INTERNATIONAL BACCALAUREATE MATHEMATICS STANDARD LEVEL 3RD EDITION 3rd Imprint Series editor: Fabio Cirrito Contributing author Patrick Tobin i MATHEMATICS Standard Level Copyright ©Patrick Tobin, Key-Strokes Pty Ltd, Mifasa Pty Ltd. First published in 1997 by IBID Press 2nd Edition published in 1999 by IBID Press, 3rd Edition published in 2004 (2nd Imprint published in 2005) by IBID Press. Reprinted 2007 Published by IBID Press, Victoria. Library Catalogue: Cirrito Fabio Editor. , Tobin 1. Mathematics, 2.

International Baccalaureate.Series Title: International Baccalaureate in Detail ISBN: 1 876659 17 3 (10-digit) 978 1 876659 17 2 (13-digit) All rights reserved except under the conditions described in the Copyright Act 1968 of Australia and subsequent amendments. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior permission of the publishers. While every care has been taken to trace and acknowledge copyright, the publishers tender their apologies for any accidental infringement where copyright has proved untraceable.They would be pleased to come to a suitable arrangement with the rightful owner in each case.

This book has been developed independently of the International Baccalaureate Organisation (IBO). The text is in no way connected with, or endorsed by, the IBO. “This publication is independently produced for use by teachers and students. Although references have been reproduced with permission of the VCAA the publication is in no way connected or endorsed by the VCAA. ” We also wish to thank the Incorporated Association of Registered Teachers of Victoria for granting us permission to reproduced questions from their October Examination Papers.Cover design by Adcore Creative.

Published by IBID Press, at www. ibid. com. au For further information contact [email protected] com. au Printed by SHANNON Books, Australia. ii PREFACE TO 3RD EDITION The 3rd edition of the Mathematics Standard Level text has been completely revised and updated.

Sections of the previous two editions are still present, but much has happened to improve the text both in content and accuracy. In response to the many requests and suggestions from teachers worldwide the text was extensively revised. There are more examples for students to refer to when learning the subject matter for the ? st time or during their revision period. There is an abundance of well-graded exercises for students to hone their skills. Of course, it is not expected that any one student work through every question in this text – such a task would be quite a feat. It is hoped then that teachers will guide the students as to which questions to attempt. The questions serve to develop routine skills, reinforce concepts introduced in the topic and develop skills in making appropriate use of the graphics calculator.

The text has been written in a conversational style so that students will ? nd that they are not simply making reference to an encyclopedia ? led with mathematical facts, but rather ? nd that they are in some way participating in or listening in on a discussion of the subject matter. Throughout the text the subject matter is presented using graphical, numerical, algebraic and verbal means whenever appropriate. Classical approaches have been judiciously combined with modern approaches re? ecting new technology – in particular the use of the graphics calculator.

The book has been speci? cally written to meet the demands of the Mathematics Standard Level course and has been pitched at a level that is appropriate for students studying this subject.The book presents an extensive coverage of the syllabus and in some areas goes beyond what is required of the student. Again, this is for the teacher to decide how best to use these sections. Sets of revision exercises are included in the text. Many of the questions in these sets have been aimed at a level that is on par with what a student could expect on an examination paper. However, some of the questions do go beyond the level that students may expect to ? nd on an examination paper. Success in examinations will depend on an individual’s preparation and they will ? d that making use of a selection of questions from a number of sources will be very bene? cial. Chapter 23 is dedicated to portfolio work.

It includes many topics that students can use, either as they stand or as a source of ideas from which to then produce their own investigation. Each suggestion has been extensively trialled and they have been included in the book because they will address the assessment criteria required for the successful completion of the profolio work. I hope that most of the suggestions and recommendations that were brought forward have been addressed in this edition.However, there is always room for improvement. As always, I welcome and encourage teachers and students to contact me with feedback, not only on their likes and dislikes but suggestions on how the book can be improved as well as where errors and misprints occur. There will be updates on the IBID Press website in relation to errors that have been located in the book – so we suggest that you visit the IBID website at www. ibid.

com. au. If you believe you have located an error or misprint please email me at [email protected] com. au. Fabio Cirrito, July 2004 iiiMATHEMATICS Standard Level PREFACE TO 2ND EDITION The principles behind this second edition are very much the same as those of the ? rst. Where possible we have made amendments based on recommendations and requests by teachers worldwide. Corrections to answers have been made as well as to the text.

