1. Slide P- 1

2. Chapter P Prerequisites

3. P.1 Real Numbers

4. Slide P- 4 Quick Review

5. Slide P- 5 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.

6. Slide P- 6 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,…}

7. Slide P- 7 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.

8. Slide P- 8 The Real Number Line

9. Slide P- 9 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.

10. Slide P- 10 Trichotomy Property Let a and b be any two real numbers. Exactly one of the following is true: a b.

11. Slide P- 11 Example Interpreting Inequalities Describe the graph of x > 2.

12. Slide P- 12 Example Interpreting Inequalities Describe the graph of x > 2. The inequality describes all real numbers greater than 2.

13. Slide P- 13 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.

14. Slide P- 14 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.

15. Slide P- 15 Graphing Inequalities x > 2 x < -3 (- ,-3] (2,  ) -1< x < 5 (-1,5]

16. Slide P- 16 Properties of Algebra

17. Slide P- 17 Properties of Algebra

18. Slide P- 18 Properties of the Additive Inverse

19. Slide P- 19 Exponential Notation

20. Slide P- 20 Properties of Exponents

21. Slide P- 21 Example Simplifying Expressions Involving Powers

22. Slide P- 22 Example Converting to Scientific Notation Convert 0.0000345 to scientific notation.

23. Slide P- 23 Example Converting from Scientific Notation Convert 1.23 × 10 5 from scientific notation. 123,000

24. P.2 Cartesian Coordinate System

25. Slide P- 25 Quick Review Solutions

26. Slide P- 26 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.

27. Slide P- 27 The Cartesian Coordinate Plane

28. Slide P- 28 Quadrants

29. Slide P- 29 Absolute Value of a Real Number

30. Slide P- 30 Properties of Absolute Value

31. Slide P- 31 Distance Formula (Number Line)

32. Slide P- 32 Distance Formula (Coordinate Plane)

33. Slide P- 33 The Distance Formula using the Pythagorean Theorem

34. Slide P- 34 Midpoint Formula (Number Line)

35. Slide P- 35 Midpoint Formula (Coordinate Plane)

36. Slide P- 36 Find the distance and midpoint for the line segment joined by A(-2,3) and B(4,1). A(-2,3) B(4,1) = (1,2) Distance and Midpoint Example

37. Slide P- 37 Show that A(4,1), B(0,3), and C(6,5) are vertices of an isosceles triangle. A(4,1) B(0,3) C(6,5) Since d(AC) = d(AB), ΔABC is isosceles Example Problem

38. Slide P- 38 P is a point on the y-axis that is 5 units from the point Q (3,7). Find P. P Q(3,7) (0,y) y = 3, y = 11 The point P is (0,3) or (0,11) Example

39. Slide P- 39 Prove that the diagonals of a rectangle are congruent. Coordinate Proofs Given ABCD is a rectangle. Prove AC = BD A(0,0) B(0,a) D(b,0) C(b,a) Since AC= BD, the diagonals of a square are congruent

40. Slide P- 40

41. Slide P- 41 Standard Form Equation of a Circle

42. Slide P- 42 Standard Form Equation of a Circle

43. Slide P- 43 Example Finding Standard Form Equations of Circles

44. P.3 Linear Equations and Inequalities

45. Slide P- 45 Quick Review

46. Slide P- 46 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.

47. Slide P- 47 Properties of Equality

48. Slide P- 48 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.

49. Slide P- 49 Operations for Equivalent Equations

50. Slide P- 50 Example Solving a Linear Equation Involving Fractions

51. Slide P- 51 Linear Inequality in x

52. Slide P- 52 Properties of Inequalities

53. P.4 Lines in the Plane

54. Slide P- 54 Quick Review

55. Slide P- 55 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.

56. Slide P- 56 Slope of a Line

57. Slide P- 57 Slope of a Line

58. Slide P- 58 Example Finding the Slope of a Line Find the slope of the line containing the points (3,-2) and (0,1).

59. Slide P- 59 Point-Slope Form of an Equation of a Line

60. Slide P- 60 Point-Slope Form of an Equation of a Line

61. Slide P- 61 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.

62. Slide P- 62 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

63. Slide P- 63 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.

64. Slide P- 64 Viewing Window

65. Slide P- 65 Parallel and Perpendicular Lines

66. Slide P- 66 Example Finding an Equation of a Parallel Line or y = mx + b

67. Slide P- 67 Determine the equation of the line (written in standard form) that passes through the point (-2, 3) and is perpendicular to the line 2y – 3x = 5. Example

68. P.5 Solving Equations Graphically, Numerically, and Algebraically

69. Slide P- 69 Quick Review Solutions

70. Slide P- 70 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.

71. Slide P- 71 Example Solving by Finding x-Intercepts

72. Slide P- 72 Example Solving by Finding x-Intercepts

73. Slide P- 73 Zero Factor Property Let a and b be real numbers. If ab = 0, then a = 0 or b = 0.

74. Slide P- 74 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.

75. Slide P- 75 Completing the Square

76. Slide P- 76 Quadratic Equation

77. Slide P- 77 Example Solving Using the Quadratic Formula

78. Slide P- 78 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

79. Slide P- 79 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.

80. Slide P- 80 Example Solving by Finding Intersections

81. Slide P- 81 Example Solving by Finding Intersections

82. P.6 Complex Numbers

83. Slide P- 83 Quick Review

84. Slide P- 84 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.

85. Slide P- 85 Find two numbers whose sum is 10 and whose product is 40. x = 1 st number 10 – x = 2 nd number x(10 – x) = 40 Complex Numbers

