ROUNDING AND ESTIMATING ORDER

1
ROUNDING AND ESTIMATING ORDER
1.4

• Round to the nearest ten, hundred, or thousand.
• Estimate sums and differences by rounding.
• Use lt or gt for to write a true sentence in a
  situation like 6 10.

2
Objective

• Round to the nearest ten, hundred, or thousand.

3
Example A

• Round 57 to the nearest ten.
• Solution
• Since 57 is closer to 60, we round up to 60.

50
60
55
57
4
Example B

• Round 52 to the nearest ten.
• Solution
• Since 52 is closer to 50, we round down to 50.

52
50
60
55
5
Example C

• Round 55 to the nearest ten.
• Solution
• We agree to round up to 60.

50
60
55
When a number is halfway between rounding
numbers, round up.
6
Rounding Whole Numbers

• To round to a certain place
• a) Locate the digit in that place.
• b) Consider the next digit to the right.
• c) If the digit to the right is 5 or higher,
  round up. If the digit to the right is 4 or
  lower, round down.
• d) Change all digits to the right of the
  rounding location to zeros.

7
Example D

• Round 7564 to the nearest hundred.
• Solution
• a) Locate the digit in the hundreds place, 5.
• 7 5 6 4
• b) Consider the next digit to the right, 6.
• 7 5 6 4
• c) Since that digit is 5 or higher, round 5
  hundreds up to 6 hundreds.
• d) Change all digits to the right of the hundreds
  digit to zeros.
• 7 6 0 0

8
Example E

• Round 88,696 to the nearest ten.
• Solution
• a) Locate the digit in the tens place, 9.
• 8 8, 6 9 6
• b) Consider the next digit to the right, 6.
• 8 8, 6 9 6
• c) Since that digit is 5 or higher, round 9 tens
  to 10 tens and carry the 1 over to the hundreds.
• d) Change the digit to the right of the tens
  digit to zeros.
• 8 8, 7 0 0

9
Objective

• Estimate sums and differences by rounding.

10
Example F

• Mario and Greta are considering buying a new
  computer. There are two models, and each has
  options beyond the basic price, as shown below.
  Mario and Greta have a budget of 1100. Make a
  quick estimate to determine if the XS with a
  monitor, memory upgrade to 80 gig and a printer
  is within their budget.

11

• Solution
• First, we list the base price and then the cost
  of each option. We then round each number to the
  nearest hundred and add.
• XS 595 600
• Monitor 220 200
• Memory 90 100
• Printer 120 100
• 1000
• The price of the computer is within their budget.

12
Example G

• Estimate the difference by first rounding to the
  nearest thousand 8426 ? 3840.

Solution 8 4 2 6 ? 3 8 4 0
8 0 0 0 ? 4 0 0 0 4 0 0 0
13
Objective

• Use lt or gt for to write a true sentence in
  a situation like 6 10.

14
Order of Whole Numbers
For any whole numbers a and b 1. a lt b (read a
is less than b) is true when a is to the left of
b on the number line. 2. a gt b (read a is
greater than b) is true when a is to the right
of b on the number line. We call lt and gt
inequality symbols.
15
Example H

• Use lt or gt for to write a true sentence 84
  94.

Solution
Since 84 is to the left of 94 on a number line,
84 lt 94.
16
Section 1.4
1. Round 62,598 to the nearest hundred.
a) 63,000
b) 62,600
c) 62,500
d) 62,000
17
Section 1.4
1. Round 62,598 to the nearest hundred.
a) 63,000
b) 62,600
c) 62,500
d) 62,000
18
Section 1.4
2. Which of the following is true?
a) 3 lt 2
b) 17 lt 15
c) 4 gt 12
d) 11 gt 9
19
Section 1.4
2. Which of the following is true?
a) 3 lt 2
b) 17 lt 15
c) 4 gt 12
d) 11 gt 9
20
EXPONENTIAL NOTATION AND ORDER OF OPERATIONS
1.7

• Write exponential notation for products such as 4
  ? 4 ? 4.
• Evaluate exponential notation.
• Simplify expressions using the rules for order of
  operations.
• Remove parentheses within parentheses.

