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
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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.
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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
Answers to the Concept Quiz
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.