Download Introductory Information and Review - Chapter 1

SECTION 1.1 Numbers
Chapter 1
Introductory Information and Review
Section 1.1:
Numbers
 Types of Numbers
 Order on a Number Line
Types of Numbers
Natural Numbers:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Example:
Solution:
Even/Odd Natural Numbers:
2
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SECTION 1.1 Numbers
Whole Numbers:
Example:
Solution:
Integers:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Example:
Solution:
Even/Odd Integers:
Example:
Solution:
4
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SECTION 1.1 Numbers
Rational Numbers:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Irrational Numbers:
6
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SECTION 1.1 Numbers
Real Numbers:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Note About Division Involving Zero:
Additional Example 1:
Solution:
8
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SECTION 1.1 Numbers
Additional Example 2:
Solution:
Natural Numbers:
Whole Numbers:
Integers:
Prime/Composite Numbers:
Positive/Negative Numbers:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Even/Odd Numbers:
Rational Numbers:
10
University of Houston Department of Mathematics
SECTION 1.1 Numbers
Additional Example 3:
Solution:
Natural Numbers:
Whole Numbers:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Integers:
Prime/Composite Numbers:
Positive/Negative Numbers:
Even/Odd Numbers:
Rational Numbers:
12
University of Houston Department of Mathematics
SECTION 1.1 Numbers
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
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SECTION 1.1 Numbers
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Order on a Number Line
The Real Number Line:
Example:
Solution:
16
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SECTION 1.1 Numbers
Inequality Symbols:
The following table describes additional inequality symbols.
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Example:
Solution:
Example:
Solution:
Additional Example 1:
Solution:
18
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SECTION 1.1 Numbers
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Additional Example 3:
Solution:
20
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SECTION 1.1 Numbers
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
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University of Houston Department of Mathematics
Exercise Set 1.1: Numbers
State whether each of the following numbers is prime,
composite, or neither. If composite, then list all the
factors of the number.
1.
2.
(a) 8
(d) 7
(b) 5
(e) 12
(c) 1
(a) 11
(d) 0
(b) 6
(e) 2
(c) 15
7.
1
9
(b)
2
9
(d)
4
9
(e)
5
9
3
9
(c)
Note:
3
9

1
3
8.
Notice the pattern above and use it as a
shortcut in (f)-(m) to write the following
fractions as decimals without performing
long division.
4.
(f)
6
9
(i)
9
9
(l)
25
9
(g)
7
9
(j)
10
9
(m)
29
9
(b) 0.6
(c)
4
(e) 
5
(h) 10
(f)  9
8
(i) 0
(k) 0.03003000300003…
8
9
(k)
14
9
Note:

Use the patterns from the problem above to
change each of the following decimals to either a
proper fraction or a mixed number.
(a) 0.4
(b) 0.7
(c) 2.3
(d) 1.2
(e) 4.5
(f) 7.6
Positive
Natural
Irrational
Negative
Whole
Real
Odd
Composite
Rational
Positive
Natural
Irrational
Negative
Whole
Real
Odd
Composite
Rational
Positive
Natural
Irrational
Negative
Whole
Real
Odd
Composite
Rational
Positive
Natural
Irrational
Negative
Whole
Real
2
9.
2
3
Odd
Composite
Rational
0.7
Even
Prime
Integer
Undefined
(h)
6
9
9
Even
Prime
Integer
Undefined
In (a)-(e), use long division to change the
following fractions to decimals.
(a)

1.3
(d)
4.7
(g) 3.1
7
(j)
9
(a)
Circle all of the words that can be used to describe
each of the numbers below.
Answer the following.
3.
6.
Even
Prime
Integer
Undefined
10. 
4
7
Even
Prime
Integer
Undefined
Answer the following.
State whether each of the following numbers is
rational or irrational. If rational, then write the
number as a ratio of two integers. (If the number is
already written as a ratio of two integers, simply
rewrite the number.)
5.
3
7
(a) 0.7
(b)
5
(c)
(d) 5
(e)
16
(f) 0.3
(g) 12
(h)
2.3
3.5
(j)  4
(k) 0.04004000400004...
(i)
e
MATH 1300 Fundamentals of Mathematics
11. Which elements of the set
8,  2.1,  0.4, 0,
7,  ,
15
4

, 5, 12 belong
to each category listed below?
(a) Even
(c) Positive
(e) Prime
(g) Natural
(i) Integer
(k) Rational
(m) Undefined
(b)
(d)
(f)
(h)
(j)
(l)
Odd
Negative
Composite
Whole
Real
Irrational
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Exercise Set 1.1: Numbers
12. Which elements of the set
6.25,  4
3
4
,  3,  5,  1,
2
5

