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A Brief Guide to Getting the Most from this Book EEDDIITTIIOONN 1 Read the Book Feature Description Benefit Applications Using From the chapter and section openers through the examples Ever wondered how you’ll use algebra? This feature will Real-World Data and exercises, interesting applications from nearly every show you how algebra can solve real problems. discipline, supported by up-to-date real-world data, are included in every section. Detailed Worked-Out Examples are clearly written and provide step-by-step The blue annotations will help you to understand the Examples solutions. No steps are omitted, and key steps are thoroughly solutions by providing the reason why the algebraic steps explained to the right of the mathematics. are true. Explanatory Voice balloons help to demystify algebra. They translate Does math ever look foreign to you? This feature translates Voice Balloons algebraic language into plain English, clarify problem-solving math into everyday English. procedures, and present alternative ways of understanding. Great Question! Answers to students’ questions offer suggestions for problem This feature should help you not to feel anxious or solving, point out common errors to avoid, and provide threatened when asking questions in class. informal hints and suggestions. Achieving Success The book’s Achieving Success boxes offer strategies for Follow these suggestions to help achieve your full success in learning algebra. academic potential in mathematics. 2 Work the Problems Feature Description Benefit Check Point Examples Each example is followed by a similar problem, called a Check You learn best by doing. You’ll solidify your understanding Point, that offers you the opportunity to work a similar exercise. of worked examples if you try a similar problem right away Answers to all Check Points are provided in the answer section. to be sure you understand what you’ve just read. Concept and Vocabulary These short-answer questions, mainly fill-in-the blank and It is difficult to learn algebra without knowing its special Checks true/false items, assess your understanding of the definitions language. These exercises test your understanding of the and concepts presented in each section. vocabulary and concepts. Extensive and An abundant collection of exercises is included in an Exercise The parallel order of the Practice Exercises lets you refer Varied Exercise Sets Set at the end of each section. Exercises are organized within to the worked examples and use them as models for several categories. Practice Exercises follow the same order solving these problems. Practice PLUS provides you with as the section’s worked examples. Practice PLUS Exercises ample opportunity to dig in and develop your problem- contain more challenging problems that often require you to solving skills. combine several skills or concepts. 3 Review for Quizzes and Tests Feature Description Benefit Mid-Chapter Near the midway point in the chapter, an integrated set of Combining exercises from the first half of the chapter gives Check Points review exercises allows you to review the skills and concepts you a comprehensive review before you continue on. you learned separately over several sections. Chapter Each chapter contains a review chart that summarizes the Review this chart and you’ll know the most important Review Charts definitions and concepts in every section of the chapter, material in the chapter. complete with examples. Chapter Tests Each chapter contains a practice test with problems that cover You can use the Chapter Test to determine whether you the important concepts in the chapter. Take the test, check have mastered the material covered in the chapter. your answers, and then watch the Chapter Test Prep Videos. Chapter Test Prep These videos contain worked-out solutions to every exercise in These videos let