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Slide P- 1. Chapter P Prerequisites P.1 Real Numbers.

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Slide P- 1

Chapter P Prerequisites

P.1 Real Numbers

Slide P- 4 Quick Review

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.

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,…}

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.

Slide P- 8 The Real Number Line

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.

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

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

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

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.

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.

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

Slide P- 16 Properties of Algebra

Slide P- 17 Properties of Algebra

Slide P- 18 Properties of the Additive Inverse

Slide P- 19 Exponential Notation

Slide P- 20 Properties of Exponents

Slide P- 21 Example Simplifying Expressions Involving Powers

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

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

P.2 Cartesian Coordinate System

Slide P- 25 Quick Review Solutions

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.

Slide P- 27 The Cartesian Coordinate Plane

Slide P- 28 Quadrants

Slide P- 29 Absolute Value of a Real Number

Slide P- 30 Properties of Absolute Value

Slide P- 31 Distance Formula (Number Line)

Slide P- 32 Distance Formula (Coordinate Plane)

Slide P- 33 The Distance Formula using the Pythagorean Theorem

Slide P- 34 Midpoint Formula (Number Line)

Slide P- 35 Midpoint Formula (Coordinate Plane)

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

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

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

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

Slide P- 40

Slide P- 41 Standard Form Equation of a Circle

Slide P- 42 Standard Form Equation of a Circle

Slide P- 43 Example Finding Standard Form Equations of Circles

P.3 Linear Equations and Inequalities

Slide P- 45 Quick Review

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.

Slide P- 47 Properties of Equality

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.

Slide P- 49 Operations for Equivalent Equations

Slide P- 50 Example Solving a Linear Equation Involving Fractions

Slide P- 51 Linear Inequality in x

Slide P- 52 Properties of Inequalities

P.4 Lines in the Plane

Slide P- 54 Quick Review

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.

Slide P- 56 Slope of a Line

Slide P- 57 Slope of a Line

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

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

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

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.

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

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.

Slide P- 64 Viewing Window

Slide P- 65 Parallel and Perpendicular Lines

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

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

P.5 Solving Equations Graphically, Numerically, and Algebraically

Slide P- 69 Quick Review Solutions

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.

Slide P- 71 Example Solving by Finding x-Intercepts

Slide P- 72 Example Solving by Finding x-Intercepts

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

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.

Slide P- 75 Completing the Square

Slide P- 76 Quadratic Equation

Slide P- 77 Example Solving Using the Quadratic Formula

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

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.

Slide P- 80 Example Solving by Finding Intersections

Slide P- 81 Example Solving by Finding Intersections

P.6 Complex Numbers

Slide P- 83 Quick Review

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.

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

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

Slide P- 87 Complex Numbers

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

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

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

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

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

Slide P- 93 Complex Numbers

Slide P- 94 Complex Numbers

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

Slide P- 96 Evaluate: Complex Numbers

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.

Slide P- 98 Example Multiplying Complex Numbers

Slide P- 99 Example Multiplying Complex Numbers

Slide P- 100 Complex Conjugate

Slide P- 101 Discriminant of a Quadratic Equation

Slide P- 102 Example Solving a Quadratic Equation

Slide P- 103 Example Solving a Quadratic Equation

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

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

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

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

Slide P- 108 Complex Numbers

Slide P- 109 Complex Numbers

P.7 Solving Inequalities Algebraically and Graphically

Slide P- 111 Quick Review

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.

Slide P- 113 Solving Absolute Value Inequalities

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

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

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

Slide P- 117 Example Solving an Absolute Value Inequality

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

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

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

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

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.

Slide P- 123 Chapter Test

Slide P- 124 Chapter Test

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