, 1, 2, 10
belong to each category listed below?
(a) Even
(c) Positive
(e) Prime
(g) Natural
(i) Integer
(k) Rational
(m) Undefined
(b)
(d)
(f)
(h)
(j)
(l)
Odd
Negative
Composite
Whole
Real
Irrational
19. Find a real number that is not a rational number.
20. Find a whole number that is not a natural
number.
21. Find a negative integer that is not a rational
number.
22. Find an integer that is not a whole number.
23. Find a prime number that is an irrational number.
24. Find a number that is both irrational and odd.
Fill in each of the following tables. Use “Y” for yes if
the row name applies to the number or “N” for no if it
does not.
Answer True or False. If False, justify your answer.j
25. All natural numbers are integers.
13.
26. No negative numbers are odd.
25
0
1
5 3
10
55
13.3
27. No irrational numbers are even.
Undefined
Natural
Whole
Integer
Rational
Irrational
Prime
Composite
Real
28. Every even number is a composite number.
29. All whole numbers are natural numbers.
30. Zero is neither even nor odd.
31. All whole numbers are integers.
14.
32. All integers are rational numbers.
2.36
0
0
5
2
2
2
7
93
Undefined
Natural
Whole
Integer
Rational
Irrational
Prime
Composite
Real
33. All nonterminating decimals are irrational
numbers.
34. Every terminating decimal is a rational number.
Answer the following.
35. List the prime numbers less than 10.
Answer the following. If no such number exists, state
“Does not exist.”
36. List the prime numbers between 20 and 30.
15. Find a number that is both prime and even.
37. List the composite numbers between 7 and 19.
16. Find a rational number that is a composite
number.
38. List the composite numbers between 31 and 41.
17. Find a rational number that is not a whole
number.
40. List the odd numbers between
39. List the even numbers between 13 and
97 .
29 and
123 .
18. Find a prime number that is negative.
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University of Houston Department of Mathematics
Exercise Set 1.1: Numbers
Fill in the appropriate symbol from the set
41.
7 ______ 7
42.
3 ______
 , ,   .
3
43.  7 ______ 7
44.
3 ______  3
45.
81 ______ 9
46. 5 ______  25
47. 5.32 ______
53
10
48.
7
______ 0.07
100
49.
1
3
______
1
4
50.
1
6
______
1
5
51. 
1
1
______ 
3
4
1
1
52.  ______ 
6
5
53.
15 ______ 4
54.
7 ______
49
55. 3 ______  9
56.
29 ______ 5
58. Find the multiplicative inverse of the following
numbers. If undefined, write “undefined.”
(a) 3
(b) 4
(c) 1
(d)  23
(e) 2 73
59. Find the multiplicative inverse of the following
numbers. If undefined, write “undefined.”
(a) 2
(b) 95
(c) 0
(d) 1 53
(e) 1
60. Find the additive inverse of the following
numbers. If undefined, write “undefined.”
(a) 2
(b) 95
(c) 0
(d) 1 53
(e) 1
61. Place the correct number in each of the following
blanks:
(a) The sum of a number and its additive
inverse is _____. (Fill in the correct
number.)
(b) The product of a number and its
multiplicative inverse is _____. (Fill in the
correct number.)
62. Another name for the multiplicative inverse is
the ____________________.
Order the numbers in each set from least to greatest
and plot them on a number line.
(Hint: Use the approximations 2  1.41 and
3  1.73 .)
0
9