21
Objective

• Write exponential notation for products such as
  4 ? 4 ? 4.

22
Exponential Notation

• The 5 is called an exponent.
• The 4 is the base.

5
4 x 4 x 4 x 4 x 4 we write as 4
1442443
factors
5
23
Example A
Write exponential notation for 7?7?7?7?7?7. Solut
ion Exponential notation is 76

24
Objective

• Evaluate exponential notation.

25
Example B

• Evaluate 84 and 104
• Solution
• 84 8 ? 8 ? 8 ? 8 4096
• 104 10 ? 10 ? 10 ? 10 10,000

26
Objective

• Simplify expressions using the rules for order
  of operations.

27
Rules for Order of Operations
1. Do all calculations within parentheses ( ),
brackets , or braces before operations
outside. 2. Evaluate all exponential
expressions. 3. Do all multiplications and
divisions in order from left to right. 4. Do all
additions and subtractions in order from left to
right.
28
Example C

• Simplify 24 ? 3 ? 4
• Solution
• There are no parentheses or exponents, so we
  start with the third step.
• 24 ? 3 ? 4 8 ? 4
• 32

Doing all multiplications and divisions in order
from left to right
29
Example D

• Simplify 20 ? 4 ? 3 ? (10 7)
• Solution
• 20 ? 4 ? 3 ? (10 7) 20 ? 4 ? 3 ? 3
• 5 ? 3 ? 3
• 15 ? 3
• 5

30
Example E

• Simplify
• Solution

31
Objective

• Remove parentheses within parentheses.

32

• When parentheses occur within parentheses, we can
  make them different shapes, such as (also
  called brackets) and (also called
  braces). All of these have the same meaning.
  When parentheses occur within parentheses,
  computations in the innermost ones are to be done
  first.

33
Example G

• Simplify

Solution
34
Section 1.7
1. Simplify. 23 56 ? 7
a) 9
b) 14
c) 16
d) 64
35
Section 1.7
1. Simplify. 23 56 ? 7
a) 9
b) 14
c) 16
d) 64
36
Section 1.7
2. Simplify. 5(24 19)(25 23) ? 12 (9
7)
a) 10
b) 50
c) 100
d) 570
37
Section 1.7
2. Simplify. 5(24 19)(25 23) ? 12 (9
7)
a) 10
b) 50
c) 100
d) 570
38
Introduction to Variables, Algebraic Expressions,
and Equations
1.8

• Evaluate Algebraic Expressions.
• Identify Solutions of Equations.
• Translate Phrases into Variable Expressions

39
Algebraic Expressions

• An algebraic expression consists of variables,
  numerals, and operation signs.
• x 38 19 y
• When we replace a variable with a number, we say
  that we are substituting for the variable.
• This process is called evaluating the expression.

40
Evaluate each expression for the given values.

• Example
• a b for
• a 21 and b 74
• Solution
• a b 21 74
• 95

• Example
• 7xy for
• x 3 and y 6
• Solution
• 7xy 7 3 6
• 126

41
Example A

• Evaluate x y for x 38 and y 62.
• Solution
• We substitute 38 for x and 62 for y.
• x y 38 62 100

42
Example B

• Evaluate and for x 72 and y 8.
• Solution
• We substitute 72 for x and 8 for y

43
Example C

• Evaluate for C 30.
• Solution
• This expression can be used to find the
  Fahrenheit temperature that corresponds to 30
  degrees Celsius.

44
Translating to Algebraic Expressions
45
Example

• Translate each phrase to an algebraic expression.
• a) 9 more than y
• b) 7 less than x
• c) the product of 3 and twice w
• Solution
• Phrase Algebraic Expression
• a) 9 more than y y 9
• b) 7 less than x x ? 7
• c) the product of 3 and twice w 32w or 2w 3

46
Translating to Equations

• The symbol (equals) indicates that the
• expressions on either side of the equals
• sign represent the same number.
• An equation is a number sentence with
• the verb .

