Chapter P Prerequisites P.1 Real Numbers LESSON OBJECTIVES To be able to Represent Real Numbers. To be able to Order real numbers using interval notation … and why These topics are fundamental in the study of Write these mathematics and science. In your notebooks Slide P- 3 Please discuss in pairs and match the following vocabularies with their meanings Terminate Notation Any number that can be written as a decimal A whole number, a number that is not fraction Has no end, endless, limitless Real numbers Bring to an end, stop, close, finish Integer Not expressed as a ratio of two integers eg (pie) Can be expressed as a ratio or repeated decimals System of symbols Slide P- 4 irrational Infinite rational In pairs please do the following Review quiz (4min) 1. List the positive integers between -4 and 4. 1,2,3 2. List all negative integers greater than -4. -3,-2,-1 3. Use a calculator to evaluate the expression 2 4.5 3 . Round the value to two decimal places. 2.73 2.3 4.5 4. Evaluate the algebraic expression for the given values of the variable. x 2 x 1, x 1,1.5 3 -4,5.375 5. List the possible remainders when the positive integer n is divided by 6. 1,2,3,4,5 Slide P- 5 Copy the following into your note books A real number is any number that can be written as a decimal. Subsets of the real numbers include: n The natural (or counting) numbers: {1,2,3…} n The whole numbers: {0,1,2,…} n The integers: {…,-3,-2,-1,0,1,2,3,…} Slide P- 6 Copy the following in your note books 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- 7 The Real Number Line Pin the worksheet in your note Slide P- 8 In pairs group the following numbers as either rational, irrational, integers, infinite repeating decimal OR terminal decimal n ⅚, -3, √6, π, ⅚, 4.34, 0.363636…, 0, 9, 35, 1.75, 0.625, 0.11111….. ⅛, 2π, 5 , 5√25, 6√7, ⅜, ⅞, 2 ⅘, 2 , 10 ⅒, -5, ⅔, 25, ⅐ Slide P- 9 Pin the worksheet your Order of Real note Numbers books in Let a and b be any two real numbers. Symbol Definition Read a>b a<b a≥b a≤b a – b is positive a – b is negative a – b is positive or zero a – b is negative or zero a is greater than b a is less than b a is greater than or equal to b a is less than or equal to b The symbols >, <, ≥, and ≤ are inequality symbols. Slide P- 10 Write the following in your note books 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. Slide P- 11 Class activity to be done in pairs 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. In pairs show the aboe statement on the graph min Slide P- 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. Pin in you Notebook Slide P- 14 Unbounded Intervals of Real Numbers Pin in you Notebook 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. Slide P- 15 In groups Graph the following Inequalities (class competition 15 min) x>2 (2,) x < -3 (-,-3] -1< x < 5 (-1,5] Slide P- 16 HOMEWORK: PRECALCULUS TEXT BOOK PAGE #1EXIT QUIZ: answer the question on the worksheet NEXT LESSON before you Basic properties exit and hand Slide P- 17 Day 2Basic Properties of Algebra Slide P- 18 Learning objectives 1To be able to describe basic properties of algebra Slide P- 19 Please show on the mwb the words that have same meaning. Use numbers only….. 1.Cumulative 2.Associative 3.Inverse 4.Distributive 1. opposite, reverse, contrary 2. divide, partition, share 3. connected, related, like 4. increasing, growing, colective Slide P- 20 Please in pairs discuss in English the following review questions. Ealuate the algebraic expression for the gien alues of the ariables Answers Slide P- 21 Pin the following worksheets in your notebooks…….. Properties of Algebra Let u , v, and w be real numbers, variables, or algebraic expressions. 