Cei03 ppt 01

CHAPTER

1

1.1
1.2
1.3
1.4
1.5
1.6
1.7

Foundations of Algebra
Number Sets and the Structure of Algebra
Fractions
Adding and Subtracting Real Numbers;
Properties of Real Numbers
Multiplying and Dividing Real Numbers;
Exponents, Roots, and Order of Operations
Translating Word Phrases to Expressions
Evaluating and Rewriting Expressions


1.1
1.
2.
3.
4.
5.

Number Sets and the Structure of
Algebra

Understand the structure of algebra.
Classify number sets.
Graph rational numbers on a number line.
Determine the absolute value of a number.
Compare numbers.


Objective 1
Understand the structure of algebra.


Slide 1- 4

Definitions
Variable: A symbol that can vary in value.
Constant: A symbol that does not vary in value.
Variables are usually letters of the alphabet, like x or y.
Usually constants are symbols for numbers, like 1, 2,
¾, 6.74.


Slide 1- 5

Expression: A constant, variable, or any combination
of constants, variables, and arithmetic operations that
describe a calculation.
Examples of expressions:
2+6

or 4x − 5

or

1 2
πr h
3


Slide 1- 6

Equation: A mathematical relationship that contains
an equal sign.
Examples of equations:
2+6=8

or 4x − 5 = 12


or

1 2
V = πr h
3

Slide 1- 7

Inequality: A mathematical relationship that contains
an inequality symbol (≠, <, >, ≤, or ≥).
Symbolic form

Translation

8≠3

Eight is not equal to three.

5<7

Five is less than seven.

7>5

Seven is greater than five.

x≤3

x is less than or equal to three.

y≥2

y is greater than or equal to two.


Slide 1- 8

Objective 2
Classify number sets.


Slide 1- 9

Set: A collection of objects.
Braces are used to indicate a set. For example, the set
containing the numbers 1, 2, 3, and 4 would be written
{1, 2, 3, 4}.
The numbers 1, 2, 3, and 4 are called the members or
elements of this set.


Slide 1- 10

Writing Sets
To write a set, write the members or elements of the
set separated by commas within braces, { }.


Slide 1- 11

Example 1
Write the set containing the first four days of the week.
Answer
{Sunday, Monday, Tuesday, Wednesday}


Slide 1- 12

Numbers are classified using number sets.
Natural numbers contain the counting numbers 1, 2, 3,
4, …and is written {1, 2, 3, …}. The three dots are
called ellipsis and indicate that the numbers continue
forever in the same pattern.
Whole numbers: natural numbers and 0 {0, 1, 2, 3,…}
Integers: whole numbers and the opposite (or negative)
of every natural number {…, −3, −2, −1, 0, 1, 2, 3…}
Rational: every real number that can be expressed as a
ratio of integers.


Slide 1- 13

Rational number: Any real number that can be
a
expressed in the form , where a and b are integers
b
and b ≠ 0.


Slide 1- 14

Example 2
Determine whether the given number is a rational
number.
5
a.
b. 0.8
c. 0.3
6

Answer
5
a.
6

Yes, because 5 and
6 are integers.

b. 0.8

c.

Yes, 0.8 can be
expressed as a
fraction 8 over 10,
and 8 and 10 are
integers.

The bar indicates
that the digit
repeats. This is the
decimal equivalent
of 1 over 3. Yes this
is a rational number.


0.3

Slide 1- 15

Irrational number: Any real number that is not
rational.
Examples:
2,

3,

π

Real numbers: The union of the rational and
irrational numbers.


Slide 1- 16

Objective 3
Graph rational numbers on a number
line.


Slide 1- 17

Example 3
4
Graph on a number line. 2
5

Answer
The number is located 4/5 of the way between 2 and 3.
-1

0

1

2

2

4 3
5

Between 2 and 3, we divide the number line
into 5 equally spaced divisions. Place a dot on
the 4th mark.

Slide 1- 18

Objective 4
Determine the absolute value of a
number.


Slide 1- 19

Absolute value: A given number’s distance from 0 on
a number line.
←5 units from 0 → ←5 units from 0 →

The absolute value of a number n is written |n|.
The absolute value
The absolute value
of 5 is 5.
of −5 is 5.
|5| = 5
|−5| = 5


Slide 1- 20

Absolute Value
The absolute value of every real number is either
positive or 0.


Slide 1- 21

Example 4
Simplify.
a. |−9.4|

2
b.
9

Answer
a. |−9.4| = 9.4
2
2
b.
=
9
9


Slide 1- 22

Objective 5
Compare numbers.


Slide 1- 23

Comparing Numbers
For any two real numbers a and b, a is greater than
b if a is to the right of b on a number line.
Equivalently, b is less than a if b is to the left of a
on a number line.


Slide 1- 24

Example 5
Use =, <, or > to write a true statement.
a. 3 ___ −3
b. −1.8 ___ −1.6
Answer
a. 3 ___ −3
3 > −3
because 3 is farther
right on a number line.

b. −1.8 ___ −1.6
−1.8 < −1.6
because –1.8 is further
to the left on a number
line.


Slide 1- 25

To which set of numbers does −6
belong?
a) Irrational
b) Natural and whole numbers
c) Natural numbers, whole numbers,
and integers
d) Integers and rational numbers
1.1


Slide 1- 26

To which set of numbers does −6
belong?
a) Irrational
b) Natural and whole numbers
c) Natural numbers, whole numbers,
and integers rational numbers
d) Integers and
1.1


Slide 1- 27

Simplify |7|.

a) 7
b) −7
c) 0
d) 1/7

1.1


Slide 1- 28

Simplify |7|.

a) 7
b) −7
c) 0
d) 1/7

1.1


Slide 1- 29

Which statement is false?

a) 7 > 4
b) −2.4 > −1.4
c) 10 < 22
d) −3.6 > −6.4

1.1


Slide 1- 30

Which statement is false?

a) 7 > 4
b) −2.4 > −1.4
c) 10 < 22
d) −3.6 > −6.4

1.1


Slide 1- 31

1.2
1.
2.
3.
4.