47
Example E

• Translate each phrase to an algebraic expression.

x 8, or 8 x
4x 2, or 2 4x
n 8
ab 5
0.25n
3w 7
48

• Solution
• A replacement or substitution that makes an
  equation true is called a solution. Some
  equations have more than one solution, and some
  have no solution. When all solutions have been
  found, we have solved the equation.

49
Example

• Determine whether 12 is a solution of x 4
  16.
• Solution
• x 4 16 Writing the equation
• 12 4 16 Substituting 12 for x
• 16 16 16 16 is TRUE.

50
Example

• Translate the problem to an equation.
• What number added to 127 is 403?
• Solution
• Let y represent the unknown number.

51
Section 1.8
1. Evaluate when a 13 and b 5.
a)
b)
c)
d) 9
52
Section 1.8
1. Evaluate when a 13 and b 5.
a)
b)
c)
d) 9
53
Section 1.8
2. Write an algebraic expression The product
of a and b.
a) ab
b) a b
c) b a
d)
54
Section 1.8
2. Write an algebraic expression The product
of a and b.
a) ab
b) a b
c) b a
d)
55
Class Exercise
56
Integers and Introduction to Solving Equations
2

• 2.1 Introduction to Integers
• 2.2 Addition of Integers
• 2.3 Subtraction of Integers
• 2.4 Multiplication Division of Integers
• 2.5 Order of Operations
• 2.6 Solving Equations

57
Integers and the Number Line
2.1

• State the integer that corresponds to a real
  world situation.
• Form a true sentence using lt or gt.
• Find the absolute value of any integer.
• Find the opposite of any integer.

58
Objective

• State the integer that corresponds to a real
  world situation.

59
Integers
Integers consist of the whole numbers and their
opposites.
Integers to the left of zero on the number line
are called negative integers and those to the
right of zero are called positive integers. Zero
is neither positive nor negative and serves as
its own opposite.
60
Integers

• The integers , ?5, ?4, ?3, ?2, ?1, 0, 1, 2, 3,
  4, 5,

61
Example A

• Tell which integer corresponds to each situation.
• 1. Death Valley is 282 feet below sea level.
• 2. Margaret owes 312 on her credit card. She
  has 520 in her checking account.

Solution
1. 282 below sea level corresponds to ?282.
2. The integers ?312 and 520 correspond to the
situation.
62
Common uses for negative integers.

• Time Before an event
• Temperature Degrees below zero
• Money Amount lost, spent, owed, withdrawn
• Elevation Depth below sea level
• Travel Motion in the backward
  (reverse) direction

63
Objective

• Form a true sentence using lt or gt.

64

• Numbers are written in order on the number line,
  increasing as we move to the right. For any two
  numbers on the line, the one to the left is less
  than the one to the right.
• The symbol lt means is less than,
• ?4 lt 8 is read ?4 is less than 8.
• The symbol gt means is greater than,
• ?6 gt ?9 is read ?6 is greater than ?9.

65
Example B

• Use either lt or gt for to form a true
  sentence.
• 1. ?7 3 2. 8 ?3 3. ?21 ?9
• Solution

Since ?7 is to the left of 3, we have ?7 lt 3.
Since ?21 is to the left of ?9, we have ? 21 lt ?9.
Since 8 is to the right of ?3, we have 8 gt ?3.
66
Objective

• Find the absolute value of any integer.

67
Absolute Value

• The absolute value of a number is its distance
  from zero on a number line. We use the symbol x
  to represent the absolute value of a number x.

68
Example C

• Find the absolute value of each number.
• a. ?5 b. 36
• c. 0 d. ?52
• Solution
• a. ?5 The distance of ?5 from 0 is 5, so ?
  5 5.
• b. 36 The distance of 36 from 0 is 36, so
  36 36.
• c. 0 The distance of 0 from 0 is 0, so 0
  0.
• d. ?52 The distance of ?52 from 0 is 52, so
  ?52 52.