1. Communative Property Addition: u v v u Multiplication uv vu 2. Associative Property Addition: (u v) w u (v w) Multiplication: (uv) w u (vw) 3. Identity Property Addition: u 0 u Multiplication: u 1 u Slide P- 22 Properties of Algebra Let u , v, and w be real numbers, variables, or algebraic expressions. 4. Inverse Property Addition: u (-u ) 0 1 1, u 0 u 5. Distributive Property Mulitiplication: u Multiplication over addition: u (v w) uv uw (u v) w uw vw Multiplication over subtraction: u (v w) uv uw (u v) w uw vw Slide P- 23 Properties of the Additive Inverse Let u , v, and w be real numbers, variables, or algebraic expressions. Property Example 1. (u ) u (3) 3 2. ( u )v u (v) uv ( 4)3 4(3) 12 3. (u )(v) uv (6)(7) 42 4. (1)u u (1)5 5 5. (u v) (u ) (v) (7 9) (7) (9) 16 Slide P- 24 Group Activity n Write the basic properties of algebra on the manila paper provided and hang it on the classroom noticeboard. Slide P- 25 Class actiity In pairs work out questions on page 34 #37-46 Slide P- 26 CLASS ACTIVITIES ANSWERS Slide P- 27 ANSWERS Slide P- 28 ANSWERS Slide P- 29 Slide P- 30 ANSWERS Slide P- 31 Please write this in your notebooks….. Exponential Notation Let a be a real number, variable, or algebraic expression and n a positive integer. Then a aaa...a, where n is the n n factors n exponent, a is the base, and a is the nth power of a, read as "a to the nth power." Slide P- 32 Properties of Exponents Let u and v be a real numbers, variables, or algebraic expressions and m and n be integers. All bases are assumed to be nonzero. Property Example 1. u u u m n 5 5 5 5 mn 3 m u 2. u u 3. u 1 1 4. u u 5. (uv) u v m mn u u 7. v v m m m 1 y y (2 z ) 2 z 32 z -3 n 6. (u ) u 5 0 -n m x 94 4 0 n 7 9 x x x 8 1 mn n m 3 4 4 3 m 5 5 5 (x ) x x 2 3 23 a a b b 7 5 6 7 7 Slide P- 33 Example Simplifying Expressions Involving Powers 2 3 1 2 uv Simplify . u v 2 3 2 1 3 1 2 2 3 5 uv uu u u v vv v Slide P- 34 In pairs workout the following ( discussion must be in English….) 1. Convert 0.0000345 to scientific notation. 0.0000345 3.45 10 -5 Slide P- 35 Converting from Scientific Notation 2. Convert 1.23 × 105 from scientific notation. 123,000 Slide P- 36 n HOMEWORK: page 34 #47-51 precalculus textbook n Exit quiz: answer the question on the worksheet provide and hand in before you leave. n Next lesson: Cartesian coordinate system. Slide P- 37 Day 3 n Cartesian coordinate system Slide P- 38 For everyone: work out the following on the mwb and swap with your neighbors to compare the answers…. -5 3 and . 2.75 4 2 Use a calculator to evaluate the expression. Round answers 1. Find the distance between to two decimal places. 2. 8 6 2 -12 8 2 4. 3 5 2 10 3. 2 5. -2 2 5.83 2 5 1 3 2 2 3.61 Slide P- 39 Learning Objectives To be able to draw a Cartesian Plane n To be able to evaluate the Absolute Value of a Real Number n To be able to find Distance and midpoint between two points. n … and why These topics provide the foundation for the material that will be covered in this textbook. Write the lesson objective in your note books Slide P- 40 Take 3 min to discuss in your groups the meaning of the following key words..! n Congruent n Cartesian plane Coordinate Quadrant n n n Absolute value identical in form; coincide/fix exactly when put together. two perpendicular number lines: the x-axis and the y-axis where x-value and y-value meet four quarters of the coordinate plane the distance of a number on the number value line from 0 without considering which direction from zero the number lies. Slide P- 41 For everyone: Draw the following in your notebooks….. The Cartesian Coordinate Plane Slide P- 42 For everyone: Draw the following in the notebooks…. Quadrants Slide P- 43 Class Activity (5 min) n In groups Draw the 4 quadrants on the manila paper and pin it on the classroom notice board. Slide P- 44 Copy this in your notebooks please… (3min) Absolute Value of a Real Number The absolute value of a real number a is a, if a 0 |a | a if a 0. 