63. 1,  2, 0.4, ,  ,
5
4


0.49 

2


64.   3 , 1 , 0.65 , ,  1.5 , 0.64 
3


Answer the following.
57. Find the additive inverse of the following
numbers. If undefined, write “undefined.”
(a) 3
(b) 4
(c) 1
3
2
(d)  3
(e) 2 7
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Section 1.2:
Integers
 Operations with Integers
Operations with Integers
Absolute Value:
26
University of Houston Department of Mathematics
SECTION 1.2 Integers
Addition of Integers:
Example:
Solution:
Subtraction of Integers:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Example:
Solution:
Multiplication of Integers:
Example:
Solution:
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SECTION 1.2 Integers
Division of Integers:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Additional Example 1:
Solution:
30
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SECTION 1.2 Integers
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Additional Example 3:
32
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SECTION 1.2 Integers
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Additional Example 4:
Solution:
34
University of Houston Department of Mathematics
Exercise Set 1.2: Integers
Evaluate the following.
1.
2.
10. (a) 1(7)
(a) 3  7
(d) 3  ( 7)
(b) 3  (7)
(e) 3  0
(c) 3  7
(a) 8  5
(d) 8  ( 5)
(b) 8  5
(e) 0  ( 5)
(c) 8  (5)
(d) 0( 7)
(g)
7
1
(j) 7(1)(1)
3.
4.
5.
6.
(a) 0  4
(d) 4  0
(b) 4  0
(a) 6  0
(d) 6  0
(b) 0  ( 6)
(a) 10  2
(d) 2  (10)
(g) 2  10
(b) 10  (2)
(e) 2  ( 10)
(h) 10  ( 2)
(a) 7  (9)
(d) 9  ( 7)
(g) 7  ( 9)
(b) 7  9
(e) 9  (7)
(f) 9  7
(c) 0  6
(c) 10  2
(f) 2  10
(c) 7  9
(f) 9  7
 , ,   .
7.
(a) 1(4) ____ 0
(c) 5(1)(2) ____ 0
(b) 7(2) ____ 0
(d) 3(1)(0) ____ 0
8.
(a) 3(2) ____ 0
(c) 5(0)( 2) ____ 0
(b) 7( 1) ____ 0
(d) 2(2)(2) ___ 0
Evaluate the following. If undefined, write
“Undefined.”
(d) 6( 1)
6
0
6(1)
(e)
(g) 6( 1)
(h)
(a) 6(0)
(j)
6
0
(b)
6
1
7
1
(c) 7( 1)
(e) 1( 7)
(f)
0
7
(i)
(h)
(k) 7(0)( 1)
(l)
(c) 0  ( 4)
Fill in the appropriate symbol from the set
9.
(b)
0
6
(f) 6( 1)
(c)
(i)
(k) 6(1)(1) (l)
6
1
0
6
MATH 1300 Fundamentals of Mathematics
11. (a) 10(2)
(d)
10
2
12. (a)
6
3
(d) 6(3)
10
2
10
(e)
2
(b)
0
7
7
0
7
0
(c) 10(2)
(f)
10
2
(b) 6( 3)
(c)
6
3
(e) 6(3)
(f)
6
3
13. (a) 2( 3)( 4) (b) (2)(3)( 4)
(c) 1(2)(3)(4)
(d) 1(2)(3)(4)
14. (a) 3(2)(5)
(b) 3(2)(5)
(c) 3(2)(1)(5)
(d) 3(2)(2)( 5)
(b) 8  (2)
(c) 8(2)
(e) 8  ( 2)
(f) (8)(0)
(g) 8(1)
(h) 8  1
(i)
(j) 0  8
(k) 2  ( 8)
(l)
15. (a) 8  2
(d)
(m)
8
2
2
8
(n)
2
0
8
1
0
8
(o) 2  8
12
3
(d) 3  12
(b) 12(3)
(c) 12  3
(e) 0( 3)
(f) 0  ( 3)
(g) (3)(12)
(h)
16. (a)
12
1
(i)
3
0
1(12)
(j)
3
12
(k) 1  ( 3)
(l)
(m)
0
3
(n) 3  ( 1)
(o) 3(1)
35
CHAPTER 1 Introductory Information and Review
Section 1.3:
Fractions
 Greatest Common Divisor and Least Common Multiple
 Addition and Subtraction of Fractions
 Multiplication and Division of Fractions
Greatest Common Divisor and Least Common Multiple
Greatest Common Divisor:
36
University of Houston Department of Mathematics
SECTION 1.3 Fractions
A Method for Finding the GCD:
Least Common Multiple:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
A Method for Finding the LCM:
Example:
Solution:
38
University of Houston Department of Mathematics
SECTION 1.3 Fractions
The LCM is
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
The LCM is 2  2  2  3  5  120 .
Additional Example 2:
Solution:
The LCM is 2  3  3  5  7  630 .
40
University of Houston Department of Mathematics
SECTION 1.3 Fractions
Additional Example 3:
Solution:
The LCM is 2  2  3  3  2  72 .
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Additional Example 4:
Solution:
The LCM is 2  3  3  2  5  180 .
42
University of Houston Department of Mathematics
SECTION 1.3 Fractions
Addition and Subtraction of Fractions
Addition and Subtraction of Fractions with Like Denominators:
a b a b
 
c c
c
and
a b a b
 
c c
c
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Addition and Subtraction of Fractions with Unlike
Denominators:
44
University of Houston Department of Mathematics
SECTION 1.3 Fractions
Example:
Solution:
Additional Example 1:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Solution:
Additional Example 2:
46
University of Houston Department of Mathematics
SECTION 1.3 Fractions
Solution:
Additional Example 3:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Solution:
(b) We must rewrite the given fractions so that they have a common denominator.
Find the LCM of the denominators 14 and 21 to find the least common denominator.
48
University of Houston Department of Mathematics
SECTION 1.3 Fractions
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Multiplication and Division of Fractions
Multiplication of Fractions:
50
University of Houston Department of Mathematics
SECTION 1.3 Fractions
Example:
Solution:
Division of Fractions:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Additional Example 1:
Solution:
52
University of Houston Department of Mathematics
SECTION 1.3 Fractions
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Additional Example 3:
Solution:
54
University of Houston Department of Mathematics
SECTION 1.3 Fractions
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
55
Exercise Set 1.3: Fractions
For each of the following groups of numbers,
(a) Find their GCD (greatest common divisor).
(b) Find their LCM (least common multiple).
1.
6 and 8
2.
4 and 5
3.
7 and 10
4.
12 and 15
5.
14 and 28
6.
6 and 22
7.
8 and 20
8.
9 and 18
9.
18 and 30
19. (a)
2 75
(b)
5 23
(c)
12 14
20. (a)
4 19
(b)
11 54
(c)
9 73
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as a mixed number.)
21. (a) 2  1
7 7
(b)
8 4 3
 