69
To find a numbers absolute value 1. If a
number is zero or positive, use the number
itself. 2. If the number is negative, make the
number positive.
70
Objective

• Find the opposite of any integer.

71
Opposites or Additive Inverses

• Given a number on one side of 0, we can get a
  number on the other side by reflecting the number
  across zero. For example, the reflection of 4 is
  ?4.

-6 -5 -4 -3 -2 -1 0
1 2 3 4 5 6
-6 -5 -4 -3 -2 -1 0
1 2 3 4 5 6
72
Notation for Opposites

• The opposite of a number x is written ?x (read
  the opposite of x).

73
Example D

• If x is ?5, find ?x.

Solution To find the opposite of x when x is ?5,
we reflect ?5 to the other side of 0.
-6 -5 -4 -3 -2 -1 0
1 2 3 4 5 6
-6 -5 -4 -3 -2 -1 0
1 2 3 4 5 6
We have ? (?5) 5. The opposite of ?5 is 5.
74
Example E

• Change the sign (find the opposite, or additive
  inverse) of each number.
• a) ?9 b) 8
• Solution
• a) ?9 ? (?9) 9
• b) 8 ?(8) ?8

75
Example F

• Evaluate ? (?x) for x ?7.
• Solution
• We replace x with ?7 and find ?(?(?7)).
• Reflecting ?7 to the other side of 0 and then
  back again gives us ?7. Thus, ?(?(?7)) ?7.

76
Example G

• Evaluate ??x for x 3.
• Solution
• We have ? ?x ? ?3
• Since ?3 3, it follows that ??3 ?3.

77
Section 2.1
1. Use lt, gt, or for to write a true
sentence ?5 ?3.
a) ? 5 lt ? 3
b) ? 5 gt ? 3
c) ? 5 ? 3
d) Cannot be determined.
78
Section 2.1
1. Use lt, gt, or for to write a true
sentence ?5 ?3.
a) ? 5 lt ? 3
b) ? 5 gt ? 3
c) ? 5 ? 3
d) Cannot be determined.
79
Section 2.1
2. Find the absolute value of ?13.
a) ? 13
b) 13
c) 169
d)
80
Section 2.1
2. Find the absolute value of ?13.
a) ? 13
b) 13
c) 169
d)
81
ADDITION OF INTEGERS
2.2

• Add integers without using a number line.

82
Objective

• Add integers without using a number line.

83
Adding Integers

• To perform the addition a b, we start at a, and
  then move according to b.
• a) If b is positive, we move to the right.
• b) If b is negative, we move to the left.
• c) If b is 0, we stay at a.

84
Example A

• Add 3 (?6).
• Solution
• 3 (?6) ?3

85
Example B

• Add ?5 8.
• Solution
• ?5 8 3

86
Adding Negative Integers

• To add two negative integers, add their absolute
  values and change the sign (making the answer
  negative).

87
Example C

• Add.
• 1. ?5 (?8) 2. ?9 (?7)
• Solution
• 1. ?5 (?8) ?13
• 2. ?9 (?7) ?16

Add the absolute values 5 8 13. Make the
answer negative.
88
Adding Positive and Negative Integers

• To add a positive integer and a negative integer,
  find the difference of their absolute values.
• a) If the negative integer has the greater
  absolute value, the answer is negative.
• b) If the positive integer has the greater
  absolute value, the answer is positive.

89
Example D

• Add.
• 1. 4 (?6) 2. 12 (?9)
• 3. ?8 5 4. ?7 5
• Solution
• 1. 4 (?6)
• 2. 12 (?9)
• 3. ?8 5
• 4. ?7 5

Think The absolute values are 4 and 6. The
difference is 2. Since the negative number has
the larger absolute value, the answer is
negative, ?2.
?2
3
?3
?2
90
Adding Opposites

• For any integer a,
• a (?a) ?a a 0.
• (The sum of any number and its additive inverse,
  or opposite, is 0.)

