Chapter 1



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Chapter 1

Introductory Information and Review

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Section 1.1: Numbers

➢ Types of Numbers

➢ Order on a Number Line

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Types of Numbers

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Natural Numbers:

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Example:

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Solution:

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Even/Odd Natural Numbers:

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Whole Numbers:

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Example:

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Solution:

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Integers:

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Example:

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Solution:

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Even/Odd Integers:

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Example:

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Solution:

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Rational Numbers:

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Example:

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Solution:

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Irrational Numbers:

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Real Numbers:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Natural Numbers:

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Whole Numbers:

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Integers:

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Prime/Composite Numbers:

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Positive/Negative Numbers:

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Even/Odd Numbers:

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Rational Numbers:

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Additional Example 3:

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Solution:

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Natural Numbers:

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Whole Numbers:

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Integers:

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Prime/Composite Numbers:

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Positive/Negative Numbers:

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Even/Odd Numbers:

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Rational Numbers:

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Additional Example 4:

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Solution:

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Order on a Number Line

The Real Number Line:

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Example:

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Solution:

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Inequality Symbols:

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The following table describes additional inequality symbols.

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Example:

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Solution:

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Example:

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Solution:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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State whether each of the following numbers is prime, composite, or neither. If composite, then list all the factors of the number.

1. (a) 8 (b) 5 (c) 1

(d) [pic] (e) 12

2. (a) 11 (b) [pic] (c) 15

(d) 0 (e) [pic]

Answer the following.

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

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] Note: [pic]

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) [pic] (g) [pic] (h) [pic]

(i) [pic] (j) [pic] (k) [pic]

(l) [pic] (m) [pic] Note: [pic]

4. Use the patterns from the problem above to change each of the following decimals to either a proper fraction or a mixed number.

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

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. (a) 0.7 (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) 12 (h) [pic] (i) [pic] (j) [pic] (k) [pic]

6. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) 3.1 (h) [pic] (i) 0 (j) [pic] (k) 0.03003000300003…

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

7. [pic]

Even Odd Positive Negative

Prime Composite Natural Whole Integer Rational Irrational Real Undefined

8. [pic]

Even Odd Positive Negative

Prime Composite Natural Whole Integer Rational Irrational Real Undefined

9. [pic]

Even Odd Positive Negative

Prime Composite Natural Whole Integer Rational Irrational Real Undefined

10. [pic]

Even Odd Positive Negative

Prime Composite Natural Whole Integer Rational Irrational Real Undefined

Answer the following.

11. Which elements of the set

[pic] belong to each category listed below?

(a) Even (b) Odd

(c) Positive (d) Negative

(e) Prime (f) Composite

(g) Natural (h) Whole

(i) Integer (j) Real

(k) Rational (l) Irrational

(m) Undefined

12. Which elements of the set [pic] belong to each category listed below?

(a) Even (b) Odd

(c) Positive (d) Negative

(e) Prime (f) Composite

(g) Natural (h) Whole

(i) Integer (j) Real

(k) Rational (l) Irrational

(m) Undefined

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.

13.

| |[pic] |[pic] |[pic] | (55 |[pic] |

|Undefined | | | | | |

|Natural | | | | | |

|Whole | | | | | |

|Integer | | | | | |

|Rational | | | | | |

|Irrational | | | | | |

|Prime | | | | | |

|Composite | | | | | |

|Real | | | | | |

14.

| |2.36 |[pic] |[pic] |[pic]|[pic] |

|Undefined | | | | | |

|Natural | | | | | |

|Whole | | | | | |

|Integer | | | | | |

|Rational | | | | | |

|Irrational | | | | | |

|Prime | | | | | |

|Composite | | | | | |

|Real | | | | | |

Answer the following. If no such number exists, state “Does not exist.”

15. Find a number that is both prime and even.

16. Find a rational number that is a composite number.

17. Find a rational number that is not a whole number.

18. Find a prime number that is negative.

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.

Answer True or False. If False, justify your answer.j

25. All natural numbers are integers.

26. No negative numbers are odd.

27. No irrational numbers are even.

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.

32. All integers are rational numbers.

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.

