Fundamentals of math

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

1

CHAPTER 1 Introductory Information and Review

Example:

Solution:

Even/Odd Natural Numbers:

2

University of Houston Department of Mathematics

SECTION 1.1 Numbers

Whole Numbers:

Example:

Solution:

Integers:


3


Example:

Solution:

Even/Odd Integers:

Example:

Solution:

4


SECTION 1.1 Numbers

Rational Numbers:

Example:

Solution:


5


Irrational Numbers:

6


SECTION 1.1 Numbers

Real Numbers:

Example:

Solution:


7


Note About Division Involving Zero:

Additional Example 1:

Solution:

8


SECTION 1.1 Numbers

Solution:

Natural Numbers:

Whole Numbers:

Integers:

Prime/Composite Numbers:

Positive/Negative Numbers:


9

Even/Odd Numbers:

Rational Numbers:

10


SECTION 1.1 Numbers


Solution:

Natural Numbers:

Whole Numbers:


11

Integers:

Prime/Composite Numbers:

Positive/Negative Numbers:

Even/Odd Numbers:

Rational Numbers:

12


SECTION 1.1 Numbers


Solution:


13


14


SECTION 1.1 Numbers


15


Order on a Number Line
The Real Number Line:

Example:

Solution:

16


SECTION 1.1 Numbers

Inequality Symbols:

The following table describes additional inequality symbols.

Example:

Solution:


17


Example:

Solution:

Example:

Solution:


Solution:

18


SECTION 1.1 Numbers


Solution:


19



Solution:

20


SECTION 1.1 Numbers


Solution:


21


22


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.
(f)
(i)

9
9

(l)

4.

6
9

25
9

(j)

10
9

(m)

29
9

8
9

(k)

7
9

(f)  9

8

(i) 0

(k) 0.03003000300003…

14
9

Note:

6
9

9



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)

(g)

(c)

4
(e) 
5
(h) 10

Even
Prime
Integer
Undefined

In (a)-(e), use long division to change the
following fractions to decimals.
(a)

(b) 0.6

(a)

Circle all of the words that can be used to describe
each of the numbers below.

Answer the following.
3.



1.3
(d)
4.7
(g) 3.1
7
(j)
9

6.

Even
Prime
Integer
Undefined

10. 

4
7

Even
Prime
Integer
Undefined

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.)
3
7

(a) 0.7

(b)

5

(c)

(d) 5

(e)

16

(f) 0.3

(g) 12

(h)

2.3
3.5

(j)  4

5.

(k) 0.04004000400004...

(i)

e


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

23

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.

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.

24



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
3
(d)  2
(e) 2 7
3
59. Find the multiplicative inverse of the following
5
(a) 2
(b) 9
(c) 0
3
(d) 1 5

(e) 1

60. Find the additive inverse of the following
5
(a) 2
(b) 9
(c) 0
3
(d) 1 5

(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



57. Find the additive inverse of the following
(a) 3
(b) 4
(c) 1
3
2
(d)  3
(e) 2 7


25


Section 1.2:

Integers

 Operations with Integers

Operations with Integers
Absolute Value:

26


SECTION 1.2 Integers

Addition of Integers:

Example:

Solution:

Subtraction of Integers:


27

Example:

Solution:

Multiplication of Integers:

Example:

Solution:

28



Division of Integers:

Example:

Solution:


29



Solution:

30




Solution:


31



32


Solution:


33


Solution:

34


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)

(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

(d) 6( 1)

6
0
6(1)
(e)

(g) 6( 1)

(h)

(a) 6(0)

(b)

6
1

(c) 7( 1)

(e) 1( 7)

(f)

0
7

(i)

(h)

(k) 7(0)( 1)

(l)

11. (a) 10(2)

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)

0
6
(f) 6( 1)

6
1
0
6

(c) 8(2)

(e) 8  ( 2)

(f) (8)(0)

(h) 8  1

(i)

(j) 0  8

(c)

(b) 8  (2)

(g) 8(1)

Evaluate the following. If undefined, write
“Undefined.”
9.

7
1

(c) 0  ( 4)

(d)

(b) 7  9
(e) 9  (7)
(f) 9  7

(b)

(k) 2  ( 8)

(l)

15. (a) 8  2
(d)

(m)

8
2

2
8

(n)

2
0

8
1
0
8

(o) 2  8

6
0

(k) 6(1)(1) (l)

12
3
(d) 3  12

(b) 12(3)

(c) 12  3

(e) 0( 3)

(f) 0  ( 3)

(g) (3)(12)

(j)

(i)

(h)

16. (a)

12
1

(i)

3
0
1(12)

(j)

(k) 1  ( 3)

(l)

(m)


3
12
0
3

(n) 3  ( 1)

(o) 3(1)

35


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


SECTION 1.3 Fractions

A Method for Finding the GCD:

Least Common Multiple:


37


A Method for Finding the LCM:

Example:

Solution:

38



The LCM is


Solution:


39


The LCM is 2  2  2  3  5  120 .


Solution:

The LCM is 2  3  3  5  7  630 .
40



Solution:

The LCM is 2  2  3  3  2  72 .


