Fundamentals of math

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Fundamentals of math

  1. 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 1
  2. 2. CHAPTER 1 Introductory Information and Review Example: Solution: Even/Odd Natural Numbers: 2 University of Houston Department of Mathematics
  3. 3. SECTION 1.1 Numbers Whole Numbers: Example: Solution: Integers: MATH 1300 Fundamentals of Mathematics 3
  4. 4. CHAPTER 1 Introductory Information and Review Example: Solution: Even/Odd Integers: Example: Solution: 4 University of Houston Department of Mathematics
  5. 5. SECTION 1.1 Numbers Rational Numbers: Example: Solution: MATH 1300 Fundamentals of Mathematics 5
  6. 6. CHAPTER 1 Introductory Information and Review Irrational Numbers: 6 University of Houston Department of Mathematics
  7. 7. SECTION 1.1 Numbers Real Numbers: Example: Solution: MATH 1300 Fundamentals of Mathematics 7
  8. 8. CHAPTER 1 Introductory Information and Review Note About Division Involving Zero: Additional Example 1: Solution: 8 University of Houston Department of Mathematics
  9. 9. SECTION 1.1 Numbers Additional Example 2: Solution: Natural Numbers: Whole Numbers: Integers: Prime/Composite Numbers: Positive/Negative Numbers: MATH 1300 Fundamentals of Mathematics 9
  10. 10. CHAPTER 1 Introductory Information and Review Even/Odd Numbers: Rational Numbers: 10 University of Houston Department of Mathematics
  11. 11. SECTION 1.1 Numbers Additional Example 3: Solution: Natural Numbers: Whole Numbers: MATH 1300 Fundamentals of Mathematics 11
  12. 12. 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
  13. 13. SECTION 1.1 Numbers Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 13
  14. 14. CHAPTER 1 Introductory Information and Review 14 University of Houston Department of Mathematics
  15. 15. SECTION 1.1 Numbers MATH 1300 Fundamentals of Mathematics 15
  16. 16. CHAPTER 1 Introductory Information and Review Order on a Number Line The Real Number Line: Example: Solution: 16 University of Houston Department of Mathematics
  17. 17. SECTION 1.1 Numbers Inequality Symbols: The following table describes additional inequality symbols. Example: Solution: MATH 1300 Fundamentals of Mathematics 17
  18. 18. CHAPTER 1 Introductory Information and Review Example: Solution: Example: Solution: Additional Example 1: Solution: 18 University of Houston Department of Mathematics
  19. 19. SECTION 1.1 Numbers Additional Example 2: Solution: MATH 1300 Fundamentals of Mathematics 19
  20. 20. CHAPTER 1 Introductory Information and Review Additional Example 3: Solution: 20 University of Houston Department of Mathematics
  21. 21. SECTION 1.1 Numbers Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 21
  22. 22. CHAPTER 1 Introductory Information and Review 22 University of Houston Department of Mathematics
  23. 23. 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 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.) 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 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 23
  24. 24. 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. 24 University of Houston Department of Mathematics
  25. 25. 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 3 (d)  2 (e) 2 7 3 59. Find the multiplicative inverse of the following numbers. If undefined, write “undefined.” 5 (a) 2 (b) 9 (c) 0 3 (d) 1 5 (e) 1 60. Find the additive inverse of the following numbers. If undefined, write “undefined.” 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   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 25
  26. 26. 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
  27. 27. SECTION 1.2 Integers Addition of Integers: Example: Solution: Subtraction of Integers: MATH 1300 Fundamentals of Mathematics 27
  28. 28. CHAPTER 1 Introductory Information and Review Example: Solution: Multiplication of Integers: Example: Solution: 28 University of Houston Department of Mathematics
  29. 29. SECTION 1.2 Integers Division of Integers: Example: Solution: MATH 1300 Fundamentals of Mathematics 29
  30. 30. CHAPTER 1 Introductory Information and Review Additional Example 1: Solution: 30 University of Houston Department of Mathematics
  31. 31. SECTION 1.2 Integers Additional Example 2: Solution: MATH 1300 Fundamentals of Mathematics 31
  32. 32. CHAPTER 1 Introductory Information and Review Additional Example 3: 32 University of Houston Department of Mathematics
  33. 33. SECTION 1.2 Integers Solution: MATH 1300 Fundamentals of Mathematics 33
