copyright © 2009 pearson education, inc. publishing as pearson addison-wesley numbers, variables,...

Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Numbers, Variables, and Expressions

Natural Numbers and Whole Numbers

Prime Numbers and Composite Numbers

Variables, Algebraic Expressions, and Equations

Translating Words to Expressions

1.1

Natural Numbers and Whole Numbers

The set of natural numbers are also known as the counting numbers.

1, 2, 3, 4, 5, 6,…Because there are infinitely many natural numbers, three dots are used to show that the list continues in the same pattern without end.The whole numbers can be expressed as

0, 1, 2, 3, 4, 5, …

Slide 3Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Prime Numbers and Composite Numbers

When two natural numbers are multiplied, the result is another natural number.The product of 6 and 7 is 42.

6 7 = 42The numbers 6 and 7 are factors of 42.A prime number has only itself and 1 as factors.A natural number greater than 1 that is not prime is a composite number.Any composite number can be written as a product of prime numbers.


Prime Factorization

The prime factorization of 120.


120 2 2 2 3 5


EXAMPLE

Solution

Classifying numbers as prime or composite

Classify each number as prime or composite. If a number is composite, write it as a product of prime numbers.a. 37 b. 3 c. 45 d. 300

a. 37 The only factors of 37 are 1 and itself. The number is prime.

b. 3 The only factors of 3 are 1 and itself. The number is prime.

c. 45 Composite because 9 and 5 are factors. Prime factorization: 32 5


EXAMPLE

Solution

Classifying numbers as prime or composite

Classify each number as prime or composite. If a number is composite, write it as a product of prime numbers.a. 37 b. 3 c. 45 d. 300

d. 300 Prime factorization

300

30 10

6 5 2 5

2 3 300 2 2 3 5 5

Variables, Algebraic Expressions, and Equations

Variables are often used in mathematics when tables of numbers are inadequate. A variable is a symbol, typically an italic letter used to represent an unknown quantity.An algebraic expression consists of numbers, variables, operation symbols, such as +, , , and , and grouping symbols, such as parentheses.An equation is a mathematical statement that two algebraic expressions are equal.A formula is a special type of equation that expresses a relationship between two or more quantities.



EXAMPLE

Solution

Evaluating algebraic expressions with one variable

Evaluate each algebraic expression for x = 6.a. x + 4 b. 4x c. 20 – x d.

a. x + 4

( 4)x

x

6 + 4 = 10

b. 4x

4(6) = 24

c. 20 – x

20 – 6 = 14 d.

( 4)x

x 4

6

(6 )

6

32


EXAMPLE Evaluating algebraic expressions with two variables

Solution

Evaluate each algebraic expression for y = 3 and z = 9a. 5yz b. z – y c.

a. 5yz

5(3)(9) = 135

b. z – y

9 – 3 = 6

z

y

c. z

y3

9

3


EXAMPLE

Solution

Evaluating formulas

Find the value of y for x = 20 and z = 5. a. y = x + 4 b. y = 9xz

a. y = x + 4

y = 20 + 4

b. y = 9xz

y = 9(20)(5)

= 24 = 900


EXAMPLE

Solution

Translating words to expressions

Translate each phrase to an algebraic expression.a. Twice the cost of a bookb. Ten less than a numberc. The product of 8 and a number

a. Twice the cost of a book

b. Ten less than a number

c. The product of 8 and a number

2c where c is the cost of the book

n – 10 where n is the number

8n where n is the number


EXAMPLE

Solution

Finding the area of a rectangle

The area A of a rectangle equals its length L times its width W.a. Write a formula that shows the relationship between these three quantities.b. Find the area of a yard that is 100 feet long and 75 feet wide.

a. The word times indicates the length and width should be multiplied. The formula is A = LW.

b. A = LW

= (100)(75)

= 7500 square feet


Fractions

Basic Concepts

Simplifying Fractions to Lowest Terms

Multiplication and Division of Fractions

Addition and Subtraction of Fractions

Applications

1.2

Basic Concepts

The parts of a fraction are named as follows.


7

8Fraction bar

Numerator

Denominator


EXAMPLE

Solution

Identifying numerators and denominators

Give the numerator and denominator of each fraction.

a. b. c.

a. The numerator is 8 and the denominator is 19.

b. The numerator is mn, and the denominator is p.

8

19

mn

p 7

c d

f

c. The numerator is c + d, and the denominator is f – 7.

Simplifying Fractions to Lowest Terms

When simplifying fractions, we usually factor out the greatest common factor (GCF) for the numerator and the denominator. The greatest common factor is the largest factor common to both the numerator and the denominator.



EXAMPLE

Solution

Finding the greatest common factor

Find the greatest common factor (GCF) for each pair of numbers.

a. 14, 21 b. 42, 90

a. Because 14 = 7 ∙ 2 and 21 = 7 ∙ 3, the number 7 is the largest factor that is common to both 14 and 21. Thus the GCF of 14 and 21 is 7.


EXAMPLE continued

b. When working with larger numbers, one way to determine the greatest common factor is to find the prime factorization of each number.

42 = 6 ∙ 7 = 2 ∙ 3 ∙ 7 and

90 = 6 ∙ 15 = 2 ∙ 3 ∙ 3 ∙ 5

The prime factorizations have one 2 and one 3 in common. Thus the GCF for 42 and 90 is 6.


EXAMPLE

Solution

Simplifying fractions to lowest terms

Simplify each fraction to lowest terms.a. b. c.

a. The GCF of 9 and 15 is 3.

9

15

20

2845

135

9

15 5

3

3

3

3

5

b. 20

28 7

4

4

5

The GCF of 20 and 28 is 4.

5

7

c. The GCF of 45 and 135 is 45. 45

135

145

5 34

1

3

Multiplication of Fractions



EXAMPLE

Solution

Multiplying fractions

Multiply each expression and simplify the result when appropriate.

a. b. c.

a.

3 4

7 9

316

4

5m

n r

b.

c.

3 4

7 9

3 4

7 9

12

63

3

4

1

3

2

4

21

316

4

16 3

1 4

16 3

1 4

48

4

4

12 4 12

5m

n r

5m

n r

5m

nr


EXAMPLE

Solution

Finding fractional parts

Find each fractional part.

a. One-third of one-fourthb. One half of three-fourths

a.

b.

1 1

3 4

1 1

3 4

1

12

1 3

2 4 1 3

2 4

3

8

Division of Fractions



EXAMPLE

Solution

Dividing fractions

Divide each expression.

a. b. c.

a.

1 2

5 3

927

2

6

d f

g

b.

c.

27 2

1 9

27 2

1 9

54

9

9

6

1

9

6

6

d g

f

1 2

5 3

1 3

5 2

1 3

5 2

3

10

927

2

6

d f

g

6

d g

f

6

dg

f

Fractions with Like Denominators



EXAMPLE

Solution

Adding and subtracting fractions with common denominators

Add or subtract as indicated. Simplify your answer to lowest terms when appropriate.

a. b.

a.

7 2

11 11

17 11

18 18

b.

7

11 1

2

1

1

7

1

2

1

9

1

8

7

18

1 11

1

1

17

8

11

8

6

1

6 1The fraction can be simplified to .

18 3

Fractions with Unlike Denominators



EXAMPLE

Solution

Rewriting fractions with the LCD

Rewrite each set of fractions using the LCD.a. b.

a. The LCD is 24

2 3,

3 8

1 4 9, ,

8 5 10

82 8 2 16

3 8 3 248

b. The LCD is 40.

33 3 3 9

8 3 8 3 24

51 5 1 5

8 5 8 5 40

84 8 4 32

5 8 5 408

49 4 9 36

10 4 10 404


EXAMPLE

Solution

Adding and subtracting fractions with unlike denominators

Add or subtract as indicated. Simplify your answer to lowest terms when appropriate.

a. b.

a.

5 1

6 9

4 1

5 2

b.

3 2

3 2

5 1

6 9

15 2

18 18

17

18

2 5

2 5

4 1

5 2

8 5

10 10

3

10


EXAMPLE

Solution

Applying fractions to carpentry

A pipe measures inches long and needs to be cut into three equal pieces. Find the length of each piece.

336

8

Begin by writing as the improper fraction 3

368

291.

8

2913

8

291 1

8 3

291 1, or 12 inches

24 8


Exponents and Order of Operations

Natural Number Exponents

Order of Operations


1.3

Natural Number Exponents

The area of a square equals the length of one of its sides times itself. If the square is 5 inches on a side, then its area is

5 5 = 52 = 25 square inches

The expression 52 is an exponential expression with base 5 and exponent 2.


Base

Exponent


EXAMPLE Writing products in exponential notation

Write each product as an exponential expression.

a.

b.

c.

8 8 8 8 8 58

2 2 2 2

3 3 3 3

y y y y y y

42

3

6y


EXAMPLE Evaluating exponential notation

Evaluate each expression.

a.

b.

45 5 5 5 5 625

32

3

2 2 2 8

3 3 3 27

Order of Operations



EXAMPLE

Solution

Evaluating arithmetic expressions

Evaluate each expression by hand.a. 12 – 6 – 2 b. 12 – (6 – 2) c.

a. 12 – 6 – 2

8

3 3

6 – 2

4

b. 12 – (6 – 2)

12 – 4

8

c. 8

3 38 4

6 3


EXAMPLE

Solution

Evaluating arithmetic expressions

Evaluate each expression.a. b. c.

a.

24 3

8 2

15 – 6

9

b. c.

15 2 3 3 4 2 (8 1)

15 2 3 3 4 2 )8 1( 3 4 2 7

3 8 7

11 7

4

24

2

3

8

4 9

8 2

13

6


EXAMPLE

Solution

Writing and evaluating expressions

Write each expression and then evaluate it. a. Two to the fifth power plus threeb. Twenty-four less two times four

a. Two to the fifth power plus three

b. Twenty-four less two times four

52 3 2 2 2 2 2 3 32 3 35

24 2 4 24 8 16


Real Numbers and the Number Line

Signed Numbers

Integers and Rational Numbers

Square Roots

Real and Irrational Numbers

The Number Line

Absolute Value

Inequality

1.4

Signed Numbers

The opposite, or additive inverse, of a number a is −a.



EXAMPLE

Solution

Finding opposites (or additive inverses)

Find the opposite of each expression.

a. 29 b. c. d. −(−13)

a. The opposite of 29 is −29.

b. The opposite of is

c.

9

11

68

2

9

11

9.

116 6

8 8 3 5, so the opposite of 8 is 5.2 2

d. −(−13) = 13, so the opposite of −(−13) is −13.


EXAMPLE

Solution

Finding an additive inverse (or opposite)

Find the additive inverse of –x, if x = .

The additive inverse of −x is x = because −(−x) = x by the double negative rule.

4

9

4

9

Integers and Rational Numbers

The integers include the natural numbers, zero, and the opposite of the natural numbers.

…,−2, −1, 0, 1, 2,…


A rational number is any number that can be expressed as the ratio of two integers, where q ≠ 0.

p

q


EXAMPLE

Solution

Classifying numbers

Classify each number as one or more of the following: natural number, whole number, integer, or rational number.

a. b. −9 c.

a. Because , the number is a natural number, whole number, integer, and rational number.

b. The number −9 is an integer and rational number, but not a natural number or a whole number.

c. The fraction is a rational number because it is the ratio of two integers. However it is not a natural number, a whole number, or an integer.

21

3

15

7

217

3

21

3

15

7

Square Roots

Square roots are frequently used in algebra. The number b is a square root of a number a if b ∙ b = a. Every positive number has one positive square root and one negative square root.



EXAMPLE

Solution

Calculating principal square roots

Evaluate each square root. Approximate to three decimal places when appropriate.

a. b. c.

a. because 8 ∙ 8 = 64 and 8 is nonnegative.

b. because 13 ∙ 13 = 169 and 13 is nonnegative.

c. is a number between 4 and 5. We can estimate the value of with a calculator.

64 169 23

64 8

169 13

2323 23 4.796

Real and Irrational Numbers



EXAMPLE

Solution

Classifying numbers

Identify the natural numbers, whole numbers, integers, rational numbers, and irrational numbers in the following list.

Natural numbers:

175.7,4, , 25, 7, and 23

9

4 and 25 5Whole numbers: 4 and 25 5

Integers:4, 25 5,and 23

Rational numbers:17

5.7,4, , 25 5, and 239

Irrational numbers: 7


EXAMPLE

Solution

Plotting numbers on a number line

Plot each real number on a number line.

a. b. c.

a. Plot a dot halfway between −2 and −3.

b. Plot a dot between 2 and 3.

c. Plot a dot halfway between 3 and 4.

5

2 7 7

2

52.5

2

7 2.65

73.5

2

Absolute Value

The absolute value of a real number equals its distance on the number line from the origin. Because distance is never negative, the absolute value of a real number is never negative.



EXAMPLE

Solution

Finding the absolute value of a real number

Write the expression without the absolute value sign.

a. b. c. d.

a. because the distance between the origin and −9 is 9.

b. because the distance is 0 between the origin and 0.

c.

9 0 16 y

9 9

0 0

16 16

d. y y


EXAMPLE

Solution

Ordering real numbers

List the following numbers from least to greatest. Then plot these numbers on a number line.

4, , 3, and 2.4

4, 3, 2.4, and


Addition and Subtraction of Real Numbers

Addition of Real Numbers

Subtraction of Real Numbers

Applications

1.5

There are four arithmetic operations: addition, subtraction, multiplication, and division.




In an addition problem the two numbers added are called addends, and the answer is called the sum. 5 + 8 = 135 and 8 are the addends13 is the sumThe opposite (or additive inverse) of a real number a is a.


EXAMPLE Adding Opposites

Find the opposite of each number and calculate the sum of the number and its opposite.

a.

b.

78 The opposite is 78.

Sum = 78 ( 78) 0

3

4 3

The opposite is .4

3 3Sum: 0

4 4


EXAMPLE Adding real numbers

Evaluate each expression.

a.

b.

3 ( 8) 11

3 9

4 10

3 15

4 209 18

10 20

The numbers are both negative, add the absolute values. The sign would be negative as well.

15 18 3

20 20 20

Subtraction of Real Numbers



EXAMPLE

Solution

Subtracting real numbers

Find each difference by hand.a. 12 – 16 b. –6 – 2

a. 12 – 16

12 + (–16)

4

b. –6 – 2

–6 + (–2)

8


EXAMPLE

Solution

Adding and subtracting real numbers

Evaluate each expression.a. b.

a. b.

6 7 ( 8) 2 6.3 5.8 10.4

6 7 ( 8) 2 ( 7) 286

1 8 2

9

6.3 5.8 10.4

6.3 5.8 ( 10.4)

0.5 ( 10.4)

10.9


EXAMPLE

Solution

Balancing a checking account

The initial balance in a checking account is $326. Find the final balance if the following represents a list of withdrawals and deposits:$20, $15, $200, and $150

Find the sum of the five numbers.

326 ( 20) ( 15) 200 ( 150)

306 200 ( 15) ( 150)

506 ( 165) 341

The final balance is $341.


Multiplication and Division of Real Numbers

Multiplication of Real Numbers

Division of Real Numbers

Applications

1.6

Multiplication of Real Numbers



EXAMPLE

Solution

Multiplying real numbers

Find each product by hand.

a. −4 ∙ 8 b. c. d.

b. The product is positive because both factors are positive.

c. Since both factors are negative, the product is positive.

4 3

9 7 3.4 60

d.

4.5 6 5 3

a. The resulting product is negative because the factors have unlike signs. Thus −4 ∙ 8 = −32.

4 3 12 4

9 7 63 21

3.4 60 204

4.5 6 5 3 27 5 3 135 3 405


EXAMPLE

Solution

Evaluating real numbers with exponents

Evaluate each expression by hand.

a. (−6)2 b. −62

b. This is the negation of an exponential expression with base 6. Evaluating the exponent before negative results in −62 = −(6)(6) = −36.

a. Because the exponent is outside of parentheses, the base of the exponential expression is −6. The expression is evaluated as (−6)2 = (−6)(−6) = 36.

Division of Real Numbers



EXAMPLE

Solution

Dividing real numbers

Evaluate each expression by hand.

a. b. c. d.

b.

c.

124

3 6

52

d. 4 ÷ 0 is undefined. The number 0 has no reciprocal.

4 0

a.

25 2 2 1 2 1

66 5 5 6 30 15

25

6

1 24 3 7224 72

3 1 1 1

6 1 6 36

52 52 52 26


EXAMPLE

Solution

Converting fractions to decimals

Convert the measurement to a decimal number.

a. Begin by dividing 5 by 16.

52 -inch washer

16

16 5.0000− 48

20− 16

40− 32

80−80

0

0.3125

Thus the mixed number

is equivalent to 2.3125.

52

16


EXAMPLE

Solution

Converting decimals to fractions

Convert each decimal number to a fraction in lowest terms.

a. 0.32 b. 0.875

b. The decimal 0.875 equals eight hundred seventy-five thousandths.

a. The decimal 0.32 equals thirty-two hundredths. 32 8 4 8

100 25 4 25

875 7 125 7

1000 8 125 8


EXAMPLE

Solution

Application

After surveying 125 pediatricians, 92 stated that they had admitted a patient to the children’s hospital in the last month for pneumonia. Write the fraction as a decimal.

92

125

92 8 7360.736

125 8 1000

One method for writing the fraction as a decimal is to divide 92 by 125 using long division. An alternative method is to multiply the fractions by so the denominator becomes 1000. Then, write the numerator in the thousandths place in the decimal.

8,

8


Properties of Real Numbers

Commutative Properties

Associative Properties

Distributive Properties

Identity and Inverse Properties

Mental Calculations

1.7


Commutative PropertiesThe commutative property for addition states that two numbers, a and b, can be added in any order and the result will be the same.

6 + 8 = 8 + 6

The commutative property for multiplication states that two numbers, a and b, can be multiplied in any order and the result will be the same.

9 4 = 4 9


EXAMPLE Applying the commutative properties

Use the commutative properties to rewrite each expression.

a.

b.

c.

72 56 can be written as 56 72

4b can be written as 4 b

( )d e g ( ) ( )

( )

d d

g d

e g e

e

g

The associative property allows us to change how numbers are grouped.


Associative Properties


EXAMPLE Applying the associative properties

Use the associative property to rewrite each expression.

a.

b.

(7 8) 9 7 (8 9)

( )a bc ( )ab c


EXAMPLE Identifying properties of real numbers

State the property that each equation illustrates.

a.

b.

7 (4 ) (7 4)w w

8 3 3 8

Associative property of multiplication because the grouping of the numbers has been changed.

Commutative property for addition because the order of the numbers has changed.

Distributive Properties


The distributive properties are used frequently in algebra to simplify expressions.

7(3 + 8) = 7 3 + 7 8

The 7 must be multiplied by both the 3 and the 8.


EXAMPLE

Solution

Applying the distributive properties

Apply a distributive property to each expression.a. 4(x + 3) b. –8(b – 5) c. 12 – (a + 2)

a. 4(x + 3)

= 4 x + 4 3

= 4x + 12

b. –8(b – 5)

c. 12 – (a + 2)

= 12 + (1)(a + 2)

= 8 b (8) 5

= 8b + 40

= 12 + (1) a + (–1) 2= 12 a – 2

= 10 a


EXAMPLE

Solution

Inserting parentheses using the distributive property

Use the distributive property to insert parentheses in the expression and then simplify the result.a. b.

a. b.

8 4a a 5 9y y

8 4a a(8 4)a

12a

5 9y y

( 5 9)y

4y


EXAMPLE Identifying properties of real numbers

State the property or properties illustrated by each equation.a.

b.

3(8 ) 24 3y y

(7 ) 8 15w w

Distributive property .

Commutative and associative properties for addition. (7 ) 8 (7 8)

15

w w

w

The identity property of 0 states that if 0 is added to any real number a, the result is a. The number 0 is called the additive identity.

3 + 0 = 3

The identity property of 1 states that if any number a is multiplied by 1, the result is a. The number 1 is called the multiplicative identity.

4 1 = 4




EXAMPLE Identifying identity and inverse properties

State the property or properties illustrated by each equation.a.

b.

0 ab ab

( ) 3 0 3 3a a

Identity property for 0.

Additive inverse property and the identity property for 0.


EXAMPLE Performing calculations mentally

Use the properties of real numbers to calculate each expression mentally.a.

b.

32 16 8 4

1 3 44

4 4 3

1632 8 4 32 8( )4) 6(1

40 20 60

3 4

4 3

14

4

14

4

3 4

4 3

1 1 1


Simplifying and Writing Algebraic Expressions

Terms

Combining Like Terms

Simplifying Expressions

Writing Expressions

1.8

Terms


A term is a number, a variable, or a product of numbers and variables raised to powers. Examples of terms include

4, z, 5x, and −6xy2.

The coefficient of a term is the number that appears in the term.


EXAMPLE

Solution

Identifying terms

Determine whether each expression is a term. If it is a term, identify its coefficient.

a. 97 b. 17x c. 4a – 6b d. 9y2

b. The product of a number and a variable is a term. The coefficient is 17.

c. The difference of two terms in not a term.

d. The product of a number and a variable with an exponent is a term. Its coefficient is 9.

a. A number is a term. The coefficient is 97.


EXAMPLE

Solution

Identifying like terms

Determine whether the terms are like or unlike.

a. 9x, −15x b. 16y2, 1 c. 5a3, 5b3 d. 11, −8z

b. The term 1 has no variable and the 16 has a variable of y2. They are unlike terms.

c. The variables are different, so they are unlike terms.

d. The term 11 has no variable and the −8 has a variable of z. They are unlike terms.

a. The variable in both terms is x, with the same power of 1, so they are like terms.


EXAMPLE

Solution

Combining like terms

Combine terms in each expression, if possible.

a. −2y + 7y b. 4x2 – 6x

b. They are unlike terms, so they can not be combined.

a. Combine terms by applying the distributive property. −2y + 7y = (−2 + 7)y = 5y


EXAMPLE

Solution

Simplifying expressions

Simplify each expression.

a. 13 + z – 9 + 7z b. 9x – 2(x – 5)

b. a. 13 + z – 9 + 7z

= 13 +(– 9) + z + 7z

= 13 +(– 9) + (1+ 7)z

= 4 + 8z

9x – 2(x – 5)

= 9x + (– 2)x + (−2)(– 5)

= 9x – 2x + 10

= 7x + 10


EXAMPLE

Solution

Simplifying expressions

Simplify each expression.

a. 6x2 – y + 9x2 – 3y b.

b. a. 6x2 – y + 9x2 – 3y

= 6x2 + 9x2 + (–1y) + (–3y)

= (6 + 9)x2 + (–1+ (– 3))y

= 15x2 –4y

18 6

3

a

18 6

3

a

18 6

3 3

a

6 2a


EXAMPLE

Solution

Writing and simplifying an expression

A sidewalk has a constant width w and comprises several short sections with lengths 11, 4, and 18 feet.

a. Write and simplify an expression that gives the number of square feet of sidewalk.

b. Find the area of the sidewalk if its width is 3 feet.

a. 11w + 4w + 18w = (11 + 4 + 18)w = 33w

b. 33w = 33 ∙ 3 = 99 square feet

11 ft

4 ft

18 ft

w

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