you review any exercises you miss on the Videos each chapter test. chapter test. Lecture Series on DVD These interactive lecture videos highlight key examples from These videos let you review each objective from the every section of the textbook. A new interface allows easy textbook that you need extra help on. navigation to sections, objectives, and examples. Prepare for Exams with Blitzer’s New Interactive Video Lecture Series with Chapter Test Prep Videos. Students can make the most of their study time by preparing for exams with the new Interactive Lecture Series with Chapter Test Prep Videos: ▶ Interactive Lectures highlight key examples and exercises from every section of the textbook. A new interface allows easy navigation to sections, objectives, and examples. These lectures are available in MyMathLab and on the Lecture Series on DVD. ▶ Chapter Test Prep Videos provide step-by-step video solutions for every problem from the Chapter Tests in the textbook. The Chapter Test Prep videos are available in MyMathLab, on the Lecture Series DVD, and on . Definitions, Rules, and Formulas The Real Numbers Set-Builder Notation and Graphs Natural Numbers: {1, 2, 3, …} {xa � x � b} a b Whole Numbers: {0, 1, 2, 3, …} {xa � x � b} Integers: {… , -3, -2, -1, 0, 1, 2, 3, …} a b Rational Numbers: a a and b are integers, b  0 {xa � x � b} b a b Irrational Numbers: {x|x is real and not rational} 5 0 6 {xx � b} a b {xx � a} a b Basic Rules of Algebra Commutative: a + b = b + a; ab = ba Associative: (a + b) + c = a + (b + c); Slope Formula (ab)c = a(bc) Distributive: aa((bb +- cc)) == aabb +- aacc; slope(m) = CChhaannggee iinn yx = yx22 -- xy11, x2 - x1  0 # Identity: a + 0 = a; a 1 = a 1. If m 7 0, the line rises from left to right. If m 6 0, the Inverse: a + (-a) = 0; a#1a = 1(a  0) line falls from left to right. Multiplication Properties: (-1)a = -a; 2. The slope of a horizontal line is 0. The slope of a (-1)(-a) = a; a#0 = 0; (-a)(b) = (a)(-b) = -ab; vertical line is undefined. (-a)(-b) = ab 3. Parallel lines have equal slopes. Order of Operations Equations of Lines 1. Perform operations above and below any fraction 1. Standard form: Ax + By = C bar, following steps (2) through (5). 2. Slope-intercept form: y = mx + b 2. Perform operations inside grouping symbols, innermost m is the line’s slope and b is its y-intercept. grouping symbols first, following steps (3) through (5). 3. Simplify exponential expressions. 3. Point-slope form: y - y1 = m(x - x1) m is the line’s slope and (x , y ) is a fixed point on the 1 1 4. Do multiplication and division as they occur, working line. from left to right. 4. Horizontal line parallel to the x-axis: y = b 5. Do addition and subtraction as they occur, working 5. Vertical line parallel to the y-axis: x = a from left to right. Definitions, Rules, and Formulas (continued) Properties of Exponents Rational Exponents 1. bm#bn = bm+n 1. an1 = n a 2. (bm)n = bmn 2. amn = 2n a m = n am 3. (ab)m = ambm 2 2 m 1 1 1 4. bm = bm-n 3. a-n =1amn =2 n a m = n am bn 2 2 a m am 1 2 5. b = bm The Quadratic Formula 6. ba0 =b 1, where b  0 The solutions of ax2 + bx + c = 0 with a  0 are 7. b-n = b1n and b1-n = bn,where b  0 x = -b { b2 - 4ac . 22a Special Factorizations Imaginary and Complex Numbers 1. Difference of two squares: A2 - B2 = A + B A - B 1. i = -1 and i 2 = -1 1 2. The set of numbers in the form a + bi is the set of 2. Perfect square trinomial1s: 21 2 complex numbers. If b = 0, the complex number a + bi A2 + 2AB + B2 = A + B 2 is a real number. If b Z 0, the complex number a + bi A2 - 2AB + B2 = A - B 2 is an imaginary number. 1 2 3. Sum of two cubes: 1 2 A3 + B3 = A + B A2 - AB + B2 The Graph of y  ax2  bx  c 4. Difference of two1 cubes: 21 2 1. The graph of y = ax2 + bx + c is called a parabola, A3 - B3 = A - B A2 + AB + B2 shaped like a bowl. If a 7 0, the parabola opens upward, and if a 6 0, the parabola opens downward. 