  34. 34. CHAPTER 1 Introductory Information and Review Additional Example 4: Solution: 34 University of Houston Department of Mathematics
  35. 35. 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 Fill in the appropriate symbol from the set  , ,   . 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) MATH 1300 Fundamentals of Mathematics 3 12 0 3 (n) 3  ( 1) (o) 3(1) 35
  36. 36. 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
  37. 37. SECTION 1.3 Fractions A Method for Finding the GCD: Least Common Multiple: MATH 1300 Fundamentals of Mathematics 37
  38. 38. CHAPTER 1 Introductory Information and Review A Method for Finding the LCM: Example: Solution: 38 University of Houston Department of Mathematics
  39. 39. SECTION 1.3 Fractions The LCM is Additional Example 1: Solution: MATH 1300 Fundamentals of Mathematics 39
  40. 40. 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
  41. 41. SECTION 1.3 Fractions Additional Example 3: Solution: The LCM is 2  2  3  3  2  72 . MATH 1300 Fundamentals of Mathematics 41
  42. 42. 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
  43. 43. 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 43
  44. 44. CHAPTER 1 Introductory Information and Review Addition and Subtraction of Fractions with Unlike Denominators: 44 University of Houston Department of Mathematics
  45. 45. SECTION 1.3 Fractions Example: Solution: Additional Example 1: MATH 1300 Fundamentals of Mathematics 45
  46. 46. CHAPTER 1 Introductory Information and Review Solution: Additional Example 2: 46 University of Houston Department of Mathematics
  47. 47. SECTION 1.3 Fractions Solution: Additional Example 3: MATH 1300 Fundamentals of Mathematics 47
  48. 48. 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
  49. 49. SECTION 1.3 Fractions Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 49
  50. 50. CHAPTER 1 Introductory Information and Review Multiplication and Division of Fractions Multiplication of Fractions: 50 University of Houston Department of Mathematics
  51. 51. SECTION 1.3 Fractions Example: Solution: Division of Fractions: Example: Solution: MATH 1300 Fundamentals of Mathematics 51
  52. 52. CHAPTER 1 Introductory Information and Review Additional Example 1: Solution: 52 University of Houston Department of Mathematics
  53. 53. SECTION 1.3 Fractions Additional Example 2: Solution: MATH 1300 Fundamentals of Mathematics 53
  54. 54. CHAPTER 1 Introductory Information and Review Additional Example 3: Solution: 54 University of Houston Department of Mathematics
  55. 55. SECTION 1.3 Fractions Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 55
  56. 56. 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 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
  57. 57. 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 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) 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) 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
  58. 58. 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
  59. 59. SECTION 1.4 Exponents and Radicals Solution: Example: Solution: MATH 1300 Fundamentals of Mathematics 59
  60. 60. 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
  61. 61. SECTION 1.4 Exponents and Radicals MATH 1300 Fundamentals of Mathematics 61
  62. 62. CHAPTER 1 Introductory Information and Review Additional Example 1: Solution: 62 University of Houston Department of Mathematics
  63. 63. SECTION 1.4 Exponents and Radicals Additional Example 2: Solution: MATH 1300 Fundamentals of Mathematics 63
  64. 64. CHAPTER 1 Introductory Information and Review Additional Example 3: Solution: 64 University of Houston Department of Mathematics
  65. 65. SECTION 1.4 Exponents and Radicals MATH 1300 Fundamentals of Mathematics 65
  66. 66. 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
  67. 67. SECTION 1.4 Exponents and Radicals Example: Solution: Example: MATH 1300 Fundamentals of Mathematics 67
  68. 68. CHAPTER 1 Introductory Information and Review Solution: Exponential Form: Additional Example 1: Solution: 68 University of Houston Department of Mathematics
  69. 69. SECTION 1.4 Exponents and Radicals Additional Example 2: MATH 1300 Fundamentals of Mathematics 69
  70. 70. CHAPTER 1 Introductory Information and Review Solution: Additional Example 3: Solution: 70 University of Houston Department of Mathematics
  71. 71. SECTION 1.4 Exponents and Radicals MATH 1300 Fundamentals of Mathematics 71