The support shown by the many teachers worldwide has been very encouraging, and for this reason we have this second edition as well as other support material relating to The Portfolio work and Examination practice papers for students as well as teachers support material.Fabio Cirrito, 1998 PREFACE TO 1ST EDITION This text has been produced independently as a resource to support the teaching of the Mathematical Methods Standard Level Course of the International Baccalaureate. The examples and questions do not necessarily re? ect the views of the of? cial senior examining team appointed by the International Baccalaureate Organisation. The notation used is, as far as possible, that speci? ed in the appropriate syllabus guidelines of the IB.

The units of physical measurements are in S. I. The language and spelling are U.K.

English. Currency quantities are speci? ed in dollars, though these could be read as any currency that is decimalised, such as Swiss francs, Lire etc. The graphic calculators covered directly in the text are the Texas TI/82 and 83. Supplementary material is available from the publisher for students using some other makes and models of calculators. As it is important that students learn to interpret graphic calculator output, the text and answers present a mixture of graphic calculator screens and conventional diagrams when discussing graphs.The text has been presented in the order in which the topics appear in the syllabus.

This does not mean that the topics have to be treated in this order, though it is generally the case that the more fundamental topics appear at the front of the book. Students are reminded that it is the IB Syllabus that speci? es the contents of the course and not this text. One of the keys to success in this course is to be thoroughly familiar with the course contents and the styles of questions that have been used in past examinations. Fabio Cirrito, August 1997. iv CONTENTS 1 1. 1 1. 1.

3 1. 3. 1 1. 3. 2 1. 3. 3 1. 3.

4 1. 4 1. 4. 1 1. 4.

2 1. 5 2 2. 1 2. 1. 1 2. 1.

2 2. 1. 3 2. 1. 4 2. 1.

5 2. 1. 6 2. 2 2. 2. 1 2. 2. 2 2.

3 2. 3. 1 2. 3.

2 2. 3. 3 2. 4 2. 4. 1 2. 4. 2 2.

4. 3 2. 4.

4 3 3. 1 3. 1. 1 3.

1. 2 3. 1. 3 3. 2 3. 2. 1 3.

2. 2 4 4. 1 4. 1. 1 4. 1.

2 4. 2 THEORY OF KNOWLEDGE Pure and Applied Mathematics Axioms Proof Rules of Inference Proof by Exhaustion Direct Proof Proof by Contradiction Paradox What is a Paradox? Russell’s Paradox? Mathematics and Other Disciplines ALGEBRA OF LINEAR AND QUADRATIC EXPRESSIONS The Real Number Line The RealNumber Line Set Builder Notation Interval Notation Number Systems Irrational Numbers The Absolute Value Linear Algebra Review of Linear Equations Linear Inequations Linear Functions Graph of the Linear Function Simultaneous Linear Equations in Two Unknowns Simultaneous Linear Equations in Three Unknowns Quadratics Quadratic Equation Quadratic Function Quadratic Inequalities Simultaneous Equations Involving Linear-Quadratic Equations Modelling Modelling – An introduction Modelling – Phase 1 Modelling – Phase 2 Modelling – Phase 3 Mathematical Models Mathematical Investigation and Modelling Open and Closed problem Solving THE BINOMIAL THEOREM The Binomial Theorem The Binomial Theorem The General term Proof v 1 1 2 4 5 6 7 8 10 10 11 12 15 15 15 15 15 17 18 20 21 21 26 28 28 33 37 39 39 44 51 54 59 59 59 65 78 89 89 90 95 95 95 100 104 MATHEMATICS Standard Level 5 5. 1 5. 1. 1 5.