1 21 2 The turning point of the parabola is the vertex. Variation 2. Graph y = ax2 + bx + c by finding any x-intercepts English Statement Equation (replace y with 0), the y-intercept (replace x with 0), y varies directly as x. y = kx the vertex, and additional points near the vertex and k y varies inversely as x. y = x intercepts. The x-coordinate of the vertex is -b . 2a -b Properties of Radicals The y-coordinate is found by substituting for x in 2a y = ax2 + bx + c and solving for y. All roots represent real numbers. 1. The product rule: 2. The quotient rule: 3. The vertex of y = ax2 + bx + c is a minimum point n # n n n when a 7 0 and a maximum point when a 6 0. a b = ab a n a = , b  0 2 2 2 2n b b 2 B Triangles 3. The Pythagorean Theorem 1. The sum of the measures of the interior angles of a In any right triangle with triangle is 180°. B hypotenuse of length c and legs 2. Similar triangles have corresponding angles with of length a and b, c2 = a2 + b2. the same measure and corresponding sides that are c a proportional. Two triangles are similar if two angles Hypotenuse Leg of one are equal in measure to two corresponding angles of the other. A b C Leg Common Formulas for Area, Perimeter, and Volume Square Rectangle Circle Triangle Trapezoid A = s2 A = lw A = pr2 A = qbh A = qh(a + b) P = 4s P = 2l + 2w C = 2pr b s w r h h s l b a Rectangular Circular Cube Solid Cylinder Sphere Cone V = s3 V = lwh V = pr2h V = dpr3 V = apr2h r s h h r h w r l s s Other Formulas 1. Temperature 5 C = (F - 32) (Fahrenheit to Celsius) 9 9 F = C + 32 (Celsius to Fahrenheit) 5 2. Percent A = PB (A is P percent of B.) 3. Simple Interest I = Prt (Interest is principal times rate times time.) 4. Distance d = rt (Distance is rate times time.) 5. Distance between Points The distance from (x1, y1) to (x2, y2) is (x2 - x1)2 + (y2 - y1)2 . 2 Definitions, Rules, and Formulas The Real Numbers 4. Horizontal line parallel to the x-axis: y = b Natural Numbers: {1, 2, 3, p} 5. Vertical line parallel to the y-axis: x = a Whole Numbers: {0, 1, 2, 3, p} Systems of Equations Integers: {p , -3, -2, -1, 0, 1, 2, 3, p} Rational Numbers: {a | a and b are integers, b Z 0} b y y y Irrational Numbers: {x | x is real and not rational} Basic Rules of Algebra x x x Commutative: a + b = b + a; ab = ba Associative: (a + b) + c = a + (b + c); (ab)c = a(bc) One solution: No solution: Infinitely many Distributive: a(b + c) = ab + ac; a(b - c) = ab - ac consistent inconsistent solutions: dependent and consistent # Identity: a + 0 = a; a 1 = a Inverse: a + (-a) = 0; a#1a = 1(a  0) A system of linear equations may be solved: Multiplication Properties: (-1)a = -a; (a) graphically, (b) by the substitution method, (c) by the # (-1)(-a) = a; a 0 = 0; (-a)(b) = (a)(-b) = -ab; addition or elimination method, (d) by matrices, or (e) by (-a)(-b) = ab determinants. a b Set-Builder Notation, Interval Notation, a1b1 = a1b2 - a2b1 and Graphs 2 2 (a, b) = {x | a 6 x 6 b} a b Cramer’s Rule: [a, b) = {x | a … x 6 b} 2 2 a b Given a system of a equations of the form (a, b] = {x | a 6 x … b} a b c b a c 1 1 1 1 [a, b] = {x | a … x … b} a b a1x + b1y = c1 , then x = c2b2 and y = a2c2 . (-, b) = {x | x 6 b} b a2x + b2y = c2 a1b1 a1b1 (-, b] = {x | x … b} a2b2 a2b2 b (a, ) = {x | x 7 a} 2 2 2 2 a Absolute Value [a, ) = {x | x Ú a} a 2 2 2 2 (-, ) = {x | x is a real number} = {x | xR} x if x Ú 0 1. x = -x if x 6 0 2. I0f |0x| = c, then x = c or x = -c. (c 7 0) Slope Formula 3. If |x| 6 c, then -c 6 x 6 c. (c 7 0) Change in y y2 - y1 4. If |x| 7bc, then x 6 -c or x 7 c. (c 7 0) slope (m) = Change in x = x2 - x1, x1  x2 Special Factorizations 1 2 Equations of Lines 1. Difference of two squares: 1. Slope-intercept form: y = mx + b A2 - B2 = (A + B)(A - B) m is the line’s slope and b is its y-intercept. 