers

• Graph

• Graph

5

3

3

7

5

30

3

12

0

0 11

5

3

2

3

7

Page 61: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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,

Page 62: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 63: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 64: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 65: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Example of GraphingIrrational Numbers

• Graph 3

3

7320508.1 7.1

02 1

Page 66: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 67: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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-

Page 68: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 69: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 70: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 71: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 72: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 73: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 74: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 75: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 76: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 77: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 78: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 79: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 80: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 81: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 82: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 83: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 84: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 85: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 86: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 87: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 88: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 89: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 90: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 91: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 92: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 93: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 94: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 95: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Properties of Real Numbers

• Commutative Property Examples:

• Associative Property Examples:

x3

32 x

2x

3x

x2

yx2

32 x

xy2

Page 96: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 97: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 98: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 99: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 100: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 101: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Properties of Real Numbers

• Distributive Property – multiplication can be distributed over addition or subtraction without changing the result

acabcbaandacabcba

Page 102: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Illustration of Distributive Property

33523 5323523

15673 2121

4434 x 34434 xx 12434 xx

Page 103: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 104: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 105: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 106: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 107: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 108: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 109: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 110: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 111: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Determine Like Terms

• Given the term:

• Which of the following are like terms?

34xy

yx32

3

5

2xy

335 yx

354. xy

Page 112: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 113: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 114: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 115: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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

Page 116: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

+ 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

Page 117: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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'

Page 118: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

Simplifying an Expression

• Get rid of parentheses by multiplying or distributing

• Combine like terms

• Example:

xxxx 422253

xxxx 441053

143 x

Page 119: Elementary Algebra Exam 1 Material. Familiar Sets of Numbers Natural numbers –Numbers used in counting: 1, 2, 3, … (Does not include zero) Whole numbers

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


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