91
Example E

• Add 16 (?2) 8 15 (?6) (?14).
• Solution
• Because of the commutative and associate laws for
  addition, we can group the positive numbers
  together and the negative numbers together and
  add them separately. Then we add the two results.
• 16 (?2) 8 15 (?6) (?14)
• 16 8 15 (?2) (?6) (?14)
• 39 (?22)
• 17

92
Section 2.2
1. Add ?6 12.
a) ?6
b) 0
c) 6
d) 72
93
Section 2.2
1. Add ?6 12.
a) ?6
b) 0
c) 6
d) 72
94
Section 2.2
2. Add ?6 12 (?5) 3.
a) ?8
b) 4
c) 14
d) 26
95
Section 2.2
2. Add ?6 12 (?5) 3.
a) ?8
b) 4
c) 14
d) 26
96
SUBTRACTION OF INTEGERS
2.3

• Subtract integers and simplify combinations of
  additions and subtractions.
• Solve applied problems involving addition and
  subtraction of integers.

97
Objective

• Subtract integers and simplify combinations of
  additions and subtractions.

98
The Difference a ? b

• The difference a ? b is the number that when
  added to b gives a.

99
Example A

• Subtract 4 ? 9.
• Solution
• Think 4 ? 9 is the number that when added to 9
  gives 4.
• What number can we add to 9 to get 4?
• The number must be negative.
• The number is ?5
• 4 9 5.
• That is, 4 ? 9 ?5 because 9 (?5) 4.

100
Subtracting by Adding the Opposite

• To subtract, add the opposite, or additive
  inverse, of the number being subtracted
• a b a ( b).

101
Example B

• Equate each subtraction with a corresponding
  addition. Then write the equation in words.
• 1. ?15 ? (?25) 2. ?13 ? 40
• Solution
• 1. ?15 ? (?25) ?15 25 Adding the opposite of
  ?25
• Negative fifteen minus negative twenty-five is
  negative fifteen plus twenty-five.
• 2. ?13 ? 40 ?13 (?40) Adding the opposite of
  40
• Negative thirteen minus forty is negative
  thirteen plus negative forty.

102
Example C

• Subtract.
• 1. 3 7 2. 5 9 3. 4 (10)
• Solution
• 1. 3 7 3 (7)
• 4
• 2. 5 9 5 ( 9)
• 14
• 3. 4 (10) 4 10
• 6

The opposite of 7 is 7. We change the
subtraction to addition and add the opposite.
Instead of subtracting 7, we add 7.
103
Example D

• Simplify ?4 ? (?6) ? 10 5 ? (?7).
• Solution
• ?4 ? (?6) ? 10 5 ? (?7) ?4 6 (?10) 5
  7
• ?4 (?10) 6 5 7
• ?14 18
• 4

Adding opposites
Using a commutative law
104
Objective

• Solve applied problems involving addition and
  subtraction of integers.

105
Example E

• The Johnsons were taking a vacation and one day
  they drove from mile marker 54 to mile marker
  376. How far did they drive?
• Solution
• 376 54 376 (?54)
• 322 miles

Adding the opposite of 54.
106
Section 2.3
1. Subtract 4 (?11).
a) ?15
b) ?7
c) 7
d) 15
107
Section 2.3
1. Subtract 4 (?11).
a) ?15
b) ?7
c) 7
d) 15
108
Section 2.3
2. Simplify 15 ( 23) 20 ( 4).
a) 16
b) 24
c) 54
d) 62
109
Section 2.3
2. Simplify 15 ( 23) 20 ( 4).
a) 16
b) 24
c) 54
d) 62
110
Class Exercise
111
MULTIPLICATION OF INTEGERS
2.4

• Multiply integers.
• Find products of three or more integers and
  simplify powers of integers.

112
Objective

• Multiply integers.

113
Multiplying a Positive and a Negative Number

• Multiplication of integers is like multiplication
  of whole numbers. The difference is that we must
  determine whether the answer is positive or
  negative.