Fractions

Write equivalent fractions.
Write equivalent fractions with the LCD.
Write the prime factorization of a number.
Simplify a fraction to lowest terms.


Fraction: A quotient of two numbers or expressions a
a
and b having the form , where b ≠ 0.
b

3
4

← Numerator
← Denominator

The top number in a fraction is called the numerator.
The bottom number is called the denominator.
Fractions indicated part of a whole.


Slide 1- 33

Objective 1
Write equivalent fractions.


Slide 1- 34

Writing Equivalent Fractions
For any fraction, we can write an equivalent
fraction by multiplying or dividing both its
numerator and denominator by the same nonzero
number.


Slide 1- 35

Example 1
Find the missing number that makes the fractions
equivalent.
a. 9 = ?
b. − 18 = ?
15

45

Solution
a. 9 ?
15

=

45

9×
3
27
=
15 ×
3 45
Multiply the numerator and
denominator by 3.

36

b.

2

18 ?
− =
36 2
18 ÷ 18
1
−
=−
36 ÷ 18
2

Divide the numerator and
denominator by 6.

Slide 1- 36

Objective 2
Write equivalent fractions with the
LCD.


Slide 1- 37

Multiple: A multiple of a given integer n is the
product of n and an integer.
We can generate multiples of a given number by
multiplying the given number by the integers.
Multiples of 2

Multiples of 3

2× = 2
1
2 ×2 = 4
2× = 6
3
2 ×4 = 8
2 × = 10
5
2 ×6 = 12

3× = 3
1
3 ×2 = 6
3× = 9
3
3 ×4 = 12
3 × = 15
5
3 ×6 = 18


Slide 1- 38

Least common multiple (LCM): The smallest number
that is a multiple of each number in a given set of
numbers.
Least common denominator (LCD): The least
common multiple of the denominators of a given set
of fractions.


Slide 1- 39

Example 2
7
5
Write and as equivalent fractions with the LCD.
8
6

Solution
The LCD of 8 and 6 is 24.
7 7×
3 21
=
=
8 8×
3 24

5 5 ×4 20
=
=
6 6 ×4 24


Slide 1- 40

Objective 3
Write the prime factorization of a
number.


Slide 1- 41

Factors: If a ⋅ b = c, then a and b are factors of c.
Example: 6 ⋅ 7 = 42, 6 and 7 are factors of 42
Prime number: A natural number that has exactly two
different factors, 1 and the number itself.
Example: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,…
Prime factorization: A factorization that contains only
prime factors.

Slide 1- 42

Example 3
Find the prime factorization of 420.
Solution
420 Factor 420 to 10 and 42. (Any
Factor 10 to 2 and 5, which are
primes. Then factor 42 to 6
and 7.
7 is prime and then factor 6
into 2 and 3, which are primes.

[
42
[ ]

6
[ ]

2

]
10
[ ]

7 2
3

two factors will work.)

5

Answer 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 7


Slide 1- 43

Objective 4
Simplify a fraction to lowest terms.


Slide 1- 44

a
Lowest terms: Given a fraction and b ≠ 0, if the
b

only factor common to both a and b is 1, then the
fraction is in lowest terms.


Slide 1- 45

Simplifying a Fraction with the Same Nonzero
Numerator and Denominator
n 1
= = 1, when n ≠ 0.
n 1

Eliminating a Common Factor in a Fraction
an a × a
1
=
= , when b ≠ 0 and n ≠ 0.
bn b × b
1


Slide 1- 46

These rules allow us to write fractions in lowest terms
using prime factorizations. The idea is to replace the
numerator and denominator with their prime
factorizations and then eliminate the prime factors that
are common to both the numerator and denominator.


Slide 1- 47

Simplifying a Fraction to Lowest Terms
To simplify a fraction to lowest terms:
1. Replace the numerator and denominator with their
prime factorizations.
2. Eliminate (divide out) all prime factors common
to the numerator and denominator.
3. Multiply the remaining factors.


Slide 1- 48

Example 4a
Simplify to lowest terms.

30
42

Solution
2× ×
3 5
30
5
=
=
42
2 × ×7
3
7
Replace the numerator and denominator with their prime
factorizations; then eliminate the common prime factors.


Slide 1- 49

Example 4b
220
Simplify to lowest terms.
2380

Solution
2 ×2 × ×
5 11
220
11
=
=
2380 2 ×2 × ×7 ×
5 17
119
Replace the numerator and denominator with their prime
factorizations; then eliminate the common prime factors.


Slide 1- 50

Example 5
At a company, 225 of the 1050 employees have
optional eye insurance coverage as part of their
benefits package. What fraction of the employees have
optional eye insurance coverage?
Solution
225
3× × ×
3 5 5
3
=
=
1050 2 × × × ×7 14
3 5 5

Answer 3 out of 14 employees have optional eye
insurance.


Slide 1- 51

What is the prime factorization of 360?

a) 6 ⋅ 6 ⋅ 5
b) 23 ⋅ 32 ⋅ 5
c) 22 ⋅ 32 ⋅ 5
d) 32 ⋅ 5 ⋅ 7

1.2


Slide 1- 52

What is the prime factorization of 360?

a) 6 ⋅ 6 ⋅ 5
b) 23 ⋅ 32 ⋅ 5
c) 22 ⋅ 32 ⋅ 5
d) 32 ⋅ 5 ⋅ 7

1.2


Slide 1- 53

Simplify to lowest terms:

a)
b)
c)
d)

1.2

112
280

14
35
1
4
2
5
21
23

Slide 1- 54

Simplify to lowest terms:

a)
b)
c)
d)

1.2

112
280

14
35
1
4
2
5
21
23

Slide 1- 55

1.3
1.
2.
3.
4.