  72. 72. 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) Fill in the appropriate symbol from the set   (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 Evaluate the following. 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 University of Houston Department of Mathematics
  73. 73. 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) 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) 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
  74. 74. Exercise Set 1.4: Exponents and Radicals 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 Evaluate the following. 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 University of Houston Department of Mathematics
  75. 75. 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
  76. 76. CHAPTER 1 Introductory Information and Review Example: Solution: Example: Solution: Additional Example 1: 76 University of Houston Department of Mathematics
  77. 77. SECTION 1.5 Order of Operations Solution: Additional Example 2: Solution: Additional Example 3: Solution: MATH 1300 Fundamentals of Mathematics 77
  78. 78. CHAPTER 1 Introductory Information and Review Additional Example 4: Solution: Additional Example 5: Solution: 78 University of Houston Department of Mathematics
  79. 79. Exercise Set 1.5: Order of Operations 7. 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. (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) MATH 1300 Fundamentals of Mathematics  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 Evaluate the following. 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
  80. 80. 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
  81. 81. 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 81
  82. 82. 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
  83. 83. SECTION 1.6 Solving Linear Equations Solution: Example: Solution: Additional Example 1: Solution: MATH 1300 Fundamentals of Mathematics 83
  84. 84. CHAPTER 1 Introductory Information and Review Additional Example 2: Solution: Additional Example 3: Solution: 84 University of Houston Department of Mathematics
  85. 85. 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 MATH 1300 Fundamentals of Mathematics 85
  86. 86. 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
  87. 87. SECTION 1.7 Interval Notation and Linear Inequalities Interval Notation: Example: Solution: MATH 1300 Fundamentals of Mathematics 87
  88. 88. CHAPTER 1 Introductory Information and Review Example: Solution: Example: 88 University of Houston Department of Mathematics
  89. 89. SECTION 1.7 Interval Notation and Linear Inequalities Solution: Additional Example 1: Solution: MATH 1300 Fundamentals of Mathematics 89
  90. 90. CHAPTER 1 Introductory Information and Review Additional Example 2: Solution: 90 University of Houston Department of Mathematics
  91. 91. SECTION 1.7 Interval Notation and Linear Inequalities Additional Example 3: Solution: Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 91
  92. 92. CHAPTER 1 Introductory Information and Review Additional Example 5: Solution: Additional Example 6: Solution: 92 University of Houston Department of Mathematics
  93. 93. SECTION 1.7 Interval Notation and Linear Inequalities Additional Example 7: Solution: MATH 1300 Fundamentals of Mathematics 93
  94. 94. 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 University of Houston Department of Mathematics
  95. 95. 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  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 MATH 1300 Fundamentals of Mathematics 95
  96. 96. 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
  97. 97. SECTION 1.8 Absolute Value and Equations Solution: Example: Solution: MATH 1300 Fundamentals of Mathematics 97
  98. 98. CHAPTER 1 Introductory Information and Review Example: Solution: Example: Solution: 98 University of Houston Department of Mathematics
  99. 99. SECTION 1.8 Absolute Value and Equations Example: Solution: MATH 1300 Fundamentals of Mathematics 99
  100. 100. CHAPTER 1 Introductory Information and Review Additional Example 1: Solution: Additional Example 2: Solution: 100 University of Houston Department of Mathematics
  101. 101. SECTION 1.8 Absolute Value and Equations Additional Example 3: Solution: Additional Example 4: Solution: MATH 1300 Fundamentals of Mathematics 101
  102. 102. CHAPTER 1 Introductory Information and Review Additional Example 5: Solution: 102 University of Houston Department of Mathematics
  103. 103. 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 MATH 1300 Fundamentals of Mathematics 103

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