To multiply a positive integer and a negative
integer, multiply their absolute values and make
the answer negative.
114
Example A

• Multiply.
• 1. (7)(?9) 2. 40(?1) 3. ?3 ? 7
• Solution
• 1. (7)(?9)
• 2. 40(?1)
• 3. ?3 ? 7

?63
?40
?21
115
Multiplying Two Negative Integers

• To multiply two negative integers, multiply their
  absolute values. The answer is positive.

116
Example B

• Multiply.
• 1. (?3)(?4) 2. (?11)(?5) 3. (?2)(?1)
• Solution
• 1. (?3)(?4)
• 2. (?11)(?5)
• 3. (?2)(?1)

12
55
2
117
To multiply two integers

• a) Multiply the absolute values.
• b) If the signs are the same, the answer is
  positive.
• c) If the signs are different, the answer is
  negative.

118
Multiplication by Zero No matter how many times 0
is added to itself, the answer is 0. This leads
to the following result.

• For any integer a,
• a ? 0 0.
• (The product of 0 and any integer is 0.)

119
Example C

• Multiply.
• 1. ?24 ? 0 2. 0(?9)
• Solution
• 1. ?24 ? 0
• 2. 0(?9)

0
0
120
Objective

• Find products of three or more integers and
  simplify powers of integers.

121
Example D

• Multiply.
• 1. ?9 ? 3(?4) 2. ?6 ? (?3) ? (?4) ? (?7)
• Solution
• 1. ?9 ? 3(?4) ?27(?4)
• 108
• 2. ?6 ? (?3) ? (?4) ? (?7) 18 ? 28
• 504

Multiplying the first two numbers
Multiplying the results
Each pair of negatives gives a positive product.
122

• The product of an even number of negative
  integers is positive.
• The product of an odd number of negative integers
  is negative.

123
Powers of Integers

• The result of raising a negative number to a
  power is positive or negative, depending on
  whether the exponent is odd or even.

124
Example E

• Simplify. (?5)4 (?6)5
• Solution
• (?5)4 (?5) (?5) (?5) (?5) 625
• (?6)5 (?6) (?6) (?6) (?6) (?6) ?7776

125

• When a negative number is raised to an even
  exponent, the result is positive.
• When a negative number is raised to an odd
  exponent, the result is negative.

126

• For any integer a,
• ?1 ? a ?a

127
Example F

• Simplify ?102.
• Solution
• Since ?102 lacks parentheses, the base is 10, not
  ?10. Thus we regard ?102 as ?1 ? 102
• ?102 ?1 ? 102
• ?1 ? 10 ? 10
• ?1 ? 100
• ?100

128
Section 2.4
1. Multiply 6 ? (?12).
a) ?72
b) ?2
c) 2
d)
129
Section 2.4
1. Multiply 6 ? (?12).
a) ?72
b) ?2
c) 2
d)
130
Section 2.4
2. Simplify (? 7)2 .
a) ? 49
b) ? 14
c) 14
d) 49
131
Section 2.4
2. Simplify (? 7)2.
a) ? 49
b) ? 14
c) 14
d) 49
132
DIVISION OF INTEGERS AND ORDER OF OPERATIONS
2.5

• Divide integers.
• Use the rules for order of operations with
  integers.

133
Objective

• Divide integers.

134
The Quotient a/b

• The quotient (or a ? b, or a/b) is the
• number, if there is one, that when multiplied by
  b gives a.

135
Example A

• Divide, if possible. Check each answer.
• 1. 15 ? (?3) 2.
• Solution
• 1. 15 ? (?3) ?5
• 2.

Think What number multiplied by 3 gives 15? The
number is 5. Check (3)(5) 15.
Think What number multiplied by 5 gives 45? The
number is 9. Check (5)(9) 45.
136

• To multiply or divide two integers
• a) Multiply or divide the absolute values.
• b) If the signs are the same, the answer is
  positive.
• c) If the signs are different, the answer is
  negative.

137
Excluding Division by 0

• Division by 0 is not defined
• a ? 0, or is undefined for all real numbers
  a.

138
Dividends of 0

• Zero divided by any nonzero real number is 0

139
Example B

• Divide, if possible ?72 ? 0.
• Solution
• is undefined.