36. List the prime numbers between 20 and 30.

37. List the composite numbers between 7 and 19.

38. List the composite numbers between 31 and 41.

39. List the even numbers between [pic] and [pic].

40. List the odd numbers between [pic] and [pic].

Fill in the appropriate symbol from the set [pic].

41. [pic] ______ [pic]

42. 3 ______ [pic]

43. [pic]______ [pic]

44. [pic]______ [pic]

45. [pic]______ 9

46. [pic] ______ [pic]

47. 5.32 ______[pic]

48. [pic]______ [pic]

49. [pic] ______ [pic]

50. [pic] ______ [pic]

51. [pic] ______ [pic]

52. [pic] ______ [pic]

53. [pic]______ 4

54. 7 ______ [pic]

55. [pic] ______ [pic]

56. [pic]______ 5

Answer the following.

57. Find the additive inverse of the following numbers. If undefined, write “undefined.”

(a) 3 (b) [pic] (c) 1

(d) [pic] (e) [pic]

58. Find the multiplicative inverse of the following numbers. If undefined, write “undefined.”

(a) 3 (b) [pic] (c) 1

(d) [pic] (e) [pic]

59. Find the multiplicative inverse of the following numbers. If undefined, write “undefined.”

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic]

60. Find the additive inverse of the following numbers. If undefined, write “undefined.”

(a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic]

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 [pic] and [pic].)

63. [pic]

64. [pic]

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Section 1.2: Integers

➢ Operations with Integers

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Operations with Integers

Absolute Value:

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Addition of Integers:

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Example:

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Solution:

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Subtraction of Integers:

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Example:

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Solution:

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Multiplication of Integers:

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Example:

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Solution:

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Division of Integers:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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Evaluate the following.

1. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic]

2. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic]

3. (a) [pic] (b) [pic] (c) [pic]

(d) [pic]

4. (a) [pic] (b) [pic] (c) [pic]

(d) [pic]

5. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic]

6. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (f) [pic]

Fill in the appropriate symbol from the set [pic].

7. (a) [pic]____ 0 (b) [pic]____ 0

(c) [pic]____ 0 (d) [pic]____ 0

8. (a) [pic] ____ 0 (b) [pic] ____ 0

(c) [pic] ____ 0 (d) [pic]___ 0

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

9. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic]

(j) [pic] (k) [pic] (l) [pic]

10. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic]

(j) [pic] (k) [pic] (l) [pic]

11. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

12. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

13. (a) [pic] (b) [pic]

(c) [pic]

(d) [pic]

14. (a) [pic] (b) [pic]

(c) [pic]

(d) [pic]

15. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic] (g) [pic] (h) [pic] (i) [pic]

(j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic]

16. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic]

(j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic]

[pic]

Section 1.3: Fractions

➢ Greatest Common Divisor and Least Common Multiple

➢ Addition and Subtraction of Fractions

➢ Multiplication and Division of Fractions

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Greatest Common Divisor and Least Common Multiple

Greatest Common Divisor:

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A Method for Finding the GCD:

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Least Common Multiple:

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A Method for Finding the LCM:

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Example:

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Solution:

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The LCM is

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Additional Example 1:

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Solution:

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The LCM is [pic].

Additional Example 2:

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Solution:

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The LCM is [pic].

Additional Example 3:

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Solution:

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The LCM is [pic].

Additional Example 4:

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Solution:

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The LCM is [pic].

Addition and Subtraction of Fractions

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Addition and Subtraction of Fractions with Like Denominators:

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[pic] and [pic]

Example:

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Solution:

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Addition and Subtraction of Fractions with Unlike Denominators:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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(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.

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Additional Example 4:

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Solution:

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Multiplication and Division of Fractions

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Multiplication of Fractions:

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Example:

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Solution:

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Division of Fractions:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Additional Example 4:

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Solution:

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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

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.

13. (a) [pic] (b) [pic] (c) [pic]

14. (a) [pic] (b) [pic] (c) [pic]

15. (a) [pic] (b) [pic] (c) [pic]

16. (a) [pic] (b) [pic] (c) [pic]

Change each of the following mixed numbers to an improper fraction.