Adding and Subtracting Real Numbers;

Add integers.
Add rational numbers.
Find the additive inverse of a number.
Subtract rational numbers.


Objective 1
Add integers.


Slide 1- 57

Parts of an addition statement: The numbers added
are called addends and the answer is called a sum.
2+3=5

Addends

Sum


Slide 1- 58

Properties
of Addition

Symbolic Form

Word Form

Additive
Identity

a+0=a

The sum of a number and 0
is that number.

Commutative
Property of
Addition

a+b=b+a

Changing the order of
addends does not affect the
sum.

Associative
Property of
Addition

a + (b + c) = (a + b) + c

Changing the grouping of
three or more addends does
not affect the sum.


Slide 1- 59

Example 1
Indicate whether each equation illustrates the additive
identity, commutative property of addition, or the
associative property of addition.
a. (5 + 6) + 3 = 5 + (6 + 3)
Answer Associative property of addition
b. 0 + (−9) = −9
Answer Additive identity
c. (−9 + 6) + 4 = 4 + (−9 + 6)
Answer Commutative property of addition

Slide 1- 60

Adding Numbers with the Same Sign
To add two numbers that have the same sign, add their
absolute values and keep the same sign.


Slide 1- 61

Example 2
Add.
a. 27 + 12

b. –16 + (– 22)

Solution
a. 27 + 12 = 39
b. –16 + (–22) = –38


Slide 1- 62

Adding Numbers with Different Signs
To add two numbers that have different signs, subtract
the smaller absolute value from the greater absolute
value and keep the sign of the number with the greater
absolute value.


Slide 1- 63

Example 3
Add.
a. 35 + (–17)

b. –29 + 7

Solution
a. 35 + (–17) = 18
b. –29 + 7 = –22


Slide 1- 64

Example 3 continued
Add.
c. 15 + (–27)

d. –32 + 6

Solution
c. 15 + (–27) = –12
d. –32 + 6 = –26


Slide 1- 65

Objective 2
Add rational numbers.


Slide 1- 66

Adding Fractions with the Same Denominator
To add fractions with the same denominator, add the
numerators and keep the same denominator; then
simplify.


Slide 1- 67

Example 4
Add.
a. 2 + 4
9

b. − 4 +  − 5 

÷
12  12 

9

Solution
a. 2 + 4 = 2 + 4 = 6
9 9
9
9
2 g3 2
=
=
3 g3 3
Replace 6 and 9 with their prime
factorizations, divide out the
common factor, 3, then multiply
the remaining factors.

4  5
b. − +  − ÷
12  12 
−4 + ( −5 )
9
=
=−
12
12
3 g3
3
=−
=−
3 g2 g 2
4
Simplify to lowest terms by dividing
out the common factor, 3.


Slide 1- 68

Example 4 continued
Add.
c. 7 +  − 3 

÷
10  10 
Solution
a. 7 +  − 3  = 7 + (−3) = 4

÷
10
10
10  10 
2 g2 2
=
=
2 g5 5
Simplify to lowest terms by dividing
out the common factor, 2.


Slide 1- 69

Adding Fractions
To add fractions with different denominators:
1. Write each fraction as an equivalent fraction with
the LCD.
2. Add the numerators and keep the LCD.
3. Simplify.


Slide 1- 70

Example 5a
1 1
Add: +
3 4

Solution
1 1
+
3 4

1( 4 ) 1(3)
=
+
3 ( 4 ) 4(3)
4 3
= +
12 12
7
=
12

Write equivalent fractions
with 12 in the denominator.

Add numerators and keep
the common denominator.
Because the addends have
the same sign, we add and
keep the same sign.


Slide 1- 71

Example 5b
5 3
Add: − +
6 4

Solution
5 ( 2 ) 3(3)
5 3
+
− + =−
6 ( 2 ) 4(3)
6 4
10 9
=− +
12 12
−10 + 9
=
12
1
=−
12


Because the addends have
different signs, we subtract and
keep the sign of the number with
the greater absolute value.


Slide 1- 72

Example 5c
7 9
Add: − +
8 30

Solution
7 9
− +
8 30

7 ( 15 ) 9(4)
=−
+
8 ( 15 ) 30(4)


105 36
=−
+
120 120
−105 + 36
Reduce to lowest terms.
=
120
−69
3 ×23
23
=
=−
=−
120
2 ×2 ×2 × ×
3 5
40

Slide 1- 73

Example 6
Anna has a balance of $378.45 and incurs a debt of
$85.42. What is Anna’s new balance?
Solution
A debt of $85.42 is −$85.42. Her balance is
378.45 + (– 85.42) = $293.03


Slide 1- 74

Objective 3
Find the additive inverse of a
number.


Slide 1- 75

Additive inverses: Two numbers whose sum is 0.
What happens if we add two numbers that have the
same absolute value but different signs, such as
5 + (–5)? In money terms, this is like making a $5
payment towards a debt of $5. Notice the payment
pays off the debt so that the balance is 0.
5 + (–5) = 0
Because their sum is zero, we say 5 and –5 are
additive inverses, or opposites.

Slide 1- 76

Example 7
Find the additive inverse of the given number.
a. 8
b. –2
c. 0
Answers
a. –8 because 8 + (–8) = 0
b. 2 because – 2 + 2 = 0
c. 0 because 0 + 0 = 0

Slide 1- 77

Example 8
Simplify.
a. – (–5)

b. –|2|

c. –| –9|

Answers
a. – (–5) = 5
b. –|2| = –2
c. –| –9| = –9

Slide 1- 78

Objective 4
Subtract rational numbers.