0, if a 0 Slide P- 45 Copy this in your notebook please……. (3 min) Properties of Absolute Value Let a and b be real numbers. 1. | a | 0 2. | -a || a | 3. | ab || a || b | a |a| 4. , b0 b |b| Slide P- 46 Pin the worksheet for distance formulas in your notebooks……… (2min) Distance Formula (Number Line) Let a and b be real numbers. The distance between a and b is | a b | . Note that | a b || b a | . Distance Formula (Coordinate Plane) The distance d between points P(x , y ) and Q(x , y ) in the 1 coordinate plane is d x x 1 2 2 1 y y 1 2 2 . 2 2 Slide P- 47 Class Actiity 1 The Distance Formula using the Pythagorean Theorem Slide P- 48 Pin the worksheet for distance formulas in your notebooks……… (2min) Midpoint Formula (Number Line) The midpoint of the line segment with endpoints a and b is ab . 2 Midpoint Formula (Coordinate Plane) The midpoint of the line segment with endpoints (a,b) and (c,d ) is ac bd , . 2 2 Slide P- 49 Class Actiity 1 Distance and Midpoint Find the distance and midpoint for the line segment joined by A(-2,3) and B(4,1). d ( A, B) (2 4) (3 1) 2 A(-2,3) 2 B(4,1) d ( A, B) (6) 2 (2) 2 d ( A, B) 40 4 10 2 10 (2 4) (3 1) , 2 2 2 4 Show all these steps in , = (1,2) Your notebook 2 2 Slide P- 50 Class Activity Show2 all Show that A(4,1), B(0,3), and C(6,5) are vertices of an isosceles triangle. these step Your notebook C(6,5) d ( A, B) (4 0) 2 (1 3) 2 2 5 d ( B, C ) (0 6) (3 5) 2 10 2 2 B(0,3) A(4,1) d ( A, C ) (4 6) 2 (5 1)2 2 5 Since d(AC) = d(AB) , ΔABC is isosceles Slide P- 51 Class Activity 3 Sole this problem a Swap your notebook P is a point on the y-axis that is 5 units from the point Q (3,7). Find P. Q(3,7) d ( P, Q) (3 0) 2 ( y 7) 2 5 P (0,y) 9 y 2 14 y 49 5 y 2 14 y 58 5 y 2 14 y 58 25 y 2 14 y 33 0 ( y 11)( y 3) 0 y = 3, y = 11 The point P is (0,3) or (0,11) Slide P- 52 Class Activity 4 Coordinate Proofs Discuss this question In your groups the Swap answers Prove that the diagonals of a rectangle are congruent. B(0,a) C(b,a) A(0,0) D(b,0) Given ABCD is a rectangle. Prove AC = BD dAC (b 0) 2 (a 0) 2 b 2 a 2 dBD (0 b) 2 (a 0) 2 b 2 a 2 Since AC= BD, the diagonals of a square are congruent Slide P- 53 n HOMEWORK: page 41 #1-28 precalculus textbook n Exit quiz: answer the question on the worksheet provide and hand in before you leave. n Next lesson: Equation of a circle Slide P- 54 Day 4 Slide P- 55 Slide P- 56 Standard Form Equation of a Circle The standard form equation of a circle with center (h, k ) and radius r is ( x h) ( y k ) r . 2 2 2 Slide P- 57 Standard Form Equation of a Circle Slide P- 58 Example Finding Standard Form Equations of Circles Find the standard form equation of the circle with center (2, 3) and radius 4. ( x h) ( y k ) r 2 2 2 where h 2, k 3, and r 4. Thus the equation is ( x 2) ( y 3) 16. 2 2 Slide P- 59 P.3 Linear Equations and Inequalities Quick Review Simplify the expression by combining like terms. 1. 2 x 4 x y 2 y 3x 3x 3 y 2. 3(2 x 2) 4( y 1) 6 x 4 y 10 Use the LCD to combine the fractions. Simplify the resulting fraction. 3 4 7 x x x x 2 x 7x 6 4. 4 3 12 2 2y 2 5. 2 y y 3. Slide P- 61 What you’ll learn about n n n n 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- 62 Properties of Equality Let u, v, w, and z be real numbers, variables, or algebraic expressions. 