86. Slide P- 86 x(10 – x) = 40 10x – x 2 = 40 x 2 – 10x = -40 x 2 – 10x + 25 = -40 +25 (x – 5) 2 = -15 Complex Numbers

87. Slide P- 87 Complex Numbers

88. Slide P- 88 The imaginary number i is the square root of –1. Complex Numbers

89. Slide P- 89 Imaginary numbers are not real numbers, so all the rules do not apply. Example: The product rule does not apply: Complex Numbers

90. Slide P- 90 If a and b are real numbers, then a + bi is a complex number. a is the real part. bi is the imaginary part. The set of complex numbers consist of all the real numbers and all the imaginary numbers Complex Numbers

91. Slide P- 91 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. Complex Numbers

92. Slide P- 92 Examples of complex numbers: 3 + 2i 8 - 2i 4 (since it can be written as 4 + 0i). The real numbers are a subset of the complex numbers. -3i (since it can be written as 0 – 3i). Complex Numbers

93. Slide P- 93 Complex Numbers

94. Slide P- 94 Complex Numbers

95. Slide P- 95 * i -1 -i 1 -i 1 i 1 i -1 1 i -i i -i 1 i -1 1 Complex Numbers

96. Slide P- 96 Evaluate: Complex Numbers

97. Slide P- 97 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.

98. Slide P- 98 Example Multiplying Complex Numbers

99. Slide P- 99 Example Multiplying Complex Numbers

100. Slide P- 100 Complex Conjugate

101. Slide P- 101 Discriminant of a Quadratic Equation

102. Slide P- 102 Example Solving a Quadratic Equation

103. Slide P- 103 Example Solving a Quadratic Equation

104. Slide P- 104 When dividing a complex number by a real number, divide each part of the complex number by the real number. Complex Numbers

105. Slide P- 105 The numbers (a + bi ) and (a – bi ) are complex conjugates. The product (a + bi )·(a – bi ) is the real number a 2 + b 2. Show: (3 + 2i) (3 – 2i) = 3 2 + 2 2. Complex Numbers

106. Slide P- 106 Show: (3 + 2i) (3 – 2i) = 3 2 + 2 2. (3 + 2i) (3 – 2i) = 3. 3 + 3(-2i) + 2i. 3 + 2i (-2i) = 3 2 – 6i + 6i – 2 2 i 2 = 3 2 – 2 2 (-1) = 3 2 + 2 2 = 9 + 4 = 13 Complex Numbers

107. Slide P- 107 When dividing a complex number by a complex number, multiply the denominator and numerator by the conjugate of the denominator. Complex Numbers

108. Slide P- 108 Complex Numbers

109. Slide P- 109 Complex Numbers

110. P.7 Solving Inequalities Algebraically and Graphically

111. Slide P- 111 Quick Review

112. Slide P- 112 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.

113. Slide P- 113 Solving Absolute Value Inequalities

114. Slide P- 114 Solve 2x – 3 < 4x + 5 -2x < 8 x > -4 -5 -4 -3 Solve |x – 2| < 1 -1 < x – 2 < 1 1 < x < 3 0 1 2 3 4 Solving Absolute Value Inequalities

115. Slide P- 115 Solve -1 < 3 – 2x < 5 -4 < -2x < 2 2 > x > -1 -1 < x < 2 Solve |x – 1| > 3 -3 > x – 1 or x – 1 > 3 -2 > x or x > 4 x 4 -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 Solving Absolute Value Inequalities

116. Slide P- 116 |2x – 6| < 4 -4 < 2x – 6 < 4 2 < 2x < 10 1< x < 5 -1 0 1 2 3 4 5 ( ) |3x – 1| > 2 3x – 1 2 3x 3 x 1 -1 0 1 2 3 4 5 ] [ Solving Absolute Value Inequalities

117. Slide P- 117 Example Solving an Absolute Value Inequality

118. Slide P- 118 Example Solving a Quadratic Inequality -2 -1 +++0--------0+++

119. Slide P- 119 Solve x 2 – x – 20 < 0 1.Find critical numbers (x + 4)(x - 5) < 0 x = -4, x = 5 2. Test Intervals (-∞,-4) (-4,5) and (5, ∞) 3. Choose a sample in each interval x = -5 (-5) 2 – (-5) – 20 = Positive x = 0 (0) 2 - (0) - 20 = Negative x = 6 (6) 2 – 3(6) = Positive Solution is (-4,5) -4 5 +++0-------0+++ Example Solving a Quadratic Inequality

120. Slide P- 120 Solve x 2 – 3x > 0 1.Find critical numbers x(x - 3) > 0 x = 0, x = 3 2. Test Intervals (-∞,0) (0,3) and (3, ∞) 3. Choose a sample in each interval x = -1 (-1) 2 – 3(-1) = Positive x = 1 (1) 2 - 3(1) = Negative x = 4 (4) 2 – 3(4) = Positive Solution is (-∞,0) or (3, ∞) 0 3 +++0------0+++ Example Solving a Quadratic Inequality

121. Slide P- 121 Solve x 3 – 6x 2 + 8x < 0 1.Find critical numbers x(x 2 – 6x + 8) < 0 x(x – 2)(x – 4) x = 0, x = 2, x = 4 2. Test Intervals (-∞,0) (0,2) (2,4) and (4, ∞) 3. Choose a sample in each interval x = -5 (-5) 3 – 6(-5) 2 + 8(-5) = Negative x = 1 (-1) 3 – 6(-1) 2 + 8(-1) = Positive x = 3 (3) 3 – 6(3) 2 + 8(3) = Negative x = 5 (5) 3 – 6(5) 2 + 8(5) = Positive Solution is (-∞,0] U [2,4] 0 2 4 -----0++0----0+++ Example Solving a Quadratic Inequality

122. Slide P- 122 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.

123. Slide P- 123 Chapter Test

124. Slide P- 124 Chapter Test