Slide 1- 79

Parts of a subtraction statement:
8–5=3
Difference

Minuend
Subtrahend


Slide 1- 80

Rewriting Subtraction
To write a subtraction statement as an equivalent
addition statement, change the operation symbol
from a minus sign to a plus sign, and change the
subtrahend to its additive inverse.


Slide 1- 81

Example 9a
Subtract
a. –17 – (–5)
Solution
Write the subtraction as an equivalent addition.
–17 – (–5)
Change the operation
from minus to plus.

= –17 + 5
= –12

Change the subtrahend
to its additive inverse.


Slide 1- 82

Example 9b
3 1
Subtract: − −
8 4

Solution
3 1
3 1
− − =− −
8 4
8 4
3  1
= − +− ÷
8  4
3  1(2) 
= − +−
÷
8  4(2) 

Write equivalent fractions with
the common denominator, 8.

3  2
5
= − +− ÷ = −
8  8
8

Slide 1- 83

Example 9c
c. 4.07 – 9.03
Solution
Write the equivalent addition statement.
4.07 – 9.03
= 4.07 + (– 9.03)
= –4.96


Slide 1- 84

Example 10
In an experiment, a mixture begins at a temperature
of 52.6°C. The mixture is then cooled to a
temperature of −29.4°C. Find the difference between
the initial and final temperatures.
Solution
52.6 – (–29.4) = 52.6 + 29.4
= 82
Answer The difference between the initial and final
temperatures is 82°C.

Slide 1- 85

Add –6 + (–9).

a) –15
b) −3
c) 3
d) 15

1.3


Slide 1- 86

Add –6 + (–9).

a) –15
b) −3
c) 3
d) 15

1.3


Slide 1- 87

Subtract 5 – (–8).

a) –13
b) −3
c) 3
d) 13

1.3


Slide 1- 88

Subtract 5 – (–8).

a) –13
b) −3
c) 3
d) 13

1.3


Slide 1- 89

Subtract

a)

16
−
21

b)

1
−
2

c)

−

d)

1.3

3  1
− −  ÷.
7 3

2
21

2
21


Slide 1- 90

Subtract

a)

16
−
21

b)

1
−
2

c)

−

d)

1.3

3  1
− −  ÷.
7 3

2
21

2
21


Slide 1- 91

1.4
1.
2.
3.
4.
5.

Multiplying and Dividing Real Numbers;

Multiply integers.
Multiply more than two numbers.
Multiply rational numbers.
Find the multiplicative inverse of a number.
Divide rational numbers.


Objective 1
Multiply integers.


Slide 1- 93

In a multiplication statement, factors are
multiplied to equal a product.
2 g 3 =

Factors

6

Product


Slide 1- 94

Properties of
Multiplication
Multiplicative
Property of 0
Multiplicative
Identity

Symbolic Form

Word Form

0 ga = 0

The product of a number
multiplied by 0 is 0.

1 ga = a

The product of a number
multiplied by 1 is the
number.

Commutative
Property of
Multiplication

ab=ba

Changing the order of
factors does not affect the
product.

Associative
Property of
Multiplication

a(bc) = (ab)c

Changing the grouping of
three or more factors does
not affect the product.

Distributive
Property of
Multiplication
over Addition

a(b + c) =ab + ac

A sum multiplied by a
factor is equal to the sum
of that factor multiplied by
each addend.


Slide 1- 95

Example 1
Give the name of the property of multiplication that is
illustrated by each equation.
a. 6(−3) = −3 ⋅ 6
Answer Commutative property of multiplication
b. 3(−9 ⋅ 5) = [3(−9)] ⋅ 5
Answer Associative property of multiplication
c. 4(4 – 2) = 4 ⋅ 4 – 4 ⋅ 2
Answer Distributive property of multiplication over
addition

Slide 1- 96

Multiplying Two Numbers with Different Signs
When multiplying two numbers that have different
signs, the product is negative.


Slide 1- 97

Example 2
Multiply.
a. 7(–4)
Solution
a. 7(–4) = –28
b. (–15)3 = –45

b. (–15)3

Warning: Make sure you see the
difference between 7(–4), which
indicates multiplication, and 7 – 4,
which indicates subtraction.


Slide 1- 98

Multiplying Two Numbers with the Same Sign
When multiplying two numbers that have the same
sign, the product is positive.


Slide 1- 99

Example 3
Multiply.
a. –5(–9)

b. (–6)(–8)

Solution
a. –5(–9) = 45
b. (–6)(–8) = 48


Slide 1- 100

Objective 2
Multiply more than two numbers.


Slide 1- 101

Multiplying with Negative Factors
The product of an even number of negative factors
is positive, whereas the product of an odd number
of negative factors is negative.


Slide 1- 102

Example 4
Multiply.
a. (–1)(–3)(–6)(7)
Solution Because there are three negative factors (an
odd number of negative factors), the result is
negative. (–1)(–3)(–6)(7) = –126
b. (–2)(–4)(2)(–5)(–3)
Solution Because there are four negative factors(an
even number of negative factors), the result is
positive. (–2)(–4)(2)(–5)(–3) = 240


Slide 1- 103

Objective 3
Multiply rational numbers.


Slide 1- 104

Multiplying Fractions
a c ac
g =
, where b ≠ 0 and d ≠ 0.
b d bd


Slide 1- 105

Example 5a
3 4
Multiply − g  ÷.
5 9

Solution
3 4
3  2 g2 
− g  ÷= − g 
÷
5 9
5  3 g3 
4
=−
15

Divide out the common factor, 3.

Because we are multiplying two
numbers that have different signs,
the product is negative.