17. (a) [pic] (b) [pic] (c) [pic]

18. (a) [pic] (b) [pic] (c) [pic]

19. (a) [pic] (b) [pic] (c) [pic]

20. (a) [pic] (b) [pic] (c) [pic]

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) [pic] (b) [pic]

22. (a) [pic] (b) [pic]

23. (a) [pic] (b) [pic]

24. (a) [pic] (b) [pic]

25. (a) [pic] (b) [pic]

26. (a) [pic] (b) [pic]

27. (a) [pic] (b) [pic]

28. (a) [pic] (b) [pic]

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) [pic] (b) [pic]

30. (a) [pic] (b) [pic]

31. (a) [pic] (b) [pic]

32. (a) [pic] (b) [pic]

33. (a) [pic] (b) [pic]

34. (a) [pic] (b) [pic]

35. (a) [pic] (b) [pic]

36. (a) [pic] (b) [pic]

37. (a) [pic] (b) [pic]

38. (a) [pic] (b) [pic]

39. (a) [pic] (b) [pic]

40. (a) [pic] (b) [pic]

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.)

41. (a) [pic] (b) [pic]

42. (a) [pic] (b) [pic]

43. (a) [pic] (b) [pic]

44. (a) [pic] (b) [pic]

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) [pic] (b) [pic] (c) [pic]

46. (a) [pic] (b) [pic] (c) [pic]

47. (a) [pic] (b) [pic] (c) [pic]

48. (a) [pic] (b) [pic] (c) [pic]

49. (a) [pic] (b) [pic] (c) [pic]

50. (a) [pic] (b) [pic] (c) [pic]

51. (a) [pic] (b) [pic] (c) [pic]

52. (a) [pic] (b) [pic] (c) [pic]

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.)

53. (a) [pic] (b) [pic]

54. (a) [pic] (b) [pic]

55. (a) [pic] (b) [pic]

56. (a) [pic] (b) [pic]

57. (a) [pic] (b) [pic]

58. (a) [pic] (b) [pic]

[pic]

Section 1.4: Exponents and Radicals

➢ Evaluating Exponential Expressions

➢ Square Roots

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Evaluating Exponential Expressions

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Two Rules for Exponential Expressions:

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Example:

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Solution:

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Example:

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Solution:

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Additional Properties for Exponential Expressions:

Two Definitions:

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Quotient Rule for Exponential Expressions:

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Exponential Expressions with Bases of Fractions:

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Example:

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Solution:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Square Roots

Definitions:

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Two Rules for Square Roots:

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Writing Radical Expressions in Simplest Radical Form:

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Example:

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Solution:

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Example:

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Solution:

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Exponential Form:

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Additional Example 1:

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Solution:

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Additional Example 2:

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Solution:

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Additional Example 3:

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Solution:

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Write each of the following products instead as a base and exponent. (For example, [pic])

1. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

2. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

Fill in the appropriate symbol from the set [pic].

3. [pic] ______ 0

4. [pic] ______ 0

5. [pic] ______ 0

6. [pic] ______ 0

7. [pic] ______ [pic]

8. [pic] ______ [pic]

Evaluate the following.

9. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic]

(j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic]

[pic]

10. (a) [pic] (b) [pic] (c) [pic]

(d) [pic] (e) [pic] (f) [pic]

(g) [pic] (h) [pic] (i) [pic]

(j) [pic] (k) [pic] (l) [pic]

(m) [pic] (n) [pic] (o) [pic]

11. (a) [pic] (b) [pic] (c) [pic]

12. (a) [pic] (b) [pic] (c) [pic]

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) [pic] (b) [pic]

14. (a) [pic] (b) [pic]

15. (a) [pic] (b) [pic]

16. (a) [pic] (b) [pic]

17. (a) [pic] (b) [pic]

18. (a) [pic] (b) [pic]

19. (a) [pic] (b) [pic]

20. (a) [pic] (b) [pic]

Rewrite each expression so that it contains positive exponent(s) rather than negative exponent(s), and then evaluate the expression.

21. (a) [pic] (b) [pic] (c) [pic]

22. (a) [pic] (b) [pic] (c) [pic]

23. (a) [pic] (b) [pic]

24. (a) [pic] (b) [pic]

25. (a) [pic] (b) [pic]

26. (a) [pic] (b) [pic]

27. (a) [pic] (b) [pic]

28. (a) [pic] (b) [pic]

Evaluate the following.