41


Solution:

The LCM is 2  3  3  2  5  180 .

42



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:


43


Addition and Subtraction of Fractions with Unlike
Denominators:

44


Example:

Solution:



45

Solution:


46


Solution:



47

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




Solution:


49


Multiplication and Division of Fractions

Multiplication of Fractions:

50


Example:

Solution:

Division of Fractions:

Example:

Solution:


51



Solution:

52




Solution:


53



Solution:

54




Solution:


55

Exercise Set 1.3: Fractions

1.

7 and 10

4.

14 and 28

6.

6 and 22

7.

9 and 18

9.

18 and 30

(c)

12 1
4

4 1
9

(b)

4
11 5

(c)

3
9 7

8 and 20

8.

5 2
3

12 and 15

5.

(b)

4 and 5

3.

5
2 7

6 and 8

2.

19. (a)
20. (a)

For each of the following groups of numbers,
(a) Find their GCD (greatest common divisor).
(b) Find their LCM (least common multiple).

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)

4
85 21
5

(b)

7 23

3 3

24. (a)

3 21

5 5

(b)

6
2
7 11  5 11

25. (a)

53 21
4
4

(b)

3
4
65 7 5

26. (a)

5
3
97 27

(b)

5
4  11

27. (a)

7 2
3

(b)

3
9
7 10  3 10

28. (a)

7
11
6 12  2 12

(b)

81 25
6
6

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.
17. (a)

(b)

4
79

(c)

82
3

18. (a)

56

51
6
31
2

(b)

10 7
8

(c)

3
65

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


Exercise Set 1.3: Fractions

8 7

15 12

1 1

6 24

(b)

35. (a)

3
4 7 51
6

(b)

7
7 10  5 1
2

36. (a)

5
10 7  3 1
4

(b)

3
1
6 12  4 8

34. (a)

49. (a) 5 

1
20

(b)

8
4
3

10

(c)

7
5

3
4
7 5 8 7

(b)

3
6
11

4
 8
(b) 20     (c) 22 
9
 5

51. (a)
37. (a)

50. (a)

12 18

35 7

(b)

52. (a)

1
4
5
16

(b) 

4
5 9 1 2
3

38. (a)

5
7 1 36
4

(b)

2 7  9 13
8
24

39. (a)

7
2
5 15  2 12

(b)

9
5
7 10  6 8

(b)

5
3
11 14  4

(c)

36 9

5 50

15 5

16 24

(c)

49 35

24 32

7
9 16  2 5
6

40. (a)

3
5
5
9


fraction, then write the answer as an improper
fraction.)

53. (a)

4
8 5    10 
77

(b)

9
1 7    10 
8

54. (a)

3
2
2 9  4 

(b)

7
4
 3 16    5 

41. (a)

2 3

9 4

(b)

4 8

15 9

55. (a)

1
 2 1   5 7 
3

(b)

3
3
 6 5    2 11 

42. (a)

7 9

16 10

(b)

11 17

14 35

56. (a)

3 1   5 1 
7
4

(b)

3
11
5 5    2 12 

43. (a)

5 1
3

(b)

7 2
3

   

(b)

 11 1   1 17 
9
18

44. (a)

2
9 5

(b)

2
6 7

5
4
 4 5   1 7 

(b)

5
1
 2 11    2 22 

5
57. (a) 5 8  2 1
4

58. (a)

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


57


Section 1.4:

Exponents and Radicals

 Evaluating Exponential Expressions
 Square Roots

Evaluating Exponential Expressions

Two Rules for Exponential Expressions:

Example:

58


SECTION 1.4 Exponents and Radicals
Solution:

Example:

Solution:


59


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




61


Solution:

62




Solution:


63



Solution:

64




65


Square Roots
Definitions:

Two Rules for Square Roots:

Writing Radical Expressions in Simplest Radical Form:

66



Example:

Solution:

Example:


67

Solution:

Exponential Form:


Solution:

68





69

Solution:


Solution:

70




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

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.
13. (a) 52  56

(b) 52  56

14. (a) 38  35

(b) 38  35

4.

 9 4 ______

69
6 2

79
75

(b)

79
7 5

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)

______ 0

(b)

7 

(b)

5 

20. (a)

7 2

69
62

17. (a)

3.

 , ,   .

15. (a)

16. (a)


 

(b)

 2  

0

5.

 8  6

6.

8

7.

10 2 ______

 10 2

8.

10 3 ______

 10 3

6

______ 0
______ 0

5

3 6

32

4



3
2 4

4
3 5

9.

1

(a) 3
(d) 3 1
(g)

 31

2

(b) 3
(e) 3 2
(h)

(c) 3
(f) 3 3

 3  2

(i)

(j) 3 0

(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)

11. (a)

12. (a)

72

 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

 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


2 2
(b) 6
2

23
29. (a)
28

30. (a)

42.

5 1
52

(b)

51
53

 2  

(b)

32. (a)

3  

(b)

 5a 2b2 
44. 
2 
 6a b 

 2  

3  

2
1 2

 c  d 0

 3a3b6 
43.   3 2 
 2a b 

31. (a)

2
3 0

c0  d 0

2
3 1

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.

35.

36.

37.

x

 45

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.

28

51. (a)

(b)

19 

50. (a)

y z

 50 

49. (a)



5 3 4 2

20

48. (a)

 6x

(b)

 36 

47. (a)

34. (a)

3 4 2 3

45. (a)

46. (a)

3x y z 

3

0
2 1

Simplify the following. No answers should contain
negative exponents.
33. (a)

2

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)


7
4
1
6

(b)

(b)

(b)

5
9
1
10
11
9

 2 2
(c)  
5

(c)

(c)

3
11
5
2

73


58. (a)

7

2

(b)

x4 y5 z 7

(b)

3

8

(b)

3

8

(c)  3 8

4

81

(b)

4

81

(c)  4 81

65. (a)

35

63. (a)
64. (a)

57. (a)

6 1, 000, 000

2 9 5

a bc

(b)

6

1,000,000

(c)  6 1,000,000

60. (a)

 5

 7

 6

(b)

2

 3

(b)

4

(c)

4

(c)

 2


(b)

5

32

(c)  5 32

4 1
16

(b)

4

1
 16

(c)  4

1
16

68. (a)

3 1
27

(b)

3

1
 27

(c)  3

1
27

69. (a)

59. (a)

2

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

 1 , since
2

1
2

6

 64 .

1
 64 is not a real number.

Evaluate the following. If the answer is not a real
number, state “Not a real number.”
61. (a)

(b)

64

(c)  64

62. (a)

74

64
25

(b)

25

(c)  25


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.


75


Example:

Solution:

Example:

Solution:


76


SECTION 1.5 Order of Operations
Solution:


Solution:


Solution:


77


Solution:


Solution:

78


Exercise Set 1.5: Order of Operations

7.

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.

(a) 3  4  5
(c) 3  4  5
(e) 3  4  5
(a) 10  6  7
(c) 10  6(7)
(e) 7  10  6

(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)

5.

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)

9.

(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)

2

 5  12

(b) 23  3

12. (a) 2  32

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)

3 7

(d)

14. (a) 24  4(2)
(b) (24  4)  2
(c) 24( 2)  4  2( 2)
15. (a) 102  5  2



2

25

(b)



(c) 3  9   3  4 

3

25

(d)

2  5

3  

18. (a)

5  

2 5

(c)




16. (a) (3  9)  3  4

7 3

(a)

10  5  2 2

(b)

(c) 2 10   2  5  5

17. (a)
6.

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)

(c) 20 10  (2) 10  5

3.

(b) 2  (7  5)
(d) 2  7( 5)
(f) 2(7)  5  7

In the abbreviation PEMDAS used for order of
operations,
(a) State what each letter stands for:
P: ____________________
E: ____________________
M: ____________________
D: ____________________
A: ____________________
S: ____________________

2.

(a) 2  7  5
(c) 2  (7)  5
(e) 2(7  ( 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




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


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 


81


Section 1.6:

Solving Linear Equations

 Linear Equations

Linear Equations
Rules for Solving Equations:

Linear Equations:

Example:

82


SECTION 1.6 Solving Linear Equations
Solution:

Example:

Solution:


Solution:


83


Solution:


Solution:

84


Exercise Set 1.6: Solving Linear Equations
Solve the following equations algebraically.
1.

x  5  12

2.

23.

x 8  9

2
5

x 1  7

24.  3 x  7  2
4
25.

5 ( x  7)
3
4
9

2
 5 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   1 ( x  12)  3
6

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


85


Section 1.7:

Interval Notation and Linear Inequalities

 Linear Inequalities

Linear Inequalities

Rules for Solving Inequalities:

86


SECTION 1.7 Interval Notation and Linear Inequalities

Interval Notation:

Example:

Solution:


87


Example:

Solution:

Example:

88


Solution:


Solution:


89



Solution:

90



Solution:


Solution:


91



Solution:


Solution:

92




Solution:


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 or equal to 3.

4.

x is not equal to 2.

6.

x is less than 1.

8.

x is greater than or equal to 4 .





x is greater than 6 .

9.



x is not equal to 5 .

7.



x is greater than or equal to 7.

5.



x is less than 4.

3.

  

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, 1 , use substitution to
3

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


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
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

 1 x  1 ( x  5)
3
2

52.

2
5

x  1    1 10  x
2
3

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


4
 3 x1510  5

58.

3
4

 562 x   5
3

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


95


Section 1.8:

Absolute Value and Equations

 Absolute Value

Absolute Value
Equations of the Form |x| = C:

Special Cases for |x| = C:

Example:

96


SECTION 1.8 Absolute Value and Equations
Solution:

Example:

Solution:


97


Example:

Solution:

Example:

Solution:

98



Example:

Solution:


99



Solution:


Solution:

100




Solution:


Solution:


101


Solution:

102


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.

22. 5  x  7  8

5x  2  8  2

5
6

23.
24.



3x  2  5 x  1
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


103

Fundamentals of math

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