Slide 1- 106

Example 5b
6  6  12
Multiply − × − ÷×
15  16  15

Solution
6  6  12
2× 
3
2 ×  2 ×2 ×
3
3
− × − ÷×
=−
× −

÷×
15  16  15
3 ×  2 ×2 ×2 ×2  3 ×
5
5
3
=
25


Divide out the common factors.
Because there are an even
number of negative factors, the
product is positive.

Slide 1- 107

Multiplying Decimal Numbers
To multiply decimal numbers:
1. Multiply as if they were whole numbers.
2. Place the decimal in the product so that it has the
same number of decimal places as the total
number of decimal places in the factors.


Slide 1- 108

Example 6a
Multiply (–7.6)(0.04).
Solution
First, calculate the value and disregard signs for now.
0.04
2 places
7.6
+ 1 place
×
024
+0280
3 places
0.3 0 4
Answer –0.304

When we multiply two numbers with
different signs, the product is negative.

Slide 1- 109

Example 6b
Multiply (−3)(5.2)(1.4)(−6.1).
Solution
First, calculate the value and disregard signs for now.
Multiply from left to right.
(−3)(5.2)(1.4)(−6.1) = (15.6)(1.4)(−6.1)
= 21.84(6.1)
= 133.224
Answer 133.224

15.6 = (3)(5.2)
21.84 = 15.6(1.4)

The product of an even number of negative
factors is positive. The factors have a total of 3
decimal places, so the product has three
decimal places.

Slide 1- 110

Objective 4
Find the multiplicative inverse of a
number.


Slide 1- 111

Multiplicative inverses: Two numbers whose product is 1.
3
2
and 2 are multiplicative inverses
3

because their product is 1.
2 3 6
g = =1
3 2 6

Notice that to write a number’s multiplicative
inverse, we simply invert the numerator and
denominator. Multiplicative inverses are also known
as reciprocals.

Slide 1- 112

Example 7
Find the multiplicative inverse.
a. 2
b. − 1
7

c. −9

8

Answer
7
a. The multiplicative inverse is .
2

b. The multiplicative inverse is −8.
1
c. The multiplicative inverse is − .
9

Slide 1- 113

Objective 5
Divide rational numbers.


Slide 1- 114

Parts of a division statement:

8 ÷ 2 =

Dividend

4

Quotient
Divisor


Slide 1- 115

Dividing Signed Numbers
When dividing two numbers that have the same
sign, the quotient is positive.
When dividing two numbers that have different
signs, the quotient is negative.


Slide 1- 116

Example 8
Divide.
a. 56 ÷ (−8)

b. −72 ÷ ( −6 )

Solution
a. 56 ÷ (−8) = −7

b.−72 ÷ ( −6 ) = 12


Slide 1- 117

Division Involving 0

0 ÷ n = 0 when n ≠ 0.
n ÷ 0 is undefined when n ≠ 0.
0 ÷ 0 is indeterminate.


Slide 1- 118

Dividing Fractions
a c a d
÷ = g , where b ≠ 0, c ≠ 0, and d ≠ 0.
b d b c


Slide 1- 119

Example 9
3 4
Divide − ÷ .
10 5

Solution
3 4
3
5
− ÷ =−
g
10 5
10 4

3
5
=−
g
5 g2 2 g2
3
=−
8

Write an equivalent multiplication.

Divide out the common factor, 5.

Because we are dividing two numbers
that have different signs, the result is
negative.

Slide 1- 120

Dividing Decimal Numbers
To divide decimal numbers, set up a long division
and consider the divisor.
Case 1: If the divisor is an integer, divide as if the
dividend were a whole number and place the
decimal point in the quotient directly above its
position in the dividend.
Case 2: If the divisor is a decimal number,
1. Move the decimal point in the divisor to the
right enough places to make the divisor an
integer.
2. Move the decimal point in the dividend the same
number of places.

Slide 1- 121

Dividing Decimal Numbers continued
3. Divide the divisor into the dividend as if both
numbers were whole numbers. Make sure you
align the digits in the quotient properly.
4. Write the decimal point in the quotient directly
above its new position in the dividend.
In either case, continue the division process until
you get a remainder of 0 or a repeating digit (or
block of digits) in the quotient.


Slide 1- 122

Example 10
Divide −44.64 ÷ (−3.6)
Solution
Because the divisor is a decimal number, we move
the decimal point enough places to the right to create
an integer—in this case, one place. Then we move
the decimal point one place to the right in the
dividend. Because we are dividing two numbers
with the same sign, the result is positive.


Slide 1- 123

Example 10 continued
Divide −44.64 ÷ (−3.6)
Solution

12.4
36 446.4
36
86
72
144
144
0

Slide 1- 124

Example 11
Martha was decorating cookies. She used 2/3 of a
container of frosting that was 3/4 full. What
fractional part of the container did she use?
Solution
To find 2/3 of 3/4, multiply.
2 3
2×
3
1
× =
=
3 4 2 ×2 × 2
3


Slide 1- 125

Multiply (–6)(–3)(7).

a) 126
b) −126
c) –63
d) 63

1.4


Slide 1- 126

Multiply (–6)(–3)(7).

a) 126
b) −126
c) –63
d) 63

1.4


Slide 1- 127

Divide −14.6 ÷ ( −0.03) .

a) −48.6
b) 48.6
c) 486.6
d) −486.6

1.4


Slide 1- 128

Divide −14.6 ÷ ( −0.03) .

a) −48.6
b) 48.6
c) 486.6
d) −486.6

1.4


Slide 1- 129

1.5

Exponents, Roots, and Order of
Operations

1. Evaluate numbers in exponential form.
2. Evaluate square roots.
3. Use the order-of-operations agreement to simplify
numerical expressions.
4. Find the mean of a set of data.