2. Perfect square trinomials: 2. Standard form: Ax + By = C 3. Point-slope form: y - y1 = m x - x1 A2 + 2AB + B2 = (A + B)2 m is the line’s slope and (x1, y1) is a fixed point on A2 - 2AB + B2 = (A - B)2 the line. 1 2 3. Sum of two cubes: 2. The complex numbers a + bi and a - bi are conjugates. Conjugates can be multiplied using the A3 + B3 = (A + B)(A2 - AB + B2) formula 4. Difference of two cubes: (A + B)(A - B) = A2 - B2. A3 - B3 = (A - B)(A2 + AB + B2) The multiplication of conjugates results in a real number. Variation 3. To simplify powers of i, rewrite the expression in English Statement Equation terms of i2. Then replace i2 with -1 and simplify. y varies directly as x. y = kx y varies directly as xn. y = kxn Quadratic Equations and Functions k y varies inversely as x. y = 1. The solutions of a quadratic equation in standard form x k ax2 + bx + c = 0, a  0, y varies inversely as xn. y = xn are given by the quadratic formula y varies jointly as x and z. y = kxz -b { b2 - 4ac x = . 22a Exponents 2. The discriminant, b2 - 4ac, of the quadratic equation Definitions of Rational Exponents ax2 + bx + c = 0 determines the number and type 1. an1 = n a 2. amn = n a m or n am of solutions. 21 2 2 3. a-mn = m 1 2 Discriminant Solutions an Positive perfect square 2 rational solutions Properties of Rational Exponents with a, b, and c rational numbers If m and n are rational exponents, and a and b are real Positive and not a perfect 2 irrational solutions numbers for which the following expressions are defined, square then Zero, with a, b, and c 1 rational solution bm rational numbers 1. bm#bn = bm+n 2. = bm-n bn Negative 2 imaginary solutions 3. (bm)n = bmn 4. (ab)n = anbn a n an 3. The graph of the quadratic function 5. = b bn f (x) = a(x - h)2 + k, a  0, a b is called a parabola. The vertex, or turning point, is Radicals (h, k). The graph opens upward if a is positive and downward if a negative. The axis of symmetry is a 1. If n is even, then n an = a . vertical line passing through the vertex. The graph 2. If n is odd, then n2an = a. can be obtained using the vertex, x-intercepts, if any, 0 0 3. The product rule2: n a# n b = n ab [set f (x) equal to zero], and the y-intercept (set x = 0). 2n a 2 a 2 4. A parabola whose equation is in the form n 4. The quotient rule: = 2n b b f (x) = ax2 + bx + c, a  0, 2 A Complex Numbers has its vertex at b b 1. The imaginary unit i is defined as - , f - . 2a 2a i = -1 , where i2 = -1. a a bb 1 If a 7 0, then f has a minimum that occurs at The set of numbers in the form a + bi is called the set b x = - . If a 6 0, then f has a maximum that occurs of complex numbers. If b = 0, the complex number 2a is a real number. If b  0 the complex number is an b imaginary number. at x = -2a. Definitions, Rules, and Formulas (continued) Exponential and Logarithmic Distance and Midpoint Formulas Functions 1. The distance from (x , y ) to (x , y ) is 1 1 2 2 1. Exponential Function: f (x) = bx, b 7 0, b  1 2 2 Graphs: (x2 - x1) + (y2 - y1) . 3 2. The midpoint of the line segment with endpoints (x , y ) and (x , y ) is y 1 1 2 2 x1 + x2 y1 + y2 , . 2 2 a b f(x) = bx f(x) = bx Conic Sections Circle 0 < b < 1 b > 1 y (0, 1) (x − h)2 + (y − k)2 = r2 x Center 2. Logarithmic Function: f (x) = logb x, b 7 0, b  1 (h, k) y = logbx is equivalent to x = by. Graphs: Radius: r Any point on the circle (x, y) y y y = x y = x x Ellipse y y Vertex (h, k + a) f(x) = bx f(x) = bx (0, 1) (x −a2 h)2+(y −b2 k)2= 1 x (0, 1) x (x −b2 h)2+(y −a2 k)2= 1 (1, 0) (1, 0) Major axis (h, k) (h, k) Vertex (h − a, k) Vertex (h + a, k) x f−1(x) = log x x b f−1(x) = logb x Vertex (h, k − a) Major axis Hyperbola b > 1 < b < 1 y2 x2 3. Properties of Logarithms xa22− by22= 1 y Ays y=mptbaotxe: a2− b2= 1 y Ays y=mptbaotxe: a. logb(MN) = logb M + logb N (0, a) (0, b) M b. logb N = logb M - logb N (−a, 0) (a, 0) (−b, 0) (b, 0) c. logb aMpb= plogb M x x log M ln M log M a d. logb M = log b = ln b = log b (0, −b) a (0, −a) ef.. lbologgbb bxx==x;x 1; 0lologg x10=x =x; exl;n l xn e=xx= x Ay s=ym −ptobatex: Ay s=ym −ptobatex: