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1: Numerical and Algebraic Expressions

# 1: Numerical and Algebraic Expressions

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1.1 • Numerical and Algebraic Expressions

3

We can modify the listing approach if the number of elements is large. For example, all of the

letters of the alphabet can be listed as

{a, b, c, . . . , z}

We begin by simply writing enough elements to establish a pattern, then the three dots indicate that the set continues in that pattern. The final entry indicates the last element of the pattern. If we write

{1, 2, 3, . . .}

the set begins with the counting numbers 1, 2, and 3. The three dots indicate that it continues

in a like manner forever; there is no last element.

A set that consists of no elements is called the null set (written л). Two sets are said to

be equal if they contain exactly the same elements. For example,

{1, 2, 3} ϭ {2, 1, 3}

because both sets contain the same elements; the order in which the elements are written

does not matter. The slash mark through the equality symbol denotes not equal to. Thus if

A ϭ {1, 2, 3} and B ϭ {1, 2, 3, 4}, we can write A ϶ B, which we read as “set A is not equal

to set B.”

Simplifying Numerical Expressions

Now let’s simplify some numerical expressions that involve the set of whole numbers, that

is, the set {0, 1, 2, 3, . . .}.

Classroom Example

Simplify 2 ϩ 6 Ϫ 3 ϩ 7 ϩ 11 Ϫ 9.

EXAMPLE 1

Simplify 8 ϩ 7 Ϫ 4 ϩ 12 Ϫ 7 ϩ 14.

Solution

The additions and subtractions should be performed from left to right in the order that they

appear. Thus 8 ϩ 7 Ϫ 4 ϩ 12 Ϫ 7 ϩ 14 simplifies to 30.

Classroom Example

Simplify 5(8 ϩ 6).

EXAMPLE 2

Simplify 7(9 ϩ 5).

Solution

The parentheses indicate the product of 7 and the quantity 9 ϩ 5. Perform the addition

inside the parentheses first and then multiply; 7(9 ϩ 5) thus simplifies to 7(14), which

becomes 98.

Classroom Example

Simplify (5 ϩ 11) Ϭ (8 Ϫ 4).

EXAMPLE 3

Simplify (7 ϩ 8) Ϭ (4 Ϫ 1).

Solution

First, we perform the operations inside the parentheses; (7 ϩ 8) Ϭ (4 Ϫ 1) thus becomes

15 Ϭ 3, which is 5.

7ϩ8

. We don’t need

4Ϫ1

parentheses in this case because the fraction bar indicates that the sum of 7 and 8 is to be

divided by the difference, 4 Ϫ 1. A problem may, however, contain parentheses and fraction

bars, as the next example illustrates.

We frequently express a problem like Example 3 in the form

4

Chapter 1 • Some Basic Concepts of Arithmetic and Algebra

EXAMPLE 4

Classroom Example

(6 Ϫ 3)(4 ϩ 1)

8

.

Simplify

ϩ

5

13 Ϫ 5

Simplify

(4 ϩ 2)(7 Ϫ 1)

4

.

ϩ

9

7Ϫ3

Solution

First, simplify above and below the fraction bars, and then proceed to evaluate as follows.

(4 ϩ 2)(7 Ϫ 1)

(6)(6)

4

4

ϩ

ϭ

ϩ

9

7Ϫ3

9

4

ϭ

EXAMPLE 5

Classroom Example

Simplify 4 # 7 ϩ 3.

36

ϩ1ϭ4ϩ1ϭ5

9

Simplify 7 и 9 ϩ 5.

Solution

If there are no parentheses to indicate otherwise, multiplication takes precedence over addition. First perform the multiplication, and then do the addition; 7 и 9 ϩ 5 therefore simplifies

to 63 ϩ 5, which is 68.

Remark: Compare Example 2 to Example 5, and note the difference in meaning.

Classroom Example

Simplify 6 Ϫ 10 Ϭ 5 ϩ 4

EXAMPLE 6

# 5.

Simplify 8 ϩ 4 и 3 Ϫ 14 Ϭ 2.

Solution

The multiplication and division should be done first in the order that they appear, from left to

right. Thus 8 ϩ 4 и 3 Ϫ 14 Ϭ 2 simplifies to 8 ϩ 12 Ϫ 7. We perform the addition and subtraction in the order that they appear, which simplifies 8 ϩ 12 Ϫ 7 to 13.

Classroom Example

Simplify 3 # 8 Ϭ 6 ϩ 5 # 2 Ϫ

21 Ϭ 3 ϩ 8 Ϭ 4 # 3.

EXAMPLE 7

Simplify 8 и 5 Ϭ 4 ϩ 7 и 3 Ϫ 32 Ϭ 8 ϩ 9 Ϭ 3

и 2.

Solution

When we perform the multiplications and divisions first in the order that they appear and then

do the additions and subtractions, our work takes on the following format.

8 и 5 Ϭ 4 ϩ 7 и 3 Ϫ 32 Ϭ 8 ϩ 9 Ϭ 3 и 2 ϭ 10 ϩ 21 Ϫ 4 ϩ 6 ϭ 33

Classroom Example

Simplify 9 ϩ 3[5(2 ϩ 4)].

EXAMPLE 8

Simplify 5 ϩ 6[2(3 ϩ 9)] .

Solution

We use brackets for the same purpose as parentheses. In such a problem we need to simplify

from the inside out; perform the operations inside the innermost parentheses first.

5 ϩ 6[2(3 ϩ 9) ] ϭ 5 ϩ 6[2(12) ]

ϭ 5 ϩ 6[24]

ϭ 5 ϩ 144

ϭ 149

Let’s now summarize the ideas presented in the previous examples regarding simplifying numerical expressions. When simplifying a numerical expression, use the following

order of operations.

1.1 • Numerical and Algebraic Expressions

5

Order of Operations

1. Perform the operations inside the symbols of inclusion (parentheses and brackets)

and above and below each fraction bar. Start with the innermost inclusion symbol.

2. Perform all multiplications and divisions in the order that they appear, from left to

right.

3. Perform all additions and subtractions in the order that they appear, from left to

right.

Evaluating Algebraic Expressions

We can use the concept of a variable to generalize from numerical expressions to algebraic

expressions. Each of the following is an example of an algebraic expression.

3x ϩ 2y

5a Ϫ 2b ϩ c

7(w ϩ z)

5d ϩ 3e

2c Ϫ d

2xy ϩ 5yz

(x ϩ y) (x Ϫ y)

An algebraic expression takes on a numerical value whenever each variable in the expression is replaced by a specific number. For example, if x is replaced by 9 and z by 4, the algebraic expression x Ϫ z becomes the numerical expression 9 Ϫ 4, which simplifies to 5. We

say that x Ϫ z “has a value of 5” when x equals 9 and z equals 4. The value of x Ϫ z, when x

equals 25 and z equals 12, is 13. The general algebraic expression x Ϫ z has a specific value

each time x and z are replaced by numbers.

Consider the following examples, which illustrate the process of finding a value of an

algebraic expression. We call this process evaluating algebraic expressions.

Classroom Example

Find the value of 5x ϩ 4y, when x is

replaced by 3 and y by 13.

EXAMPLE 9

Find the value of 3x ϩ 2y , when x is replaced by 5 and y by 17.

Solution

The following format is convenient for such problems.

3x ϩ 2y ϭ 3(5) ϩ 2(17) when x ϭ 5 and y ϭ 17

ϭ 15 ϩ 34

ϭ 49

Note that in Example 9, for the algebraic expression 3x ϩ 2y , the multiplications “3 times x”

and “2 times y” are implied without the use of parentheses. Substituting the numbers switches

the algebraic expression to a numerical expression, and then parentheses are used to indicate the

multiplication.

Classroom Example

Find the value of 11m Ϫ 5n, when

m ϭ 4 and n ϭ 7.

EXAMPLE 10

Find the value of 12a Ϫ 3b, when a ϭ 5 and b ϭ 9.

Solution

12a Ϫ 3b ϭ 12(5) Ϫ 3(9) when a ϭ 5 and b ϭ 9

ϭ 60 Ϫ 27

ϭ 33

6

Chapter 1 • Some Basic Concepts of Arithmetic and Algebra

Classroom Example

Evaluate 6xy Ϫ 3xz ϩ 5yz,

when x ϭ 2, y ϭ 5, and z ϭ 3.

Evaluate 4xy ϩ 2xz Ϫ 3yz, when x ϭ 8, y ϭ 6, and z ϭ 2.

EXAMPLE 11

Solution

4xy ϩ 2xz Ϫ 3yz ϭ 4(8)(6) ϩ 2(8)(2) Ϫ 3(6)(2) when x ϭ 8, y ϭ 6, and z ϭ 2

ϭ 192 ϩ 32 Ϫ 36

ϭ 188

Classroom Example

6a ϩ b

Evaluate

for a ϭ 4 and b ϭ 6.

4a Ϫ b

EXAMPLE 12

Evaluate

Solution

5(12) ϩ 4

5c ϩ d

ϭ

3c Ϫ d

3(12) Ϫ 4

ϭ

Classroom Example

Evaluate (4x ϩ y)(7x Ϫ 2y),

when x ϭ 2 and y ϭ 5.

5c ϩ d

for c ϭ 12 and d ϭ 4.

3c Ϫ d

for c ϭ 12 and d ϭ 4

60 ϩ 4 64

ϭ

ϭ2

36 Ϫ 4 32

EXAMPLE 13

Evaluate (2x ϩ 5y)(3x Ϫ 2y), when x ϭ 6 and y ϭ 3.

Solution

(2x ϩ 5y)(3x Ϫ 2y) ϭ (2 ؒ 6ϩ 5 ؒ 3)(3 ؒ 6 Ϫ 2 ؒ 3) when x ϭ 6 and y ϭ 3

ϭ (12 ϩ 15)(18 Ϫ 6)

ϭ (27)(12)

ϭ 324

Concept Quiz 1.1

For Problems 1–10, answer true or false.

1. The expression “ab” indicates the sum of a and b.

2. Any of the following notations, (a)b, a ؒ b, a(b), can be used to indicate the product of a

and b.

3. The phrase 2x ϩ y Ϫ 4z is called “an algebraic expression.”

4. A set is a collection of objects, and the objects are called “terms.”

5. The sets {2, 4, 6, 8} and {6, 4, 8, 2} are equal.

6. The set {1, 3, 5, 7, . . . } has a last element of 99.

7. The null set has one element.

8. To evaluate 24 Ϭ 6 ؒ 2, the first operation that should be performed is to multiply

6 times 2.

9. To evaluate 6 ϩ 8 ؒ 3, the first operation that should be performed is to multiply

8 times 3.

10. The algebraic expression 2(x ϩ y) simplifies to 24 if x is replaced by 10, and y is

replaced by 0.

1.1 • Numerical and Algebraic Expressions

7

Problem Set 1.1

For Problems 1– 34, simplify each numerical expression.

40. x ϩ 8y ϩ 5xy

(Objective 2)

41. 14xz ϩ 6xy Ϫ 4yz

1. 9 ϩ 14 Ϫ 7

2. 32 Ϫ 14 ϩ 6

3. 7(14 Ϫ 9)

4. 8(6 ϩ 12)

5. 16 ϩ 5 ؒ 7

6. 18 Ϫ 3(5)

7. 4(12 ϩ 9) Ϫ 3(8 Ϫ 4)

8. 7(13 Ϫ 4) Ϫ 2(19 Ϫ 11)

9. 4(7) ϩ 6(9)

10. 8(7) Ϫ 4(8)

11. 6 ؒ 7 ϩ 5 ؒ 8 Ϫ 3 ؒ 9

12. 8(13) Ϫ 4(9) ϩ 2(7)

13. (6 ϩ 9)(8 Ϫ 4)

14. (15 Ϫ 6)(13 Ϫ 4)

15. 6 ϩ 4[3(9 Ϫ 4)]

16. 92 Ϫ 3[2(5 Ϫ 2)]

17. 16 Ϭ 8 ؒ 4 ϩ 36 Ϭ 4 ؒ 2

8 ϩ 12

9 ϩ 15

Ϫ

19.

4

8

19 Ϫ 7

38 Ϫ 14

ϩ

20.

6

3

21. 56 Ϫ [3(9 Ϫ 6)]

22. 17 ϩ 2[3(4 Ϫ 2)]

23. 7 ؒ 4 ؒ 2 Ϭ 8 ϩ 14

24. 14 Ϭ 7 и 8 Ϫ 35 Ϭ 7 и 2

25. 32 Ϭ 8 ؒ 2 ϩ 24 Ϭ 6 Ϫ 1

43.

n

54

ϩ

n

3

44.

n

n

60

Ϫ

ϩ

n

4

6

45.

y ϩ 16

50 Ϫ y

ϩ

6

3

46.

w ϩ 57

90 Ϫ w

ϩ

9

7

for n ϭ 9

for n ϭ 12

for y ϭ 8

for w ϭ 6

48. (x ϩ 2y)(2x Ϫ y) for x ϭ 7 and y ϭ 4

49. (5x Ϫ 2y)(3x ϩ 4y) for x ϭ 3 and y ϭ 6

50. (3a ϩ b)(7a Ϫ 2b) for a ϭ 5 and b ϭ 7

51. 6 ϩ 3[2(x ϩ 4)] for x ϭ 7

52. 9 ϩ 4[3(x ϩ 3)] for x ϭ 6

54. 78 Ϫ 3[4(n Ϫ 2)] for n ϭ 4

27. 4 ؒ 9 Ϭ 12 ϩ 18 Ϭ 2 ϩ 3

28. 5 ؒ 8 Ϭ 4 Ϫ 8 Ϭ 4 ؒ 3 ϩ 6

12(7 Ϫ 4)

6(8 Ϫ 3)

ϩ

3

9

For Problems 55– 60, find the value of

6(21 Ϫ 9)

32. 78 Ϫ

4

и6ϩ5и3ϩ7и9ϩ6и5

7ϩ2и3

3и5ϩ8и2

9и6Ϫ4

7и8ϩ4

ϩ

34.

5 и 8 Ϫ 10

6 и 5 Ϫ 20

4

55. b ϭ 8 and h ϭ 12

56. b ϭ 6 and h ϭ 14

57. b ϭ 7 and h ϭ 6

58. b ϭ 9 and h ϭ 4

59. b ϭ 16 and h ϭ 5

60. b ϭ 18 and h ϭ 13

h(b1 ϩ b2 )

for

2

each set of values for the variables h, b1, and b2. (Subscripts are

used to indicate that b1 and b2 are different variables.)

For Problems 61– 66, find the value of

61. h ϭ 17, b1 ϭ 14, and b2 ϭ 6

62. h ϭ 9, b1 ϭ 12, and b2 ϭ 16

For Problems 35–54, evaluate each algebraic expression for

the given values of the variables. (Objective 3)

63. h ϭ 8, b1 ϭ 17, and b2 ϭ 24

35. 7x ϩ 4y

for x ϭ 6 and y ϭ 8

65. h ϭ 18, b1 ϭ 6, and b2 ϭ 11

36. 8x ϩ 6y

for x ϭ 9 and y ϭ 5

66. h ϭ 14, b1 ϭ 9, and b2 ϭ 7

37. 16a Ϫ 9b

for a ϭ 3 and b ϭ 4

38. 14a Ϫ 5b

for a ϭ 7 and b ϭ 9

39. 4x ϩ 7y ϩ 3xy

bh

for each set of val2

ues for the variables b and h. (Objective 3)

3(17 Ϫ 9)

9(16 Ϫ 7)

ϩ

30.

4

3

33.

for x ϭ 7, y ϭ 3, and z ϭ 2

53. 81 Ϫ 2[5(n ϩ 4)] for n ϭ 3

26. 48 Ϭ 12 ϩ 7 ؒ 2 Ϭ 2 Ϫ 1

4(12 Ϫ 7)

31. 83 Ϫ

5

42. 9xy Ϫ 4xz ϩ 3yz

for x ϭ 8, y ϭ 5, and z ϭ 7

47. (x ϩ y)(x Ϫ y) for x ϭ 8 and y ϭ 3

18. 7 ؒ 8 Ϭ 4 Ϫ 72 Ϭ 12

29.

for x ϭ 12 and y ϭ 3

for x ϭ 4 and y ϭ 9

64. h ϭ 12, b1 ϭ 14, and b2 ϭ 5

67. You should be able to do calculations like those in Problems

1– 34 with and without a calculator. Be sure that you can do

Problems 1– 34 with your calculator, and make use of the

parentheses key when appropriate.

8

Chapter 1 • Some Basic Concepts of Arithmetic and Algebra

Thoughts Into Words

68. Explain the difference between a numerical expression

and an algebraic expression.

69. Your friend keeps getting an answer of 45 when simplifying 3 ϩ 2(9). What mistake is he making and how

would you help him?

Further Investigations

Grouping symbols can affect the order in which the arithmetic operations are performed. For the following problems,

insert parentheses so that the expression is equal to the given

value.

71. Insert parentheses so that 36 ϩ 12 Ϭ 3 ϩ 3 ϩ 6

equal to 50.

# 2 is

72. Insert parentheses so that 36 ϩ 12 Ϭ 3 ϩ 3 ϩ 6

equal to 38.

# 2 is

70. Insert parentheses so that 36 ϩ 12 Ϭ 3 ϩ 3 ϩ 6

equal to 20.

# 2 is

73. Insert parentheses so that 36 ϩ 12 Ϭ 3 ϩ 3 ϩ 6

equal to 55.

# 2 is

5. True

6. False

1. False

2. True

3. True

4. False

9. True

10. False

1.2

7. False

8. False

Prime and Composite Numbers

OBJECTIVES

1

Identify whole numbers greater than one as prime or composite

2

Factor a whole number into a product of prime numbers

3

Find the greatest common factor of two or more whole numbers

4

Find the least common multiple of two or more whole numbers

Occasionally, terms in mathematics are given a special meaning in the discussion of a particular topic. Such is the case with the term “divides” as it is used in this section. We say that

6 divides 18, because 6 times the whole number 3 produces 18; but 6 does not divide 19,

because there is no whole number such that 6 times the number produces 19. Likewise,

5 divides 35, because 5 times the whole number 7 produces 35; 5 does not divide 42, because

there is no whole number such that 5 times the number produces 42. We present the following

general definition.

Deﬁnition 1.1

Given that a and b are whole numbers, with a not equal to zero, a divides b if and only if there

exists a whole number k such that a и k ϭ b.

Remark: Notice the use of variables, a, b, and k, in the statement of a general definition. Also

note that the definition merely generalizes the concept of divides, which was introduced in the

specific examples prior to the definition.

1.2 • Prime and Composite Numbers

9

The following statements further clarify Definition 1.1. Pay special attention to the italicized words, because they indicate some of the terminology used for this topic.

1.

2.

3.

4.

5.

6.

8 divides 56, because 8 и 7 ϭ 56.

7 does not divide 38, because there is no whole number, k, such that 7 и k ϭ 38.

3 is a factor of 27, because 3 и 9 ϭ 27.

4 is not a factor of 38, because there is no whole number, k, such that 4 и k ϭ 38.

35 is a multiple of 5, because 5 и 7 ϭ 35.

29 is not a multiple of 7, because there is no whole number, k, such that 7 и k ϭ 29.

We use the factor terminology extensively. We say that 7 and 8 are factors of 56 because

7 и 8 ϭ 56; 4 and 14 are also factors of 56 because 4 и 14 ϭ 56. The factors of a number

are also divisors of the number.

Now consider two special kinds of whole numbers called prime numbers and composite

numbers according to the following definition.

Deﬁnition 1.2

A prime number is a whole number, greater than 1, that has no factors (divisors) other than

itself and 1. Whole numbers, greater than 1, which are not prime numbers, are called composite numbers.

The prime numbers less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, and 47.

Notice that each of these has no factors other than itself and 1. The set of prime numbers is an

infinite set; that is, the prime numbers go on forever, and there is no largest prime number.

We can express every composite number as the indicated product of prime numbers—

also called the prime factored form of the number. Consider the following examples.

4ϭ2

и2

6ϭ2

и3

8ϭ2

и2и2

10 ϭ 2

и5

12 ϭ 2 и 2 и 3

In each case we expressed a composite number as the indicated product of prime numbers.

There are various procedures to find the prime factors of a given composite number. For

our purposes, the simplest technique is to factor the given composite number into any two easily recognized factors and then to continue to factor each of these until we obtain only prime

factors. Consider these examples.

18 ϭ 2 и 9 ϭ 2 и 3

24 ϭ 4 и 6 ϭ 2 и 2

и3

и2и3

27 ϭ 3 и 9 ϭ 3 и 3 и 3

150 ϭ 10 и 15 ϭ 2 и 5 и 3

и5

It does not matter which two factors we choose first. For example, we might start by expressing 18 as 3 и 6 and then factor 6 into 2 и 3, which produces a final result of 18 ϭ 3 и 2 и 3.

Either way, 18 contains two prime factors of 3 and one prime factor of 2. The order in which

we write the prime factors is not important.

Greatest Common Factor

We can use the prime factorization form of two composite numbers to conveniently find their

greatest common factor. Consider the following example.

42 ϭ 2 и 3

70 ϭ 2 и 5

и7

и7

Notice that 2 is a factor of both, as is 7. Therefore, 14 (the product of 2 and 7) is the greatest common factor of 42 and 70. In other words, 14 is the largest whole number that divides

both 42 and 70. The following examples should further clarify the process of finding the

greatest common factor of two or more numbers.

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