11 11 11
22. (a)
3 1

5 5
(b)
4 5 2
 
9 9 9
23. (a)
8 54  2 15
(b)
7 23

3 3
24. (a)
3 21

5 5
(b)
7 116  5 112
25. (a)
5 34  2 14
(b)
6 53  7 54
26. (a)
9 75  2 73
(b)
4  115
27. (a)
7  23
(b)
7 103  3 109
28. (a)
11
6 127  2 12
(b)
8 16  2 56
10. 60 and 210
11. 16, 20, and 24
12. 15, 21, and 27
Change each of the following improper fractions to a
mixed number.
23
5
19
3
13. (a) 9
7
(b)
14. (a) 10
3
17
(b)
6
15. (a)  27
4
(b) 
32
11
(c) 
73
10
16. (a)  15
13
(b) 
43
8
(c) 
57
7
(c)
49
(c)
9
Change each of the following mixed numbers to an
improper fraction.
56
17. (a)
5 16
(b)
7 94
(c)
8 23
18. (a)
3 12
(b)
10 78
(c)
6 53
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as a mixed number.)
29. (a)
1 1

4 2
(b)
1 1

3 7
30. (a)
1 1

8 10
(b)
1 1

6 5
31. (a)
1 1 1
 
4 5 6
(b)
2 3

7 5
32. (a)
1 1 1
 
2 7 5
(b) 
33. (a)
1 1

35 10
(b)
4 3

11 7
3 5

4 6
University of Houston Department of Mathematics
Exercise Set 1.3: Fractions
8 7

15 12
1 1

6 24
(b)
35. (a)
4 73  5 16
(b)
7 107  5 12
36. (a)
10 75  3 14
(b)
6 121  4 83
34. (a)
37. (a)
7 53  8 74
(b)
49. (a) 5 
7 14  3 65
(b)
2 78  9 13
24
39. (a)
5 152  2 127
(b)
9 167  2 56
40. (a)
7 109  6 85
(b)
11 145  43
(b)
8
4
3
10
(c)
7
5
50. (a)
3
6
11
4
 8
(b) 20     (c) 22 
9
 5
51. (a)
12 18

35 7
(b)
52. (a)
1
4
5
16
(b) 
5 94  1 23
38. (a)
1
20
 53
 95
36 9

5 50
(c)
15 5

16 24
(c)
49 35

24 32
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as a mixed number.)
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as an improper
fraction.)
53. (a)
8 54    1077 
(b)
1 78    109 
54. (a)
 2 92    43 
(b)
 3 167    54 
41. (a)
2 3

9 4
(b)
4 8

15 9
55. (a)
 2 13   5 71 
(b)
 6 53    2 113 
42. (a)
7 9

16 10
(b)
11 17

14 35
56. (a)
 3 17    5 14 
(b)
5 53    2 1211 
43. (a)
5  13
(b)
7  23
57. (a) 5 85  2 14
   
(b)
 11 19   1 1718 
44. (a)
9  52
(b)
6  72
58. (a)
 4 54   1 75 
(b)
 2 115    2 221 
Evaluate the following. Write all answers in simplest
form. (If the answer is a mixed number/improper
fraction, then write the answer as an improper
fraction.)
45. (a) 5 
1
3
(b) 21 
5
6
(c) 16 
5
4
46. (a) 8 
3
7
(b) 24 
1
18
(c) 25 
11
10
47. (a)
1 25

7 11
(b) 
48. (a)
36  1 
  
25  8 
(b)
3 16
10  9 

    (c)
20 15
21  8 
8 7

19 3
(c)
1 42

14 5
MATH 1300 Fundamentals of Mathematics
57
CHAPTER 1 Introductory Information and Review
Section 1.4:
Exponents and Radicals
 Evaluating Exponential Expressions
 Square Roots
Evaluating Exponential Expressions
Two Rules for Exponential Expressions:
Example:
58
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
Solution:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
59
CHAPTER 1 Introductory Information and Review
Additional Properties for Exponential Expressions:
Two Definitions:
Quotient Rule for Exponential Expressions:
Exponential Expressions with Bases of Products:
Exponential Expressions with Bases of Fractions:
Example:
Evaluate each of the following:
(a) 2 3
(b)
59
56
2
(c)  
5
3
Solution:
60
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics
61
CHAPTER 1 Introductory Information and Review
Additional Example 1:
Solution:
62
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
Additional Example 2:
Solution:
MATH 1300 Fundamentals of Mathematics
63
CHAPTER 1 Introductory Information and Review
Additional Example 3:
Solution:
64
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics
65
CHAPTER 1 Introductory Information and Review
Square Roots
Definitions:
Two Rules for Square Roots:
Writing Radical Expressions in Simplest Radical Form:
66
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
Example:
Solution:
Example:
MATH 1300 Fundamentals of Mathematics
67
CHAPTER 1 Introductory Information and Review
Solution:
Exponential Form:
Additional Example 1:
Solution:
68
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
Additional Example 2:
MATH 1300 Fundamentals of Mathematics
69
CHAPTER 1 Introductory Information and Review
Solution:
Additional Example 3:
Solution:
70
University of Houston Department of Mathematics
SECTION 1.4 Exponents and Radicals
MATH 1300 Fundamentals of Mathematics
71
Exercise Set 1.4: Exponents and Radicals
Write each of the following products instead as a base
and exponent. (For example, 6  6  62 )
1.
2.
(a) 7  7  7
(c) 8  8  8  8  8  8
(b) 10 10
(d) 3  3  3  3  3  3  3
(a) 9  9  9
(c) 5  5  5  5
(b) 4  4  4  4  4
(d) 17 17
Fill in the appropriate symbol from the set
3.
7 2
4.
 9 4 ______
 , ,   .
______ 0
13. (a) 52  56
(b) 52  56
14. (a) 38  35
(b) 38  35
15. (a)
69
62
(b)
69
6 2
16. (a)
79
75
(b)
79
7 5
17. (a)
4 7  43
48
(b)
411  43
48  45
18. (a)
812
8  84
(b)
84  89
84  81
19. (a)
7 
(b)
5 
20. (a)
 
(b)
 2  
0
5.
 8  6
6.
8
7.
10 2 ______
 10 2
8.
10 3 ______
 10 3
6
Write each of the following products instead as a base
and exponent. (Do not evaluate; simply write the base
and exponent.) No answers should contain negative
exponents.
______ 0
______ 0
5
3 6
32
4

3
2 4
4
3 5
Evaluate the following.
9.
1
(a) 3
(d) 3 1
(g)
(b) 3
(e) 3 2
(h)
(c) 3
(f) 3 3
 3  2
(i)
(k) 30
(l)
(m) 34
(n) 34
(o)
10. (a) 5
0
(b)
(d) 5
1
(e)
(g) 5 2
(h)
(j) 5 3
(k)
(m) 5 4
(n)
12. (a)
 0.5  2
 0.03 2
Rewrite each expression so that it contains positive
exponent(s) rather than negative exponent(s), and then
evaluate the expression.
3
(j) 3 0
11. (a)
72
 31
2
 5 
 5 1
 5  2
 5  3
 5  4
0
1
(b)  
5
2
1
(b)  
3
4
 3  3
 3  0
 3  4
(c) 5
21. (a) 5 1
(b) 5  2
(c) 5  3
22. (a) 3 1
(b) 3  2
(c) 3  3
23. (a) 2 3
(b) 2 5
24. (a) 7  2
(b) 10  4
0
(f) 5 1
(i)
5 2
(l)
5 3
1
25. (a)  
5
(o) 5 4
 1
(c)   
 9
2
(b)  
3
1
1
6
(b)  
5
1
1
26. (a)  
7
2
 1 
(c)   
 12 
1
27. (a) 5  2
(b)
 52
2
28. (a)
 82
(b) 8  2
University of Houston Department of Mathematics
Exercise Set 1.4: Exponents and Radicals
Evaluate the following.
2 2
(b) 6
2
23
29. (a)
28
30. (a)
42.
5 1
52
(b)
51
53
 2  
(b)
 2  
32. (a)
3  
(b)
3  
2
1 2
 5a 2b2 
44. 
2 
 6a b 
2
3 1
34. (a)
35.
36.
37.
x
3x y z 
3 4 2 3
 6x
y z
3
Write each of the following expressions in simplest
radical form or as a rational number (if appropriate).
If it is already in simplest radical form, say so.
(b)
(b)
45. (a)
 36 
46. (a)
20
47. (a)
 50 
48. (a)
19 
49. (a)
28
50. (a)
 45
51. (a)
1
2
(b)
7
(c)
18
(b)
49
(c)
 32 
(b)
14
(c)
81
16
(b)
16
49
(c)
55
(b)
72
(c)
 27 
(b)
48
(c)
500
54
(b)
 80 
(c)
60
52. (a)
120
(b)
(c)
 84 
53. (a)
1
5
 3 2
(b)  
4
(c)
2
7
54. (a)
1
3
3x y z 
3 4 2 3
 6x

y z
1
2

x x
x
7
1
2
x 2 x 3 x 4

x 4 x 1

1
k 3m2
 
k 1 m 2
 
1
2
5 3 4 2
3 4 6 1
a 4 b 3
38.

5 3 4 2
2
0
2 1
Simplify the following. No answers should contain
negative exponents.
33. (a)
 c  d 0
 3a3b6 
43.   3 2 
 2a b 
31. (a)
2
3 0
c0  d 0
1
2
1
2
3
4
c7
3 5 9
1
2
180
ab c
1
2
1
2a 4 b 3
39. 1 0 9
4 ab
5d 7 e0
40.
31 d 2 e4
41.
a 0  b0
a  b
0
1
55. (a)
56. (a)
MATH 1300 Fundamentals of Mathematics
7
4
1
6
(b)
(b)
(b)
5
9
1
10
11
9
 2 2
(c)  
5
(c)
(c)
3
11
5
2
73
Exercise Set 1.4: Exponents and Radicals
57. (a)
35
58. (a)
7
2
(b)
x4 y5 z 7
(b)
2 9 5
63. (a)
3
8
(b)
3
8
(c)  3 8
64. (a)
4
81
(b)
4
81
(c)  4 81
65. (a)
6 1, 000, 000
a bc
(b)
6
1,000,000
(c)  6 1,000,000
Evaluate the following.
59. (a)
60. (a)
 5
2
 7
 6
(b)
2
 3
(b)
4
(c)
4
(c)
 2

(b)
5
32
(c)  5 32
4 1
16
(b)
4
 161
(c)  4
1
16
68. (a)
3 1
27
(b)
3
 271
(c)  3
1
27
69. (a)
5
(b)
5
1
 100,000
(b)
6
1
66. (a)
5
67. (a)
32
6
10

6
We can evaluate radicals other than square roots.
With square roots, we know, for example, that
49  7 , since 7 2  49 , and  49 is not a real
number. (There is no real number that when squared
1
100,000
(c)  5
70. (a)
6
1
1
100,000
(c)  6 1
gives a value of 49 , since 7 and  7  give a value
2
2
of 49, not 49 . The answer is a complex number,
which will not be addressed in this course.) In a
similar fashion, we can compute the following:
Cube Roots
3
125  5 , since 53  125 .
3
125  5 , since  5   125 .
3
Fourth Roots
4 10, 000  10 , since 104  10, 000 .
4
10, 000 is not a real number.
Fifth Roots
5
32  2 , since 25  32 .
5
32  2 , since  2   32 .
5
Sixth Roots
6 1
64
6
 12 , since
 12 
6
 64 .
 641 is not a real number.
Evaluate the following. If the answer is not a real
number, state “Not a real number.”
74
61. (a)
64
(b)
64
(c)  64
62. (a)
25
(b)
25
(c)  25
University of Houston Department of Mathematics
SECTION 1.5 Order of Operations
Section 1.5:
Order of Operations
 Evaluating Expressions Using the Order of Operations
Evaluating Expressions Using the Order of Operations
Rules for the Order of Operations:
1) Operations that are within parentheses and other grouping symbols are performed
first. These operations are performed in the order established in the following steps.
If grouping symbols are nested, evaluate the expression within the innermost
grouping symbol first and work outward.
2) Exponential expressions and roots are evaluated first.
3) Multiplication and division are performed next, moving left to right and performing
these operations in the order that they occur.
4) Addition and subtraction are performed last, moving left to right and performing
these operations in the order that they occur.
Upon removing all of the grouping symbols, repeat the steps 2 through 4 until the
final result is obtained.
MATH 1300 Fundamentals of Mathematics
75
CHAPTER 1 Introductory Information and Review
Example:
Solution:
Example:
Solution:
Additional Example 1:
76
University of Houston Department of Mathematics
SECTION 1.5 Order of Operations
Solution:
Additional Example 2:
Solution:
Additional Example 3:
Solution:
MATH 1300 Fundamentals of Mathematics
77
CHAPTER 1 Introductory Information and Review
Additional Example 4:
Solution:
Additional Example 5:
Solution:
78
University of Houston Department of Mathematics
Exercise Set 1.5: Order of Operations
Answer the following.
1.
(b) If choosing between multiplication and
division, which operation should come first?
(Circle the correct answer.)
Multiplication
Division
Whichever appears first
(c) If choosing between addition and
subtraction, which operation should come
first? (Circle the correct answer.)
Addition
Subtraction
Whichever appears first
When performing order of operations, which of
the following are to be viewed as if they were
enclosed in parentheses? (Circle all that apply.)
Absolute value bars
Radical symbols
Fraction bars
4.
5.
6.
(b) 2  (7  5)
(d) 2  7( 5)
(f) 2(7)  5  7
8.
(a) 6  2  (4)
(b) 6   2  (4) 
(c) 6  2( 4)
(e) 2  (6)  4
(d) (6  2)( 4)
(f) 2  4(6  2)
(a) 3  4  5
(c) 3  4  5
(e) 3  4  5
(a) 10  6  7
(c) 10  6(7)
(e) 7  10  6
9.
2 1 1
  
5 3 4
2 1 1
(d)
  
5 3 4
2 1 1
 
5 3 4
 2 11 1
(c)    
 5 3 4 4
(a)
(b)
35 
  1
26 
 3 5
(c)  1   
 2 6
35
  1
26
3 5
(d) 1  
2 6
10. (a)
11. (a) 5  4  7 
(b)
(b) 1 7 
2
2
(c) 5  1 4  7 
(d) 7  4 1  5
(e) 52  12
(f)
 5  12
(b) 23  3
12. (a) 2  32
2
2
(c) 2  3(1  4)
(d) ( 2  3) 1  4 
(e) 2 2  32
(f)
3
 2  32
(b) 20   2 10 
13. (a) 20  2(10)
(c) 20 10  (2) 10  5
Evaluate the following.
3.
(a) 2  7  5
(c) 2  (7)  5
(e) 2(7  ( 5))
In the abbreviation PEMDAS used for order of
operations,
(a) State what each letter stands for:
P: ____________________
E: ____________________
M: ____________________
D: ____________________
A: ____________________
S: ____________________
2.
7.
(b) (3  4)  5
(d) (3  4)  5
(f) 3  (4  5)
(b) (10  6)  7
(d) 10 (6  7)
(f) 7  (10  6)
(a)
37
(b)
73
(c)
3 7
(d)
7 3
(a)
25
(b)
2 5
(c)
25
(d)
2  5
MATH 1300 Fundamentals of Mathematics
14. (a) 24  4(2)
(b) (24  4)  2
(c) 24( 2)  4  2( 2)
15. (a) 102  5  2

10  5  2 2
(b)

(c) 2 10   2  5  5
2
16. (a) (3  9)  3  4

(c) 3  9   3  4 
3
17. (a)
3  
18. (a)
5  
1
6
2
3

(b) 3  (9  3)  4
(b)
3  
(b)
5  
1
1
1
6
2
3
(c) 3

(c) 5

1
1
1
6
2
3
1
1
79
Exercise Set 1.5: Order of Operations

19. 7  41  51

20. 8 31  7 1


35.  81  2 4  32  2 

36.