29. (a) [pic] (b) [pic]

30. (a) [pic] (b) [pic]

31. (a) [pic] (b) [pic]

32. (a) [pic] (b) [pic]

Simplify the following. No answers should contain negative exponents.

33. (a) [pic] (b) [pic]

34. (a) [pic] (b) [pic]

35. [pic]

36. [pic]

37. [pic]

38. [pic]

39. [pic]

40. [pic]

41. [pic]

42. [pic]

43. [pic]

44. [pic]

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.

45. (a) [pic] (b) [pic] (c) [pic]

46. (a) [pic] (b) [pic] (c) [pic]

47. (a) [pic] (b) [pic] (c) [pic]

48. (a) [pic] (b) [pic] (c) [pic]

49. (a) [pic] (b) [pic] (c) [pic]

50. (a) [pic] (b) [pic] (c) [pic]

51. (a) [pic] (b) [pic] (c) [pic]

52. (a) [pic] (b) [pic] (c) [pic]

53. (a) [pic] (b) [pic] (c) [pic]

54. (a) [pic] (b) [pic] (c) [pic]

55. (a) [pic] (b) [pic] (c) [pic]

56. (a) [pic] (b) [pic] (c) [pic]

57. (a) [pic] (b) [pic]

58. (a) [pic] (b) [pic]

Evaluate the following.

59. (a) [pic] (b) [pic] (c) [pic]

60. (a) [pic] (b) [pic] (c) [pic]

We can evaluate radicals other than square roots. With square roots, we know, for example, that [pic], since [pic], and [pic] is not a real number. (There is no real number that when squared gives a value of [pic], since [pic] and [pic] give a value of 49, not [pic]. 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

[pic], since [pic].

[pic], since [pic].

Fourth Roots

[pic], since [pic].

[pic] is not a real number.

Fifth Roots

[pic], since [pic].

[pic], since [pic].

Sixth Roots

[pic], since [pic].

[pic] is not a real number.

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

61. (a) [pic] (b) [pic] (c) [pic]

62. (a) [pic] (b) [pic] (c) [pic]

63. (a) [pic] (b) [pic] (c) [pic]

64. (a) [pic] (b) [pic] (c) [pic]

65. (a) [pic] (b) [pic]

(c) [pic]

66. (a) [pic] (b) [pic] (c) [pic]

67. (a) [pic] (b) [pic] (c) [pic]

68. (a) [pic] (b) [pic] (c) [pic]

69. (a) [pic] (b) [pic]

(c) [pic]

70. (a) [pic] (b) [pic] (c) [pic]

[pic]

Section 1.5: Order of Operations

➢ Evaluating Expressions Using the Order of Operations

[pic]

Evaluating Expressions Using the Order of Operations

[pic]

[pic]

[pic]

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.

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

Example:

[pic]

Solution:

[pic]

Additional Example 1:

[pic]

Solution:

[pic]

Additional Example 2:

[pic]

Solution:

[pic]

Additional Example 3:

[pic]

Solution:

[pic]

Additional Example 4:

[pic]

Solution:

[pic]

Additional Example 5:

[pic]

Solution:

[pic]

Answer the following.

1. In the abbreviation PEMDAS used for order of operations,

(a) State what each letter stands for:

P: ____________________

E: ____________________

M: ____________________

D: ____________________

A: ____________________

S: ____________________

(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

2. 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

Evaluate the following.

3. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

4. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

5. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

6. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

7. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

8. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

9. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

10. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

11. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

12. (a) [pic] (b) [pic]

(c) [pic] (d) [pic]

(e) [pic] (f) [pic]

13. (a) [pic] (b) [pic]

(c) [pic]

14. (a) [pic] (b) [pic]

(c) [pic]

15. (a) [pic] (b) [pic]

(c) [pic]

16. (a) [pic] (b) [pic]

(c) [pic]

17. (a) [pic] (b) [pic] (c) [pic]

18. (a) [pic] (b) [pic] (c) [pic]

19. [pic]

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

25. [pic]

26. [pic]

27. [pic]

28. [pic]

29. [pic]

30. [pic]

31. [pic]

32. [pic]

33. [pic]

34. [pic]

35. [pic]

36. [pic]

37. [pic]

38. [pic]

39. [pic]

40. [pic]

41. [pic]

42. [pic]

43. [pic]

44. [pic]

45. [pic]

46. [pic]

Evaluate the following expressions for the given values of the variables.

47. [pic] for [pic].

48. [pic] for [pic].

49. [pic] for [pic] and [pic].

50. [pic] for [pic].

[pic]

Section 1.6: Solving Linear Equations

➢ Linear Equations

[pic]

Linear Equations

Rules for Solving Equations:

[pic]

[pic]

[pic]

[pic]

Linear Equations:

[pic]

Example:

[pic]

Solution:

[pic]

Example:

[pic]

Solution:

[pic]

Additional Example 1:

[pic]

Solution:

[pic]

Additional Example 2:

[pic]

Solution:

[pic]

Additional Example 3:

[pic]

Solution:

[pic]

Solve the following equations algebraically.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic][pic]

18. [pic]

19. [pic]

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

25. [pic]

26. [pic]

27. [pic]

28. [pic]

[pic]

Section 1.7: Interval Notation and Linear Inequalities

➢ Linear Inequalities

[pic]

Linear Inequalities

[pic]

Rules for Solving Inequalities:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Interval Notation:

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

Additional Example 1:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 2:

[pic]

Solution:

[pic]

[pic]

[pic]

Additional Example 3:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

Additional Example 4:

[pic]

Solution:

[pic]

[pic]

Additional Example 5:

[pic]

Solution:

[pic]

[pic]

Additional Example 6:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

Additional Example 7:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

[pic]

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.

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 [pic].

7. x is less than [pic]

8. x is greater than [pic].

9. x is greater than or equal to [pic].

10. x is less than or equal to [pic].

11. x is not equal to [pic].

12. x is not equal to 3.

13. x is not equal to 2 and x is not equal to 7.

14. x is not equal to [pic] and x is not equal to 0.

Write each of the following inequalities in interval notation.

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

21. [pic]

22. [pic]

Write each of the following inequalities in interval notation.

23.

24.

25.

26.

27.

28.

Given the set [pic], use substitution to determine which of the elements of S satisfy each of the following inequalities.

29. [pic]

30. [pic]

31. [pic]

32. [pic]

33. [pic]

34. [pic]

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. [pic]

36. [pic]

37. [pic]

38. [pic]

39. [pic]

40. [pic]

41. [pic]

42. [pic]

43. [pic]

44. [pic]

45. [pic]

46. [pic]

47. [pic]

48. [pic]

49. [pic]

50. [pic]

51. [pic]

52. [pic]

53. [pic]

54. [pic]

55. [pic]

56. [pic]

57. [pic]

58. [pic]

Which of the following inequalities can never be true?

59. (a) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

60. (a) [pic]

(b) [pic]

(c) [pic]

(d) [pic]

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?

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 cost-effective than Plan 2?

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”.)

[pic]

Section 1.8: Absolute Value and Equations

➢ Absolute Value

[pic]

Absolute Value

Equations of the Form |x| = C:

[pic]

[pic]

[pic]

Special Cases for |x| = C:

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

[pic]

Example:

[pic]

Solution:

[pic]

[pic]

[pic]

[pic]

Additional Example 1:

[pic]

Solution:

[pic]

[pic]

[pic]

Additional Example 2:

[pic]

Solution:

[pic]

[pic]

Additional Example 3:

[pic]

Solution:

[pic]

[pic]

[pic]

Additional Example 4:

[pic]

Solution:

[pic]

[pic]

[pic]

Additional Example 5:

[pic]

Solution:

[pic]

[pic]

[pic]

Solve the following equations.

1. [pic]

2. [pic]

3. [pic]

4. [pic]

5. [pic]

6. [pic]

7. [pic]

8. [pic]

9. [pic]

10. [pic]

11. [pic]

12. [pic]

13. [pic]

14. [pic]

15. [pic]

16. [pic]

17. [pic]

18. [pic]

19. [pic]

20. [pic]

21. [pic]

22. [pic]

23. [pic]

24. [pic]

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