Objective 1
Evaluate numbers in exponential
form.


Slide 1- 131

Sometimes problems involve repeatedly
multiplying the same number. In such problems,
we can use an exponent to indicate that a base
number is repeatedly multiplied.
Exponent: A symbol written to the upper
right of a base number that indicates how
many times to use the base as a factor.
Base: The number that is repeatedly
multiplied.


Slide 1- 132

When we write a number with an exponent, we
say the expression is in exponential form. The
24 is in exponential form, where the
expression
base is 2 and the exponent is 4. To evaluate 24 ,
write 2 as a factor 4 times, then multiply.
Four 2s

2 = 2 g 2 g 2 g 2 = 16
4

Base

Exponent


Slide 1- 133

Evaluating an Exponential Form
To evaluate an exponential form raised to a
natural number exponent, write the base as a
factor the number of times indicated by the
exponent; then multiply.


Slide 1- 134

Example 1a
Evaluate. (–9)2
Solution
The exponent 2 indicates we have two factors of –9.
Because we multiply two negative numbers, the result
is positive.
(–9)2 = (–9)(–9) = 81


Slide 1- 135

Example 1b
3

Evaluate.

 3
− ÷
 5

Solution
The exponent 3 means we must multiply the base by
itself three times.
3

 3
 3  3   3 
− ÷ =  − ÷ − ÷ − ÷

 5
 5  5   5 
27
=−
125

Slide 1- 136

Evaluating Exponential Forms with Negative
Bases
If the base of an exponential form is a negative
number and the exponent is even, then the
product is positive.
If the base is a negative number and the exponent is
odd, then the product is negative.


Slide 1- 137

Example 2
Evaluate.
(−3) 4
−34
(−2)3
a.
b.
c.
Solution
a. (−3)4 = (−3)(−3)(−3)(−3) = 81
b.

−34 = −3 ×3 ×3 ×3 = −81

c.

(−2)3 = ( −2)( −2)(−2) = −8

d.

d.

−23

−23 = −2 ×2 ×2 = −8

Slide 1- 138

Objective 2
Evaluate square roots.


Slide 1- 139

Roots are inverses of exponents. More
specifically, a square root is the inverse of a
square, so a square root of a given number is a
number that, when squared, equals the given
number.
Square Roots
Every positive number has two square roots, a
positive root and a negative root.
Negative numbers have no real-number square
roots.


Slide 1- 140

Example 3
Find all square roots of the given number.
Solution
a. 49
Answer ± 7
b. −81
Answer No real-number square roots exist.


Slide 1- 141

The symbol, , called the radical, is used to
indicate finding only the positive (or principal)
square root of a given number. The given number or
expression inside the radical is called the radicand.
Radical

Principal Square Root

25 = 5
Radicand


Slide 1- 142

Square Roots Involving the Radical Sign
The radical symbol denotes only the positive
(principal) square root.
a
=
b

a
b

, where a ≥ 0 and b > 0.


Slide 1- 143

Example 4
Evaluate the square root.
a. 169
b. 64

c.

81

0.64

d.

−25

Solution
a. 169 = 13

b.

c.

d. −25 = not a real number

0.64 = 0.8

64 8
=
81 9


Slide 1- 144

Objective 3
Use the order-of-operations
agreement to simplify numerical
expressions.


Slide 1- 145

Order-of- Operations Agreement
Perform operations in the following order:
1. Within grouping symbols: parentheses ( ),
brackets [ ], braces { }, above/below fraction
bars, absolute value | |, and radicals .
2. Exponents/Roots from left to right, in order as
they occur.
3. Multiplication/Division from left to right, in order
as they occur.
4. Addition/Subtraction from left to right, in order as
they occur.

Slide 1- 146

Example 5a
Simplify.

−26 + 15 ÷ (−5) ×2

Solution
−26 + 15 ÷ (−5) ×2
= −26 + (−3) ×2

Divide 15 ÷ (−5) = –3

= −26 + (−6)

Multiply (–3) ⋅ 2 = –6

= −32

Add –26 + (–6) = –32


Slide 1- 147

Example 5b
Simplify.
Solution

−34 + 2 12 − 20

−34 + 2 12 − 20
= −34 + 2 −8

Subtract inside the absolute value.

= −34 + 2 ×
8

Simplify the absolute value.

= −81 + 2 ×
8

Evaluate the exponent.

= −81 + 16

Multiply.

= −65

Add.


Slide 1- 148

Example 5c
Simplify. ( −3)

2

+ 5 6 − ( 2 + 1)  − 49



Solution ( −3) 2 + 5 6 − ( 2 + 1)  −


= ( −3) + 5 [ 6 − 3] − 49
2

= 9 + 5 ( 3) − 7

49
Calculate within the innermost
parenthesis.
Evaluate the exponential form,
brackets, and square root.

= 9 + 15 − 7

Multiply 5(3).

= 24 − 7

Add 9 + 15.

= 17

Subtract 24 – 7.


Slide 1- 149

Square Root of a Product or Quotient
If a square root contains multiplication or
division, we can multiply or divide first, then find
the square root of the result, or we can find the
square roots of the individual numbers, then
multiply or divide the square roots.
Square Root of a Sum or Difference
When a radical contains addition or subtraction, we
must add or subtract first, then find the root of the
sum or difference.


Slide 1- 150

Example 6a
Simplify.

13.5 ÷ 5 ( −4 ) − 3 142 − 21

Solution

13.5 ÷ 5 ( −4 ) − 3 142 − 21

2

2

= 13.5 ÷ 5 ( −4 ) − 3 121
2

= 13.5 ÷ 5 ( 16 ) − 3(11)

Subtract within the radical.
Evaluate the exponential form and
root.

