2. Chapter P
Prerequisites

3. P.1
Real Numbers

4. Quick Review

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. 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. 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. The Real Number Line

9. Order of Real Numbers Let a and b be any two real numbers.
Symbol Definition Read
a>b a – b is positive a is greater than b
a<b a – b is negative a is less than b
a≥b a – b is positive or zero a is greater than or equal to b
a≤b a – b is negative or zero a is less than or equal to b
The symbols >, <, ≥, and ≤ are inequality symbols.

10. Trichotomy Property
Let a and b be any two real numbers.
Exactly one of the following is true:
a < b, a = b, or a > b.

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

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

13. Bounded Intervals of Real Numbers
Let a and b be real numbers with a < b.
Interval Notation Inequality 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. Unbounded Intervals of Real Numbers
Let a and b be real numbers.
Interval Notation Inequality 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. Graphing Inequalities
x > 2
(2,)
x < -3
(-,-3]
-1< x < 5
(-1,5]

16. Properties of Algebra

17. Properties of Algebra

18. Properties of the Additive Inverse

19. Exponential Notation

20. Properties of Exponents

21. Example Simplifying Expressions Involving Powers

22. Example Converting to Scientific Notation
Convert to scientific notation.

23. Example Converting from Scientific Notation
Convert 1.23 × 105 from scientific notation.
123,000

24. Cartesian Coordinate System
P.2
Cartesian Coordinate System

25. Quick Review Solutions

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. The Cartesian Coordinate Plane

28. Quadrants

29. Absolute Value of a Real Number

30. Properties of Absolute Value

31. Distance Formula (Number Line)

32. Distance Formula (Coordinate Plane)

33. The Distance Formula using the Pythagorean Theorem

34. Midpoint Formula (Number Line)

35. Midpoint Formula (Coordinate Plane)

36. Distance and Midpoint Example
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)

37. Example Problem 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

38. Example 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)

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

41. Standard Form Equation of a Circle

42. Standard Form Equation of a Circle

43. Example Finding Standard Form Equations of Circles

44. Linear Equations and Inequalities
P.3
Linear Equations and Inequalities

45. Quick Review

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. Properties of Equality

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. Operations for Equivalent Equations

50. Example Solving a Linear Equation Involving Fractions

51. Linear Inequality in x

52. Properties of Inequalities

53. P.4
Lines in the Plane

54. Quick Review

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. Slope of a Line

57. Slope of a Line

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

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

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

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. 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 – y1 = m(x – x1)
Vertical line: x = a
Horizontal line: y = b

63. Graphing with a Graphing Utility
To draw a graph of an equation using a grapher:
Rewrite the equation in the form y = (an expression in x).
Enter the equation into the grapher.
Select an appropriate viewing window.
Press the “graph” key.

64. Viewing Window

65. Parallel and Perpendicular Lines

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

67. Example
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.

68. Solving Equations Graphically, Numerically, and Algebraically

69. Quick Review Solutions

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. Example Solving by Finding x-Intercepts

72. Example Solving by Finding x-Intercepts

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

74. Quadratic Equation in x
A quadratic equation in x is one that can be written in the form ax2 + bx + c = 0, where a, b, and c are real numbers with a ≠ 0.

75. Completing the Square

76. Quadratic Equation

77. Example Solving Using the Quadratic Formula

78. Solving Quadratic Equations Algebraically
These are four basic ways to solve quadratic equations
algebraically.
Factoring
Extracting Square Roots
Completing the Square
Using the Quadratic Formula

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. Example Solving by Finding Intersections

81. Example Solving by Finding Intersections

82. P.6
Complex Numbers

83. Quick Review

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

86. Complex Numbers 10x – x2 = 40 x2 – 10x = -40 x2 – 10x + 25 = -40 +25

87. Complex Numbers

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

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

90. If a and b are real numbers, then a + bi is a complex number.
Complex Numbers
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

91. Complex Numbers
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.

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).

93. Complex Numbers

94. Complex Numbers

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

96. Complex Numbers
Evaluate:

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. Example Multiplying Complex Numbers

99. Example Multiplying Complex Numbers

100. Complex Conjugate

101. Discriminant of a Quadratic Equation

102. Example Solving a Quadratic Equation

103. Example Solving a Quadratic Equation

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

105. Complex Numbers
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) =

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

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

108. Complex Numbers

109. Complex Numbers

110. Solving Inequalities Algebraically and Graphically

111. Quick Review

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. Solving Absolute Value Inequalities

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

115. Solving Absolute Value Inequalities
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 < -2 or x > 4

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

117. Example Solving an Absolute Value Inequality

118. Example Solving a Quadratic Inequality

119. Example Solving a Quadratic Inequality
Solve x2 – x – 20 < 0
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)2 – (-5) – 20 = Positive
x = (0)2 - (0) = Negative
x = (6)2 – 3(6) = Positive
Solution is (-4,5)

120. Example Solving a Quadratic Inequality
Solve x2 – 3x > 0
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)2 – 3(-1) = Positive
x = (1)2 - 3(1) = Negative
x = (4)2 – 3(4) = Positive
Solution is (-∞,0) or (3, ∞)

121. Example Solving a Quadratic Inequality
Solve x3 – 6x2 + 8x < 0
Find critical numbers x(x2 – 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)3 – 6(-5)2 + 8(-5) = Negative
x = (-1)3 – 6(-1)2 + 8(-1) = Positive
x = (3)3 – 6(3)2 + 8(3) = Negative
x = (5)3 – 6(5)2 + 8(5) = Positive
Solution is (-∞,0] U [2,4]

122. Projectile Motion
Suppose an object is launched vertically from a point so feet above the ground with an initial velocity of vo feet per second. The vertical position s (in feet) of the object t seconds after it is launched is
s = -16t2 + vot + so.

123. Chapter Test

124. Chapter Test