Think What number multiplied by 0 gives ?72?
There is no such number because anything times 0
is 0.
140
Objective

• Use the rules for order of operations with
  integers.

141
Rules For Order of Operations

• 1. Do all calculations within parentheses,
  brackets, braces, absolute value symbols,
  numerators, or denominators.
• 2. Evaluate all exponential expressions.
• 3. Do all multiplications and divisions in order
  from left to right.
• 4. Do all additions and subtractions in order
  from left to right.

142
Example C

• Simplify.
• 1. 2.
• Solution
• 1.

2.
143
Example D

• Simplify
• Solution

144
Section 2.5
1. Divide, if possible, .
a) ? 64
b) 0
c) undefined
d) 64
145
Section 2.5
1. Divide, if possible, .
a) ? 64
b) 0
c) undefined
d) 64
146
Section 2.5
2. Simplify ?3(12) 5(?4) 23.
a) 8
b) ?8
c) 24
d) ?64
147
Section 2.5
2. Simplify ?3(12) 5(?4) 23.
a) 8
b) ?8
c) 24
d) ?64
148
SOLVING EQUATIONS
2.8

• Use the addition principle to solve equations.
• Use the division principle to solve equations.
• Decide which principle should be used to solve an
  equation.
• Solve equations that require the use of both the
  addition principle and the division principle.

149
Objective

• Use the addition principle to solve equations.

150
Equivalent Equations
Equations with the same solutions are called
equivalent equations.
151
Example A

• Classify each pair as either equivalent equations
  or equivalent expressions
• 1. 6x 5 3x 3 3x 8
• 2. x 5 x 3 2
• Solution
• 1. Combining like terms, we have
• 3x 3 3x 8 (3 3)x ( 3 8 )
• 6x 5
• 6x 5 and 3x 3 3x 8 are equivalent
  expressions.

152
continued example A

• 2. x 5 x 3 2
• If we subtract 3 from both sides of the
  equation
• x 3 3 2 3
• x 0 5, or
• x 5
• x 5 and x 3 2 are equivalent
  equations.

153
The Addition Principle
For any numbers a, b, and c, a b is equivalent
to a c b c.
154
Example B

• Solve 34 t 12
• Solution
• 34 t 12
• 34 12 t 12 12
• 22 t 0
• 22 t
• The solution is 22.

155
Objective

• Use the division principle to solve equations.

156
The Division Principle
For any numbers a, b, and c, (c ? 0), a b is
equivalent to
157
Example C

• Solve 8x 72
• Solution
• 8x 72

Divide both sides by 8
158
Example D

• Solve 56 7y
• Solution
• 56 7y

Divide both sides by 7
159
Objective

• Decide which principle should be used to solve
  an equation.

160
Example E

• Solve.
• 1. 42 4 w 2. 42 3h
• Solution
• 1. 42 4 w
• 42 4 4 4 w
• 46 w
• 2. 42 3h

To undo the addition of 4, we subtract 4 or
simply add 4 to both sides.
Using the division principle
161
Objective

• Solve equations that require the use of both the
  addition principle and the division principle.

162
Example F

• Solve 6a 4 14.
• Solution
• We isolate 6a by adding 4 on both sides.
• 6a 4 4 14 4
• 6a 0 18
• 6a 18
• Now, isolate a by dividing by 6 on both sides.

163
Example G

• Solve 47 7x 5.
• Solution
• 47 7x 5
• 47 5 7x 5 5
• 42 7x

Check 47 7(6) 5 47 42 5 47 47 true
164
Section 2.6

• Solve 9a 7 43.

a) 4
b) 2
c) 2
d) 4
165
Section 2.6

• Solve 9a 7 43.

a) 4
b) 2
c) 2
d) 4
166
Section 2.6
2. Solve 7x 5 61.
a) 8
b) 6
c) 6
d) 8
167
Section 2.6
2. Solve 7x 5 61.
a) 8
b) 6
c) 6
d) 8
168
Class Exercise