1. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 1

2. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Chapter P Prerequisites

3. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.1 Real Numbers

4. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 4 Quick Review

5. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 5 Quick Review Solutions

6. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 6 What you’ll learn about Representing Real Numbers Order and Interval Notation Basic Properties of Algebra Integer Exponents Scientific Notation … and why These topics are fundamental in the study of mathematics and science.

7. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 7 Real Numbers A real number is any number that can be written as a decimal. Subsets of the real numbers include: The natural (or counting) numbers: {1,2,3…} The whole numbers: {0,1,2,…} The integers: {…,-3,-2,-1,0,1,2,3,…}

8. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 8 Rational Numbers Rational numbers can be represented as a ratio a/b where a and b are integers and b≠0. The decimal form of a rational number either terminates or is indefinitely repeating.

9. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 9 The Real Number Line

10. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 10 Order of Real Numbers Let a and b be any two real numbers. SymbolDefinitionRead a>ba – b is positivea is greater than b a<ba – b is negativea is less than b a≥ba – b is positive or zeroa is greater than or equal to b a≤ba – b is negative or zeroa is less than or equal to b The symbols >, <, ≥, and ≤ are inequality symbols.

11. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 11 Trichotomy Property Let a and b be any two real numbers. Exactly one of the following is true: a b.

12. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 12 Example Interpreting Inequalities Describe the graph of x > 2.

13. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 13 Example Interpreting Inequalities Describe the graph of x > 2. The inequality describes all real numbers greater than 2.

14. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 14 Bounded Intervals of Real Numbers Let a and b be real numbers with a < b. Interval NotationInequality Notation [a,b]a ≤ x ≤ b (a,b)a < x < b [a,b)a ≤ x < b (a,b]a < x ≤ b The numbers a and b are the endpoints of each interval.

15. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 15 Unbounded Intervals of Real Numbers Let a and b be real numbers. Interval NotationInequality Notation [a,∞)x ≥ a (a, ∞)x > a (-∞,b]x ≤ b (-∞,b)x < b Each of these intervals has exactly one endpoint, namely a or b.

16. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 16 Properties of Algebra

17. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 17 Properties of Algebra

18. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 18 Properties of the Additive Inverse

19. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 19 Exponential Notation

20. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 20 Properties of Exponents

21. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 21 Example Simplifying Expressions Involving Powers

22. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 22 Example Simplifying Expressions Involving Powers

23. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 23 Example Converting to Scientific Notation Convert 0.0000345 to scientific notation.

24. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 24 Example Converting to Scientific Notation Convert 0.0000345 to scientific notation.

25. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 25 Example Converting from Scientific Notation Convert 1.23 × 10 5 from scientific notation.

26. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 26 Example Converting from Scientific Notation Convert 1.23 × 10 5 from scientific notation. 123,000

27. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.2 Cartesian Coordinate System

28. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 28 Quick Review

29. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 29 Quick Review Solutions

30. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 30 What you’ll learn about Cartesian Plane Absolute Value of a Real Number Distance Formulas Midpoint Formulas Equations of Circles Applications … and why These topics provide the foundation for the material that will be covered in this textbook.

31. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 31 The Cartesian Coordinate Plane

32. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 32 Quadrants

33. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 33 Absolute Value of a Real Number

34. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 34 Properties of Absolute Value

35. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 35 Distance Formula (Number Line)

36. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 36 Distance Formula (Coordinate Plane)

37. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 37 The Distance Formula using the Pythagorean Theorem

38. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 38 Midpoint Formula (Number Line)

39. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 39 Midpoint Formula (Coordinate Plane)

40. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 40 Standard Form Equation of a Circle

41. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 41 Standard Form Equation of a Circle

42. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 42 Example Finding Standard Form Equations of Circles

43. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 43 Example Finding Standard Form Equations of Circles

44. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.3 Linear Equations and Inequalities

45. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 45 Quick Review

46. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 46 Quick Review Solutions

47. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 47 What you’ll learn about Equations Solving Equations Linear Equations in One Variable Linear Inequalities in One Variable … and why These topics provide the foundation for algebraic techniques needed throughout this textbook.

48. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 48 Properties of Equality

49. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 49 Linear Equations in x A linear equation in x is one that can be written in the form ax + b = 0, where a and b are real numbers with a ≠ 0.

50. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 50 Operations for Equivalent Equations

51. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 51 Example Solving a Linear Equation Involving Fractions

52. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 52 Example Solving a Linear Equation Involving Fractions

53. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 53 Linear Inequality in x

54. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 54 Properties of Inequalities

55. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.4 Lines in the Plane

56. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 56 Quick Review

57. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 57 Quick Review Solutions

58. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 58 What you’ll learn about Slope of a Line Point-Slope Form Equation of a Line Slope-Intercept Form Equation of a Line Graphing Linear Equations in Two Variables Parallel and Perpendicular Lines Applying Linear Equations in Two Variables … and why Linear equations are used extensively in applications involving business and behavioral science.

59. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 59 Slope of a Line

60. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 60 Slope of a Line

61. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 61 Example Finding the Slope of a Line Find the slope of the line containing the points (3,-2) and (0,1).

62. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 62 Example Finding the Slope of a Line Find the slope of the line containing the points (3,-2) and (0,1).

63. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 63 Point-Slope Form of an Equation of a Line

64. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 64 Point-Slope Form of an Equation of a Line

65. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 65 Slope-Intercept Form of an Equation of a Line The slope-intercept form of an equation of a line with slope m and y-intercept (0,b) is y = mx + b.

66. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 66 Forms of Equations of Lines General form: Ax + By + C = 0, A and B not both zero Slope-intercept form: y = mx + b Point-slope form: y – y 1 = m(x – x 1 ) Vertical line: x = a Horizontal line: y = b

67. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 67 Graphing with a Graphing Utility To draw a graph of an equation using a grapher: 1. Rewrite the equation in the form y = (an expression in x). 2. Enter the equation into the grapher. 3. Select an appropriate viewing window. 4. Press the “graph” key.

68. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 68 Viewing Window

69. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 69 Parallel and Perpendicular Lines

70. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 70 Example Finding an Equation of a Parallel Line

71. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 71 Example Finding an Equation of a Parallel Line

72. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.5 Solving Equations Graphically, Numerically, and Algebraically

73. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 73 Quick Review

74. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 74 Quick Review Solutions

75. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 75 What you’ll learn about Solving Equations Graphically Solving Quadratic Equations Approximating Solutions of Equations Graphically Approximating Solutions of Equations Numerically with Tables Solving Equations by Finding Intersections … and why These basic techniques are involved in using a graphing utility to solve equations in this textbook.

76. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 76 Example Solving by Finding x-Intercepts

77. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 77 Example Solving by Finding x-Intercepts

78. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 78 Zero Factor Property Let a and b be real numbers. If ab = 0, then a = 0 or b = 0.

79. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 79 Quadratic Equation in x A quadratic equation in x is one that can be written in the form ax 2 + bx + c = 0, where a, b, and c are real numbers with a ≠ 0.

80. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 80 Completing the Square

81. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 81 Quadratic Equation

82. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 82 Example Solving Using the Quadratic Formula

83. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 83 Example Solving Using the Quadratic Formula

84. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 84 Solving Quadratic Equations Algebraically These are four basic ways to solve quadratic equations algebraically. 1. Factoring 2. Extracting Square Roots 3. Completing the Square 4. Using the Quadratic Formula

85. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 85 Agreement about Approximate Solutions For applications, round to a value that is reasonable for the context of the problem. For all others round to two decimal places unless directed otherwise.

86. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 86 Example Solving by Finding Intersections

87. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 87 Example Solving by Finding Intersections

88. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.6 Complex Numbers

89. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 89 Quick Review

90. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 90 Quick Review Solutions

91. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 91 What you’ll learn about Complex Numbers Operations with Complex Numbers Complex Conjugates and Division Complex Solutions of Quadratic Equations … and why The zeros of polynomials are complex numbers.

92. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 92 Complex Number A complex number is any number that can be written in the form a + bi, where a and b are real numbers. The real number a is the real part, the real number b is the imaginary part, and a + bi is the standard form.

93. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 93 Addition and Subtraction of Complex Numbers If a + bi and c + di are two complex numbers, then Sum: (a + bi ) + (c + di ) = (a + c) + (b + d)i, Difference: (a + bi ) – (c + di ) = (a - c) + (b -d)i.

94. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 94 Example Multiplying Complex Numbers

95. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 95 Example Multiplying Complex Numbers

96. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 96 Complex Conjugate

97. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 97 Discriminant of a Quadratic Equation

98. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 98 Example Solving a Quadratic Equation

99. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 99 Example Solving a Quadratic Equation

100. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley P.7 Solving Inequalities Algebraically and Graphically

101. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 101 Quick Review

102. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 102 Quick Review Solutions

103. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 103 What you’ll learn about Solving Absolute Value Inequalities Solving Quadratic Inequalities Approximating Solutions to Inequalities Projectile Motion … and why These techniques are involved in using a graphing utility to solve inequalities in this textbook.

104. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 104 Solving Absolute Value Inequalities

105. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 105 Example Solving an Absolute Value Inequality

106. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 106 Example Solving an Absolute Value Inequality

107. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 107 Example Solving a Quadratic Inequality

108. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 108 Example Solving a Quadratic Inequality

109. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 109 Projectile Motion Suppose an object is launched vertically from a point s o feet above the ground with an initial velocity of v o feet per second. The vertical position s (in feet) of the object t seconds after it is launched is s = -16t 2 + v o t + s o.

110. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 110 Chapter Test

111. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 111 Chapter Test

112. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 112 Chapter Test Solutions

113. Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide P- 113 Chapter Test Solutions