= 2.7 ( 16 ) − 3 ( 11)

Divide.

= 43.2 − 33

Multiply.

= 10.2

Subtract.


Slide 1- 151

Sometimes fraction lines are used as grouping
symbols. When they are, we simplify the
numerator and denominator separately, then
divide the results.


Slide 1- 152

Example 7a
Simplify.
Solution

8(−5) − 23
4(8) − 8
8( −5) − 23
4(8) − 8
8(−5) − 8
=
4(8) − 8
=

−40 − 8
32 − 8

Evaluate the exponential form in
the numerator and multiply in the
denominator.
Multiply in the numerator and
subtract in the denominator.

−48
=
24

Subtract in the numerator.

= −2

Divide.


Slide 1- 153

Example 7b
Simplify.

9(4) + 12
43 + (8)(−8)
9(4) + 12
43 + (8)(−8)

Solution
=

36 + 12
64 + (8)(−8)

48
=
64 + (−64)
48
=
0

Because the denominator or divisor is 0, the answer is
undefined.


Slide 1- 154

Objective 4
Find the mean of a set of data.


Slide 1- 155

Finding the Arithmetic Mean
To find the arithmetic mean, or average, of n
numbers, divide the sum of the numbers by n.
Arithmetic mean =

x1 + x2 + ... + xn
n


Slide 1- 156

Example 8
Bruce has the following test scores in his biology
class: 92, 96, 81, 89, 95, 93. Find the average of his
test scores.
Solution
92 + 96 + 81 + 89 + 95 + 93
546
=
6
6

Divide the sum of the 6
scores by 6.

= 91


Slide 1- 157

Simplify using order of operations.

( −6 )

2

− 18 ÷ ( 9 − 6 )

a) − 18
b) 6
c) 30
d) 36

1.5


Slide 1- 158


( −6 )

2

− 18 ÷ ( 9 − 6 )

a) − 18
b) 6
c) 30
d) 36

1.5


Slide 1- 159

2 ( 4 + 23 )

a)

8
300

b)

250
361

c)

6 + 30 − ( 2 + 4 )

2

2
11

d) undefined

1.5


Slide 1- 160

2 ( 4 + 23 )

a)

8
300

b)

250
361

c)

6 + 30 − ( 2 + 4 )

2

2
11

d) undefined

1.5


Slide 1- 161

1.6

Translating Word Phrases to Expressions

1. Translate word phrases to expressions.


Objective 1
Translating word phrases to
Expressions


Slide 1- 163

Translating Basic Phrases
Addition

Translation

Subtraction

Translation

The sum of x
and three

x+3

The difference of x
and three

x–3

h plus k

h+k

h minus k

h–k

seven added to t

7+t

seven subtracted
from t

t–7

three more than
a number

n+3

three less than a
number

n–3

y increased by
two

y+2

y decreased by two

y–2

Note: Since addition is a
commutative operation, it does not
matter in what order we write the
translation.

Note: Subtraction is not a
commutative operation; therefore,
the way we write the translation
matters.


Slide 1- 164

Multiplication

Translation

The product of
x and three

Translation

The quotient of x
and three

hk

x ÷ 3 or x

h divided by k

3x

h times k

Division

h ÷ k or h

Twice a number

2n

h divided into k

Triple the
number

3n

The ratio of a to b

Two-thirds of a
number

3

k
k ÷ h or k
h
a ÷ b or a
b

2
n
3

Note: Like addition, multiplication
is a commutative operation: it does
not matter in what order we write
the translation.

Note: Division is like subtraction in
that it is not a commutative
operation; therefore, the way we
write the translation matters.


Slide 1- 165

Exponents

Translation

c squared

c2

The square of b

k3

The cube of b

b3

n to the fourth
power

n4

y raised to the
fifth power

The square root of
x

Translation

b2

k cubed

Roots

y5


x

Slide 1- 166

The key words sum, difference, product, and quotient
indicate the answer for their respective operations.
sum of x and 3

difference of x and 3

x+3

x–3

product of x and 3

quotient of x and 3
x÷3

x⋅3

Slide 1- 167

Example 1
Translate to an algebraic expression.
a. five more than two times a number
Translation: 5 + 2n or 2n + 5
b. seven less than the cube of a number
Translation: n3 – 7
c. the sum of h raised to the fourth power and twelve
Translation: h4 + 12

Slide 1- 168

Translating Phrases Involving Parentheses
Sometimes the word phrases imply an order of
operations that would require us to use parentheses in
the translation.
These situations arise when the phrase indicates that a
sum or difference is to be calculated before performing
a higher-order operation such as multiplication,
division, exponent, or root.


Slide 1- 169

Example 2
Translate to an algebraic expression.
a. seven times the sum of a and b
Translation: 7(a + b)
b. the product of a and b divided by the sum
of w2 and 4
ab
2
Translation: ab ÷ (w + 4) or
2
w +4


Slide 1- 170

Translate the phrase to an algebraic expression.
Twelve less than three times a number

a) 3n + 12
b) 12 – 3n
c) 3n – 12
d) 3n ⋅ 12

1.6


Slide 1- 171

Twelve less than three times a number

a) 3n + 12
b) 12 – 3n
c) 3n – 12
d) 3n ⋅ 12

1.6


Slide 1- 172

The difference of a and b decreased by the sum
of w and z

a) (a – b) – (w + z)
b) a – b – (w + z)
c) ab – (w + z)
d) (b – a) – (w + z)

1.6


Slide 1- 173

The difference of a and b decreased by the sum
of w and z

a) (a – b) – (w + z)
b) a – b – (w + z)
c) ab – (w + z)
d) (b – a) – (w + z)

1.6


Slide 1- 174

1.7

Evaluating and Rewriting Expressions

1. Evaluate an expression.
2. Determine all values that cause an expression to be
undefined.
3. Rewrite an expression using the distributive property.
4. Rewrite an expression by combining like terms.


Objective 1
Evaluate an expression.


Slide 1- 176

Evaluating an Algebraic Expression
To evaluate an algebraic expression:
1. Replace the variables with their corresponding
given values.
2. Calculate the numerical expression using the order
of operations.


Slide 1- 177

Example 1a
Evaluate 3w – 4(a – 6) when w = 5 and a = 7.
Solution
3w – 4(a − 6)
3(5) – 4(7 – 6)
= 3(5) – 4(1)
= 15 – 4
= 11

Replace w with 5 and a with 7.
Simplify inside the parentheses first.
Multiply.
Subtract.


Slide 1- 178

Example 1b
Evaluate x2 – 0.4xy + 9, when x = 7 and y = –2.
Solution
x2 – 0.4xy + 9
(7)2 – 0.4(7)(–2) + 9
= 49 – 0.4(7)(–2) + 9
= 49 – (–5.6) + 9
= 49 + 5.6 + 9
= 63.6

Replace x with 7 and y with –2.
Begin calculating by simplifying the exponential
form.
Multiply.
Write the subtraction as an equivalent addition.
Add from left to right.


Slide 1- 179

Objective 2
Determine all values that cause an
expression to be undefined.


Slide 1- 180

When evaluating a division expression in which the
divisor or denominator contains a variable or variables,
we must be careful about what values replace the
variable(s).
We often need to know what values could replace the
variable(s) and cause the expression to be undefined or
indeterminate.


Slide 1- 181

Example 2
Determine all values that cause the expression to be
undefined.
−2
a. 8
b.
( x + 2)( x − 9)

x+4

Answer
8
8
a. If x = −4, we have −4 + 4 = 0 , which is undefined
because the denominator is 0.
b. If x = −2 or 9 the fraction will be undefined since
the denominator will = 0.
−2
−2
=
( −2 + 2)(−2 − 9) 0

−2
−2
=
(9 + 2)(9 − 9) 0


Slide 1- 182

Objective 3
Rewrite an expression using the
distributive property.


Slide 1- 183

The Distributive Property of Multiplication over
Addition
a(b + c) = ab + ac

This property gives us an alternative to the order of
operations.
2(5 + 6) = 2(11)

2(5 + 6) = 2⋅5 + 2⋅6

= 22

= 10 + 12
= 22

Slide 1- 184

Example 3
Use the distributive property to write an equivalent
expression and simplify.
a. 3(x + 3)
b. –5(w – 4)
Solution
a. 3(x + 3) = 3 ⋅ x + 3 ⋅ 3
= 3x + 9
b. –5(w – 4) = –5 ⋅ w – (–5) ⋅ 4
= –5w + 20

Slide 1- 185

Objective 4
Rewrite an expression by combining
like terms.


Slide 1- 186

Terms: Expressions that are the addends in an
expression that is a sum.
Coefficient: The numerical factor in a term.
The coefficient of 5x3 is 5.
The coefficient of –8y is –8.
Like terms: Variable terms that have the same
variable(s) raised to the same exponents, or constant
terms.
Like terms
Unlike terms
4x and 7x
2x and 8y different variables
5y2 and 10y2
7t3 and 3t2 different exponents
8xy and 12xy
x2y and xy2 different exponents
7 and 15
13 and 15x different variables

Slide 1- 187

Combining Like Terms
To combine like terms, add or subtract the
coefficients and keep the variables and their
exponents the same.


Slide 1- 188

Example 4
Combine like terms.
a. 10y + 8y
Solution
10y + 8y = 18y
b. 8x – 3x
Solution

8x – 3x = 5x

c. 13y2 – y2
Solution

13y2 – y2 = 12y2

Slide 1- 189

Example 5
Combine like terms in 5y2 + 6 + 4y2 – 7.
Solution
5y2 + 6 + 4y2 – 7
= 5y2 + 4y2 + 6 – 7

Combine like terms.

= 9y2 – 1


Slide 1- 190

Example 6
Combine like terms in 18y + 7x – y – 7x.
Solution
18y + 7x – y – 7x
= 17y + 0
= 17y


Slide 1- 191

Example 7
1
1
Combine like terms in a + 4b + 3 + a − b.
12
6

Solution

1
1
a + 4b + 3 + a − b
12
6
1
1
= a + a + 4b − b + 3
Collect like terms.
12
6
1
1(2)
Write the fraction coefficients as
= a+
a + 4b − b + 3 equivalent fractions with their
12
6(2)
LCD, 12.
1
2
= a + a + 4b − b + 3
12
12
1
3
Combine like terms.
= a + 3b + 3 = a + 3b + 3
4
12
Slide 1- 192

Evaluate the expression 4(a + b) when
a = 3 and b = –2.
a) 4
b) −4
c) 12
d) 20

1.7


Slide 1- 193

Evaluate the expression 4(a + b) when
a = 3 and b = –2.
a) 4
b) −4
c) 12
d) 20

1.7


Slide 1- 194

For which values is the
expression undefined?

8m
(m + 2)(m − 5)

a) 8
b) −2
c) −2 and 5
d) 2 and −5

1.7


Slide 1- 195

For which values is the
expression undefined?

8m
(m + 2)(m − 5)

a) 8
b) −2
c) −2 and 5
d) 2 and −5

1.7


Slide 1- 196

Simplify: 7x + 8 – 2x – 4

a) 9x – 4
b) 9x + 4
c) 5x – 4
d) 5x + 4

1.7


Slide 1- 197

Simplify: 7x + 8 – 2x – 4

a) 9x – 4
b) 9x + 4
c) 5x – 4
d) 5x + 4

1.7


Slide 1- 198

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