1. Reflexive uu 2. Symmetric If u v, then v u. 3. Transitive If u v, and v w, then u w. 4. Addition If u v and w z, then u w v z. 5. Multiplication If u v and w z , then uw vz. Slide P- 63 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- 64 Operations for Equivalent Equations An equivalent equation is obtained if one or more of the following operations are performed. Operation Given Equation 1. Combine like terms, 2x x 3 9 Equivalent Equation 3x 1 3 reduce fractions, and remove grouping symbols 2. Perform the same operation on both sides. (a) Add ( 3) x3 7 (b) Subtract (2x ) 5x 2x 4 x4 3x 4 (c) Multiply by a nonzero constant (1/3) 3 x 12 x4 (d) Divide by a constant nonzero term (3) 3 x 12 x4 Slide P- 65 Example Solving a Linear Equation Involving Fractions Solve for y. 10 y 4 y 2 4 4 10 y 4 y 2 4 4 10 y 4 y 4 2 4 4 4 10y 4 y 8 Distributive Property 9 y 12 Simplify y Multiply by the LCD 4 3 Slide P- 66 Linear Inequality in x A linear inequality in x is one that can be written in the form ax b 0, ax b 0, ax b 0, or ax b 0, where a and b are real numbers with a 0. Slide P- 67 Properties of Inequalities Let u , v, w, and z be real numbers, variables, or algebraic expressions, and c a real number. 1. Transitive If u v, and v w, then u w. 2. Addition If u v then u w v w. If u v and w z then u w v z. 3. Multiplication If u v and c 0, then uc vc. If u v and c 0, then uc vc. The above properties are true if < is replaced by . There are similar properties for > and . Slide P- 68 P.4 Lines in the Plane Quick Review Solve for x. 1. 50 x 100 200 x 2 2. 3(1 2 x) 4( x 2) 10 x 1 2 Solve for y. 2x 5 3 x 4. 2 x 3( x y ) y y 4 72 5 5. Simplify the fraction. 10 (3) 7 3. 2 x 3 y 5 y Slide P- 70 What you’ll learn about n n n n n n 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- 71 Slope of a Line Slide P- 72 Slope of a Line The slope of the nonvertical line through the points ( x , y ) 1 1 y y y and ( x , y ) is m . x x x 2 2 1 2 1 2 If the line is vertical, then x x and the slope is undefined. 1 2 Slide P- 73 Example Finding the Slope of a Line Find the slope of the line containing the points (3,-2) and (0,1). y y 1 (2) 3 m 1 x x 03 3 2 1 2 1 Thus, the slope of the line is 1. Slide P- 74 Point-Slope Form of an Equation of a Line The point - slope form of an equation of a line that passes through the point ( x , y ) and has slope m is y y m( x x ). 1 1 1 1 Slide P- 75 Point-Slope Form of an Equation of a Line Slide P- 76 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- 77 Forms of Equations of Lines General form: Slope-intercept form: Point-slope form: Vertical line: Horizontal line: Ax + By + C = 0, A and B not both zero y = mx + b y – y1 = m(x – x1) x=a y=b Slide P- 78 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- 79 Viewing Window Slide P- 80 Parallel and Perpendicular Lines 1. Two nonvertical lines are parallel if and only if their slopes are equal. 2. Two nonvertical lines are perpendicular if and only if their slopes m and m are opposite reciprocals. 1 2 1 That is, if and only if m . m 1 2 Slide P- 81 Example Finding an Equation of a Parallel Line Find an equation of a line through (2, 3) that is parallel to 4 x 5 y 10. Find the slope of 4 x 5 y 10. 5 y 4 x 10 or y = mx + b 4 y xb 5 4 3 ( 2) b 5 8 3 b 5 1 4 1 b y x 5 5 5 4 4 y x 2 The slope of this line is . 5 5 Use point-slope form: 4 y 3 x 2 5 Slide P- 82 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. Slide P- 83 P.5 Solving Equations Graphically, Numerically, and Algebraically Quick Review Solutions Expand the product. 1. x 2 y 2 x 4 xy 4 y 2 2. 2 x 1 4 x 3 2 8x 2 x 3 2 Factor completely. 3. x 2 x x 2 3 2 4. y 5 y 36 4 2 x 1 x 1 x 2 y 9 y 2 y 2 2 5. Combine the fractions and reduce the resulting fraction to lowest terms. x 2 2x 1 x 1 x 5x 2 2 x 1 x 1 2 Slide P- 85 What you’ll learn about n n n n n 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- 86 Example Solving by Finding x-Intercepts Solve the equation 2 x 3 x 2 0 graphically. 2 Slide P- 87 Example Solving by Finding x-Intercepts Solve the equation 2 x 3 x 2 0 graphically. 2 Find the x-intercepts of y 2 x 3x 2. 2 Use the Trace to see that ( 0.5, 0) and (2,0) are x-intercepts. Thus the solutions are x 0.5 and x 2. Slide P- 88 Zero Factor Property Let a and b be real numbers. If ab = 0, then a = 0 or b = 0. Slide P- 89 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. Slide P- 90 Completing the Square To solve x bx c by completing the square, add (b / 2) to 2 2 both sides of the equation and factor the left side of the new equation. b b x bx c 2 2 2 2 2 b b x c 2 4 2 2 Slide P- 91 Quadratic Equation The solutions of the quadratic equation ax bx c 0, where 2 a 0, are given by the quadratic formula b b 4ac x . 2a 2 Slide P- 92 Example Solving Using the Quadratic Formula Solve the equation 2 x 3x 5 0. 2 a 2, b 3, c 5 b b 4ac x 2a (2 x 5)( x 1) 0 2 3 3 4 2 5 2 x 5 0 or x 1 0 2 2 2 3 49 4 3 7 4 5 x or x 1. 2 5 x or x 1 2 Slide P- 93 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- 94 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- 95 Example Solving by Finding Intersections Solve the equation | 2 x 1| 6. Slide P- 96 Example Solving by Finding Intersections Solve the equation | 2 x 1| 6. Graph y | 2 x 1| and y 6. Use Trace or the intersect feature of your grapher to find the points of intersection. The graph indicates that the solutions are x 2.5 and x 3.5. Slide P- 97 P.6 Complex Numbers Quick Review Add or subtract, and simplify. 1. (2 x 3) ( x 3) x6 2. (4 x 3) ( x 4) 3x 7 Multiply and simplify. 3. ( x 3)( x 2) x x6 4. x 3 x 3 5. (2 x 1)(3 x 5) 2 x 3 2 6 x 13 x 5 2 Slide P- 99 What you’ll learn about n n n n 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- 100 Complex Numbers Find two numbers whose sum is 10 and whose product is 40. n n n x = 1st number 10 – x = 2nd number x(10 – x) = 40 Slide P- 101 Complex Numbers n n n n n x(10 – x) = 40 10x – x2 = 40 x2 – 10x = -40 x2 – 10x + 25 = -40 +25 (x – 5)2 = -15 x 5 15 x 5 15, 5 15 Slide P- 102 Complex Numbers (5 15 ) (5 15 ) 10 (5 15 ) (5 15 ) 25 5 15 5 15 (15) 40 Slide P- 103 Complex Numbers n The imaginary number i is the square root of –1. i 1 i 1 2 16 1 16 4i 10 i 10 Slide P- 104 Complex Numbers n Imaginary numbers are not real numbers, so all the rules do not apply. Example: The product rule does not apply: 5 20 100 5 20 i 5 i 20 i 2 100 1 10 10 Slide P- 105 Complex Numbers n If a and b are real numbers, then a + bi is a complex number. n n n 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 Slide P- 106 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. Slide P- 107 Complex Numbers n Examples of complex numbers: 3 + 2i n 8 - 2i n 4 (since it can be written as 4 + 0i). The real numbers are a subset of the complex numbers. n -3i (since it can be written as 0 – 3i). n Slide P- 108 Complex Numbers i 1 2 i 2 1 1 i i i 1 i i 4 2 2 2 i i 1 1 3 2 Slide P- 109 Complex Numbers i i i 1 i i 5 4 i i i 1 1 1 6 4 2 i i i 1 i i 8 4 4 i i i 11 1 7 4 3 Slide P- 110 Complex Numbers * i -1 -i 1 i -1 -i 1 i -1 -i 1 i -1 -i 1 i 1 i -1 -1 -i -i 1 Slide P- 111 Complex Numbers Evaluate: i 35 i 24 i 1 i i i 1 1 i 4 8 4 6 8 3 6 Slide P- 112 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- 113 Example Multiplying Complex Numbers Find 3 2i 4 i . Slide P- 114 Example Multiplying Complex Numbers Find 3 2i 4 i . 3 2i 4 i 12 3i 8i 2i 2 12 5i 2( 1) 12 5i 2 14 5i Slide P- 115 Complex Conjugate The complex conjugate of the complex number z a bi is z a bi a bi. Slide P- 116 Discriminant of a Quadratic Equation For a quadratic equation ax bx c 0, where a, b, and c are 2 real numbers and a 0. if b 4ac 0, there are two distinct real solutions. 2 if b 4ac 0, there is one repeated real solution. 2 if b 4ac 0, there is a complex pair of solutions. 2 Slide P- 117 Example Solving a Quadratic Equation Solve x x 2 0. 2 Slide P- 118 Example Solving a Quadratic Equation Solve x x 2 0. 2 a b 1, and c 2. 1 1 4 1 2 2 x 2 1 1 7 2 1 i 7 2 So the solutions are x 1 i 7 1 i 7 and x . 2 2 Slide P- 119 Complex Numbers n When dividing a complex number by a real number, divide each part of the complex number by the real number. 6 8i 6 8i 3 2i 4 4 4 2 Slide P- 120 Complex Numbers n n n 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. Slide P- 121 Complex Numbers n 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 2i 2 = 3 2 – 2 2(-1) = 32 + 22 =9+4 = 13 Slide P- 122 Complex Numbers n When dividing a complex number by a complex number, multiply the denominator and numerator by the conjugate of the denominator. 2 3i 1 i 2 3i 1 i 1 i 1 i 2 2 i 3 i 3i 2 1 i 2 2 i 3 1 1 1 Slide P- 123 Complex Numbers 2 i 3 1 1 1 5i 2 5 1 i 2 2 Slide P- 124 Complex Numbers Slide P- 125 P.7 Solving Inequalities Algebraically and Graphically Quick Review Solve for x. 1. 3 2 x 1 9 2. | 2 x 1| 3 2 x 4 x 2 or x 1 3. Factor completely. 4 x 9 2 2 x 3 2 x 3 x 49 4. Reduce the fraction to lowest terms. x 7x x x2 5. Add the fractions and simplify. x 1 x 2 2 x7 x 2 x 3x 2 x x 2 2 Slide P- 127 What you’ll learn about n n n n 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- 128 Solving Absolute Value Inequalities Let u be an algebraic expression in x and let a be a real number with a 0. 1. If | u | a, then u is in the interval ( a, a ). That is, |u | a if and only if a u a. 2. If | u | a, then u is in the interval ( , a ) or ( a, ). That is, |u | a if and only if u a or u a. The inequalities < and > can be replaced with and , respectively. Slide P- 129 Solving Absolute Value Inequalities Solve 2x – 3 < 4x + 5 -2x < 8 x > -4 n n n -5 -4 -3 Solve |x – 2| < 1 -1 < x – 2 < 1 1<x<3 n n n 0 1 2 3 4 Slide P- 130 Solving Absolute Value Inequalities Solve -1 < 3 – 2x < 5 -4 < -2x < 2 2 > x > -1 -1 < x < 2 n n n n Solve |x – 1| > 3 -3 > x – 1 or x – 1 > 3 -2 > x or x > 4 x < -2 or x > 4 n n n n -2 -1 0 1 2 3 -2 -1 0 1 2 3 4 Slide P- 131 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 ( ) -1 0 1 2 3 4 5 ] [ -1 0 1 2 3 4 5 Slide P- 132 Example Solving an Absolute Value Inequality Solve | x 3 | 5. | x 3 | 5 5 x 3 5 8 x 2 As an interval the solution in ( 8, 2). Slide P- 133 Example Solving a Quadratic Inequality Solve x 3 x 2 0 2 ( x 2)( x 1) 0 x 2 or x 1. +++0--------0+++ -2 -1 Use these solutions and a sketch of the equation y x 3 x 2 to find the solution to the inequality 2 in interval form ( 2, 1). Slide P- 134 Example Solving a Quadratic Inequality Solve x2 – 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 +++0-------0+++ -4 5 Solution is (-4,5) Slide P- 135 Example Solving a Quadratic Inequality Solve x2 – 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 +++0------0+++ 0 3 Solution is (-∞,0) or (3, ∞) Slide P- 136 Example Solving a Quadratic Inequality Solve x3 – 6x2 + 8x < 0 1. 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 (-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 -----0++0----0+++ 0 2 4 Solution is (-∞,0] U [2,4] Slide P- 137 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. Slide P- 138 Chapter Test 1. Write the number in scientific notation. The diameter of a red blood corpuscle is about 0.000007 meter. 7 10 -6 2. Find the standard form equation for the circle with center (5, 3) and radius 4. x 5 y 3 16 2 2 3. Find the slope of the line through the points ( 1, 2) and (4, 5). 3 5 4. Find the equation of the line through (2, 3) and perpendicular to the line 2 x 5 y 3. y 5 x 8 2 x2 x5 1 5. Solve the equation algebraically. 3 2 3 9 x 5 1 3 6. Solve the equation algebraically. 6 x 7 x 3 x or x 3 2 2 Slide P- 139 Chapter Test 1 7. Solve the equation algebraically. | 4 x 1 | 3 x or x 1 2 2 8. Solve the inequality. | 3 x 4 | 2 (, 2] , 3 9. Solve the inequality. 4 x 12 x 9 0 2 , 10. Perform the indicated operation, and write the result in standard form. (5 7i ) (3 2i) 2 5i Slide P- 140