 
64  52  4 23
 
4
21. 7  5  2 3
2
 
22. 3 23  3 2  4
23.

37. 42  121  52  4  3

38.  144  52  2 62  12  3
1 1  3
  
2 3  4
39.
25
5  33
40.
26.
3  2 16

41.

4 1
30.

3 49
3 49  22
3 49
 

4 1
 2  3 
9  16
 
9  16 12
42.
28. 2  3 4  1
29. 2  3



4 1

32. 2  3

4 1

9  16
 2  3
2
43.
4 1
31. 2  3
5

2
2 8 24
2  32  5
2
44.
2  8   2  4
2
5  32  32  7
33.
34.
80

9  16 12
16
27. 2  3

49 3  22
3 3 10
24.   
5 10 3
25.

 3  7    7  3
45.
12  2  3  3

 2  4 3  15  1
5  12  6  3
4  2  2  1
5  2  25
46.

2

23  42  14

81  2 3

2
2
81  16  22

23  2  2  3  32
1 3 1 1 4  2
University of Houston Department of Mathematics
Exercise Set 1.5: Order of Operations
Evaluate the following expressions for the given values
of the variables.
r
k
for P  5, r  1, and k  7 .
48.
x y

y z
for x  4, y  3, and z  8 .
49.
b  b 2  8c
c2
for b  4 and c  2 .
50.
b  b 2  4ac
2a
for a  1, b  3, and c  18 .
47. P 
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Section 1.6:
Solving Linear Equations
 Linear Equations
Linear Equations
Rules for Solving Equations:
Linear Equations:
Example:
82
University of Houston Department of Mathematics
SECTION 1.6 Solving Linear Equations
Solution:
Example:
Solution:
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics
83
CHAPTER 1 Introductory Information and Review
Additional Example 2:
Solution:
Additional Example 3:
Solution:
84
University of Houston Department of Mathematics
Exercise Set 1.6: Solving Linear Equations
Solve the following equations algebraically.
1.
x  5  12
2.
x 8  9
23.
2
5
x 1  7
24.  34 x  7  2
25.
5 ( x  7)
3
4
9
 52 x  1
3.
x  4  7
4.
x  2  8
26.
5.
6 x  30
27. 2 
2x x  5

 3x
3
7
6.
4 x  28
7.
6 x  10
28. x 
x  7 5 x 1


8
6 12
8.
8 x  26
9.
3x  7  13
x  12   16 ( x  12)  3
10. 5 x  11  6
11. 2 x  3  4 x  7
12. 5 x  2  4 x  6
13. 3( x  2)  9  5( x  8)  3
14. 4( x  3)  5  2( x  4)  3
15. 3(2  5 x)  4(7 x  3)
16. 7  23  8 x   4  6(1  5 x)
17.
x
 7
5
18.
x
 10
3
19.
3
x9
2
20.
4
x  12
7
5
21.  x  3
6
8
22.  x  4
9
MATH 1300 Fundamentals of Mathematics
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CHAPTER 1 Introductory Information and Review
Section 1.7:
Interval Notation and Linear Inequalities
 Linear Inequalities
Linear Inequalities
Rules for Solving Inequalities:
86
University of Houston Department of Mathematics
SECTION 1.7 Interval Notation and Linear Inequalities
Interval Notation:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
87
CHAPTER 1 Introductory Information and Review
Example:
Solution:
Example:
88
University of Houston Department of Mathematics
SECTION 1.7 Interval Notation and Linear Inequalities
Solution:
Additional Example 1:
Solution:
MATH 1300 Fundamentals of Mathematics
89
CHAPTER 1 Introductory Information and Review
Additional Example 2:
Solution:
90
University of Houston Department of Mathematics
SECTION 1.7 Interval Notation and Linear Inequalities
Additional Example 3:
Solution:
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
91
CHAPTER 1 Introductory Information and Review
Additional Example 5:
Solution:
Additional Example 6:
Solution:
92
University of Houston Department of Mathematics
SECTION 1.7 Interval Notation and Linear Inequalities
Additional Example 7:
Solution:
MATH 1300 Fundamentals of Mathematics
93
Exercise Set 1.7: Interval Notation and Linear Inequalities
For each of the following inequalities:
(a) Write the inequality algebraically.
(b) Graph the inequality on the real number line.
(c) Write the inequality in interval notation.
Write each of the following inequalities in interval
notation.
23.
1.
x is greater than 5.
2.
x is less than 4.
3.
x is less than or equal to 3.
4.
x is greater than or equal to 7.
5.
x is not equal to 2.
6.
x is not equal to 5 .
7.
x is less than 1.
8.
x is greater than 6 .
9.
x is greater than or equal to 4 .
24.
25.
26.
27.
  



















   




   




   




10. x is less than or equal to 2 .
28.
11. x is not equal to 8 .
12. x is not equal to 3.


13. x is not equal to 2 and x is not equal to 7.
Given the set S  2, 4,  3, 13 , use substitution to
14. x is not equal to 4 and x is not equal to 0.
determine which of the elements of S satisfy each of
the following inequalities.
29. 2 x  5  10
Write each of the following inequalities in interval
notation.
15. x  3
16. x  5
17. x  2
18. x  7
30. 4 x  2  14
31. 2 x  1  7
32. 3x  1  0
33. x 2  1  10
34.
1 2

x 5
19. 3  x  5
20. 7  x  2
21. x  7
22. x  9
For each of the following inequalities:
(a) Solve the inequality.
(b) Graph the solution on the real number line.
(c) Write the solution in interval notation.
35. 2 x  10
36. 3x  24
94
University of Houston Department of Mathematics
Exercise Set 1.7: Interval Notation and Linear Inequalities
37. 5 x  30
38. 4 x  40
39. 2 x  5  11
40. 3x  4  17
41. 8  3x  20
42. 10  x  0
43. 4 x  11  7 x  4
44. 5  9 x  3x  7
45. 10 x  7  2 x  6
46. 8  4 x  6  5 x
60. (a) 3  x  5
(b) 8  x  1
(c) 2  x  8
(d) 7  x  10
Answer the following.
61. You go on a business trip and rent a car for $75
per week plus 23 cents per mile. Your employer
will pay a maximum of $100 per week for the
rental. (Assume that the car rental company
rounds to the nearest mile when computing the
mileage cost.)
(a) Write an inequality that models this
situation.
(b) What is the maximum number of miles
that you can drive and still be
reimbursed in full?
47. 5  8 x  4 x  1
48. x  10  8 x  9
49. 3(4  5 x)  2(7  x)
50. 4(3  2 x)  ( x  20)
51.
5
6
 13 x  12 ( x  5)
52.
2
5
x  12    13 10  x
53. 10  3x  2  8
54. 9  2 x  3  13
55. 4  3  7 x  17
62. Joseph rents a catering hall to put on a dinner
theatre. He pays $225 to rent the space, and pays
an additional $7 per plate for each dinner served.
He then sells tickets for $15 each.
(a) Joseph wants to make a profit. Write an
inequality that models this situation.
(b) How many tickets must he sell to make
a profit?
63. A phone company has two long distance plans as
follows:
Plan 1: $4.95/month plus 5 cents/minute
Plan 2: $2.75/month plus 7 cents/minute
How many minutes would you need to talk each
month in order for Plan 1 to be more costeffective than Plan 2?
56. 19  5  4 x  3
57.
2
3
 3 x1510  54
58.
3
4
 562 x   53
Which of the following inequalities can never be true?
59. (a) 5  x  9
(b) 9  x  5
64. Craig’s goal in math class is to obtain a “B” for
the semester. His semester average is based on
four equally weighted tests. So far, he has
obtained scores of 84, 89, and 90. What range of
scores could he receive on the fourth exam and
still obtain a “B” for the semester? (Note: The
minimum cutoff for a “B” is 80 percent, and an
average of 90 or above will be considered an
“A”.)
(c) 3  x  7
(d) 5  x  3
MATH 1300 Fundamentals of Mathematics
95
CHAPTER 1 Introductory Information and Review
Section 1.8:
Absolute Value and Equations
 Absolute Value
Absolute Value
Equations of the Form |x| = C:
Special Cases for |x| = C:
Example:
96
University of Houston Department of Mathematics
SECTION 1.8 Absolute Value and Equations
Solution:
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
97
CHAPTER 1 Introductory Information and Review
Example:
Solution:
Example:
Solution:
98
University of Houston Department of Mathematics
SECTION 1.8 Absolute Value and Equations
Example:
Solution:
MATH 1300 Fundamentals of Mathematics
99
CHAPTER 1 Introductory Information and Review
Additional Example 1:
Solution:
Additional Example 2:
Solution:
100
University of Houston Department of Mathematics
SECTION 1.8 Absolute Value and Equations
Additional Example 3:
Solution:
Additional Example 4:
Solution:
MATH 1300 Fundamentals of Mathematics
101
CHAPTER 1 Introductory Information and Review
Additional Example 5:
Solution:
102
University of Houston Department of Mathematics
Exercise Set 1.8: Absolute Value and Equations
Solve the following equations.
1.
x 7
2.
x 5
3.
x  9
4.
x  10
5.
2 x  12
6.
 3x  30
7.
x4 5
8.
x7  2
9.
x 45
10.
x 7  2
11.
3x  4  8
12.
5x  4  3
13.
3x  4  8
14.
5x  4  3
15.
2
3
x 7 1
16.
1
2
x
17.
4  3x  7  10
18.
5x  2  8  2
5
6

22. 5  x  7  8
23.
3x  2  5 x  1
24.
x  4  7x  6
1
3
19. 3 2 x  1  5  11
20.  2 2  9 x  6  4
21.  4
1
2
x  1  3  11
MATH 1300 Fundamentals of Mathematics
103