1. 2 5. 1. 3 5. 1.

4 5. 1. 5 5. 2 5.

2. 1 5. 3 5. 3. 1 5. 3.

2 5. 3. 3 5. 3. 4 5. 3.

5 5. 4 5. 4. 1 5. 4. 2 6 6. 1 6.

1. 1 6. 1. 2 6. 2 6. 2. 1 6.

2. 2 6. 3 6. 4 7 7.

1 7. 1. 1 7. 1. 2 7. 1.

3 7. 1. 4 7.

1. 5 7. 2 7. 3 7.

3. 1 7. 3. 2 7. 4 7. 5 FUNCTIONS AND RELATIONS Relations Relations The Cartesian Plane Implied Domain Types of Relations Sketching with the Graphics Calculator Functions De? itions Some Standard Functions Hybrid Functions and Continuity The Absolute Value Function The Exponential Function The Logarithmic Function Equations of the Form y = x n, n = – 1, – 2 Algebra of Functions Basic Operations and Composite Functions Identity and Inverse Functions TRANSFORMATIONS OF GRAPHS Translations Horizontal Translation Vertical Translation Dilations Dilation from the x-axis Dilation from the y-axis Re? ections Reciprocal of a Function EXPONENTIAL AND LOGARITHMIC FUNCTIONS Exponents Basic Rules of Indices Indicial Equations Equations of the form b f ( x ) = b g ( x ) What if the Base is Not The Same? A Special Base (e) Exponential Modelling Logarithms What are Logarithms Can we ? nd the Logarithm of a Negative Number?The Algebra of Logarithms Logarithmic Modelling 105 105 105 107 108 109 110 115 115 122 122 126 131 138 144 148 148 157 167 167 167 170 177 177 179 183 188 197 197 197 201 203 205 207 209 217 217 219 221 230 233 241 241 241 245 249 252 REVISION SET A – PAPER 1 AND PAPER 2 STYLE QUESTIONS 8 8.

1 8. 1. 1 8. 1. 2 8. 1.

3 8. 2 SEQUENCES AND SERIES Arithmetic Sequences and Series Arithmetic Sequences Arithmetic Series Sigma Notation Geometric Sequences and Series vi 8. 2. 1 8. 2. 2 8. 2. 3 8.

2. 4 8. 3 8. 3. 1 8.

3. 2 9 9. 1 9. 1. 1 9. 1.

2 9. 2 9. 2. 1 9. 2. 2 9. 3 9.

4 9. 5 9. 5.

1 9. 5. 2 9. 5.

3 9. 5. 4 9. 5. 5 9. 6 9.

7 9. 7. 1 9. 7. 2 9. 7. 3 10 10. 1 10.

1. 1 10. 1. 2 10. 2 10.

2. 1 10. 2. 2 10. 3 10. 3. 1 10. 3.

2 10. 4 10. 4. 1 10. 5 Geometric Sequences Geometric Series Combined A. Ps and G.Ps Convergent Series Compound Interest and Superannuation Compound Interest Superannuation MENSURATION Trigonometric Ratios Review of Trigonometric Functions for Right-angled Triangles Exact Values Applications Angle of Elevation and Depression Bearings Right Angles in 3–Dimensions Area of a Triangle Non Right-Angled Triangles The Sine Rule The Ambiguous Case Applications of The Sine Rule The Cosine Rule Applications of The Cosine Rule More Applications in 3–D Arc, Sectors and Segments Radian Measure of an Angle Arc Length Area of Sector CIRCULAR TRIGONOMETRIC FUNCTIONS Trigonometric Ratios The Unit Circle Angle of Any Magnitude Trigonometric Identities The Fundamental Identity Compound Angle Identities Trigonometric Functions The Sine, Cosine and Tangent Functions Transformations of Trigonometric Functions Trigonometric Equations Solution to sin x = a, cos x = a and tan x = a Applications 252 257 261 263 267 267 268 273 273 273 273 278 278 280 283 287 290 290 294 298 300 302 307 309 309 310 311 315 315 315 316 327 327 332 337 337 340 351 351 361 367 375 375 375 381 388 388 391 REVISION SET B – PAPER 1 AND PAPER 2 STYLE QUESTIONS 11 11. 1 11.

. 1 11. 1. 2 11. 2 11.

2. 1 11. 2. 2 MATRICES Introduction to Matrices De? nitions Matrix Multiplication Inverses and Determinants Inverse and Determinant of 2 by 2 Matrices Inverse and Determinant of 3 by 3 Matrices vii MATHEMATICS Standard Level 11. 3 12 12. 1 12. 1.

1 12. 2 12. 2. 1 12. 2. 2 12.

2. 3 12. 2. 4 12. 2. 5 12. 2. 6 12.

3 12. 3. 1 12. 3. 2 12. 3. 3 12. 3.

4 12. 3. 5 12.

4 12. 4. 1 12. 4.

2 12. 4. 3 12. 4. 4 12. 5 12. 5. 1 12.

5. 2 12. 6 12. 6. 1 12. 6.

2 12. 6. 3 12. 6. 4 12.

6. 5 12. 7 12. 7. 1 12. 7.

2 12. 7. 3 Simultaneous Equations VECTORS Introduction to Vectors Scalar and Vector Quantities Representing Vectors Directed LineSegment Magnitude of a Vector Equal Vectors Negative Vectors Zero Vector Orientation and Vectors Algebra and Geometry of Vectors Addition of Vectors Subtraction of Vectors Multiplication of Vectors by Scalars Angle Between Vectors Applications to Geometry Cartesian Representation of Vectors in 2-D and 3-D Representation in 2-D Unit Vector and Base Vector Notation Representation in 3-D Vector Operations Further Properties of Vectors in 2-D and 3-D Magnitude of a Vector Unit Vectors Scalar Product of 2 Vectors De? nition of the Scalar Product Properties of the Scalar Product Special Cases of the Scalar Product Direction Cosine Using a Graphics Calculator Vector Equation of a Line Vector Equation of a Line in 2-D Application of the vector equation r = a + tb Lines in Three Dimensions 400 409 409 409 410 410 410 410 411 411 411 415 415 416 416 417 417 423 423 423 424 425 429 429 430 432 432 434 435 438 443 444 444 452 456 465 469 469 469 469 470 471 471 472 474 474 474 474 476REVISION SET C – PAPER 1 AND PAPER 2 STYLE QUESTIONS 13 13. 1 13. 1. 1 13.

1. 2 13. 1.

3 13. 2 13. 2.

1 13. 2. 2 13. 3 13. 3. 1 13. 3.

2 13. 3. 3 13.

3. 4 STATISTICS Describing Data Data Collection Types of Data Discrete and Continuous Data Frequency Diagrams Producing a Frequency Diagram Using a Graphics Calculator Statistical Measures 1 Measure of Central Tendency Mode Mean Median viii 13. 4 13.

4. 1 13. 4.

2 13. 4. 3 13.

5 13. 5. 1 13. 5. 2 14 14. 1 14. 1.

1 14. 1. 2 14. 1. 3 14. 2 15 15. 1 15.

1. 1 15. 1. 2 15. 1. 3 15. 1. 4 15.

1. 5 15. 2 15. 3 15.

3. 1 15. 3. 2 15. 3. 3 15. 4 15.

4. 1 15. 4. 2 15. 5 16 16. 1 16. 1. 1 16.

1. 2 16. 1. 3 16. 1. 4 16.

1. 5 16. 2 16. 2. 1 16. 2. 2 16. .

3 16. 2. 4 16. 2. 5 16.

3 16. 3. 1 16.

3. 2 16. 3. 3 16. 3.

4 16. 3. 5 Statistical Measures 2 Measures of Spread Variance and Standard Deviation Using a Graphics Calculator Statistical Measures 3 Quartiles Box Plot COUNTING PRINCIPLES Multiplication Principle De? nition Multiplication Principle Permutations Combinations PROBABILITY Probability Probability as a Long–Term Relative Frequency Theoretical Probability Laws of Probability De? nition of Probability Problem Solving Strategies in Probability Probability & Venn Diagrams Conditional Probability Informal De? nition of Conditional Probability Formal De? nition of Conditional ProbabilityIndependence Bayes’ Theorem Law of Total Probability Bayes’ Theorem for Two Elements Using Permutations and Combinations in Probability DISCRETE RANDOM VARIABLES Discrete Random Variables Concept of a Random Variable Discrete Random Variable Probability Distributions Properties of The Probability Function Constructing Probability Functions Mean and Variance Central Tendency and Expectation So, what exactly does E ( X ) measure? Properties of The Expectation (Function) Variance Properties of The Variance The Binomial Distribution The Binomial Experiment Bernoulli Trials Properties of The Binomial Experiment Binomial Distribution Expectation, Mode and Variance for the Binomial Distribution ix 477 477 478 480 482 482 484 491 491 491 491 493 498 503 503 503 503 504 504 506 508 512 512 512 514 519 519 520 524 527 527 527 527 528 529 531 535 535 535 537 539 540 544 544 544 544 544 547 MATHEMATICS Standard Level 17 17. 1 17. 1. 1 17. 1.

2 17. 1. 3 17. 2 17. 2. 1 17.

2. 2 17. 2. 3 17. 2. 4 17. 2.

5 17. 2. 6 17. 2. 7 17.

2. THE NORMAL DISTRIBUTION The Normal Distribution Why The Normal Distribution? The Standard Normal Curve Using The Standard Normal Table Formalising The De? nition of The Normal Distribution The Normal Distribution Properties of This Curve Finding Probabilities Using The Normal Distribution The Standard Normal Distribution Finding Probabilities Standardising any Normal Distribution Inverse Problems Finding Quantiles 553 553 553 553 554 557 557 558 558 558 559 559 564 566 573 581 581 581 582 582 583 584 588 588 588 588 591 591 597 597 598 601 601 601 604 606 609 609 614 617 618 618 619 620 620 622 623 630 635 REVISION SET D – PAPER 1 AND PAPER 2 STYLE QUESTIONS 18 18. 18. 1. 1 18.

1. 2 18. 1. 3 18. 1. 4 18. 1.

5 18. 2 18. 2. 1 18. 2.

2 18. 2. 3 18. 3 18.

3. 1 18. 4 18. 4. 1 18. 4. 2 19 19. 1 19.

1. 1 19. 1.

2 19. 1. 3 19. 2 19. 2. 1 19. 2. 2 19.

2. 3 19. 3 19. 3.

1 19. 3. 2 19. 3. 3 19. 3. 4 19.

3. 5 19. 3. 6 19. 3.

7 19. 4 RATES OF CHANGE Quantitative Measure Functional Dependence Quantitative Aspects of Change Average Rate of Change Determining the Average Rate of Change Velocity as a Measure of Rate of Change of Displacement Qualitative Measure Qualitative Aspects of Change Describing the Behaviour of a Graph Producing a Graph from a Physical Situation Instantaneous Rate of Change Informal Idea of LimitsDifferentiation Process The Derivative and The Gradient Function Notation and Language DIFFERENTIAL CALCULUS Differentiation Review Power Rule for Differentiation Derivative of a Sum or Difference Graphical Interpretation of The Derivative The Value of The Derivative at a Particular Point on a Curve Gradient Function from a Graph Differentiating with Variables other than x and y Derivative of Transcendental Functions Derivative of Circular Trigonometric Functions Derivative of The Exponential Functions Derivative of The Natural Log Functions Derivative of a Product of Functions Derivative of a Quotient of Functions The Chain Rule Derivative of Reciprocal Circular Functions Second Derivative x 19. 5 19. 5.

1 19. 5. 2 19. 5. 3 19. 5. 4 19. 5.

5 20 20. 1 20. 1. 1 20. 1.

2 20. 2 20. 2. 1 20. 2. 2 20. 2. 3 20.

2. 4 20. 2. 5 20. 3 20. 3. 1 20.

4 20. 4. 1 20. 4. 2 21 21.

1 21. 1. 1 21. 1.

2 21. 2Proofs Derivative of y = x n where n is a Negative Integer Derivative of y = x n where n is a Fraction Product Rule and Quotient Rule Derivative of Some Trigonometric Functions Exponential and y = x n where n is Real DIFFERENTIAL CALCULUS AND CURVE SKETCHING Tangents and Normals Equation of Tangent Equation of Normal Curve Sketching Increasing and Decreasing Functions Stationary Points Global Maxima and Minima Vertical Tangents and Cusps Summary The Second Derivative and its Application De? nition Rational Functions ax + b Sketching the Graph of x ————– , cx + d ? 0 cx + d Other Rational Functions APPLICATIONS OF DIFFERENTIAL CALCULUS Rates Of Change De? nitions Rates of Change and Their Sign Applied Rates of Change Application to Economics Application to Kinematics Kinematics Motion Along a Straight Line Applied Maxima and Minima Problems Maxima–Minima Problems End point Maxima/Minima Problems Optimisation for Integer Valued Variables INTEGRATION AND ITS APPLICATIONS Integration Antidifferentiation & The Inde? nite Integral Language and Notation Determining The Inde? ite Integral Properties of The Inde? nite Integral Solving for ‘c’ Standard Integrals The De? nite Integral Why The De? nite Integral? Language and Notation Properties of The De? nite Integral Applications of Integration xi 637 637 637 638 638 641 643 643 643 645 649 649 651 662 666 668 672 672 679 679 681 687 687 687 688 689 690 691 694 694 702 702 713 715 723 723 723 723 724 726 728 731 740 740 740 743 748 21. 3 20.

3. 1 21. 4 21. 4. 1 21. 4. 2 21. 4. 3 22 22. 1 22. 1. 1 22. 1. 2 22. 1. 3 22. 1. 4 22. 2 22. 3 22. 4 22. 4. 1 22. 4. 2 22. 4. 3 22. 5 MATHEMATICS Standard Level 22. 5. 1 22. 5. 2 22. 5. 3 22. 5. 4 22. 5. 5 22. 5. 6 22. 5. 7 22. 5. 8 22. 6 22. Introduction to the Area Beneath a Curve In Search of a Better Approximation Towards an Exact Value The De? nite Integral and Areas Further Observation about Areas The Signed Areas Steps for Finding Areas Area Between Two Curves Applications to Kinematics Volumes (Solid of Revolution) 748 749 750 751 752 753 754 755 762 766 771 781 781 781 783 809 809 843 843 844 REVISION SET E – PAPER 1 AND PAPER 2 STYLE QUESTIONS 23 23. 1 23. 1. 1 23. 1. 2 23. 2 23. 2. 1 23. 3 23. 3. 1 23. 3. 2 PORTFOLIO WORK Introducing The Portfolio Purpose of the Portfolio Problem Solving Portfolio Work Type I – Mathematical Investigation All is ? ne and well . . . but where do I start?Portfolio Work Type II – Mathematical Modelling Carrying Out a Modelling Investigation Modelling Investigation Tasks ANSWERS to Exercises ANSWERS to Revision Sets Answers 1 Answers 67 xii NOTATION The list below represents the signs and symbols which are recommended by the International Organization for Standardization as well as other symbols that are used in the text. the set of positive integers and zero, {0, 1, 2, 3,… } the set of integers, {0, ±1, ±2, ±3… } the set of positive integers, {1, 2, 3,… } ? ? a the set of rational numbers ? x|x = –, b ? 0, a, b ? ? b ? ? the set of positive rational numbers, { x|x ? , x > 0 } the set of real numbers + he set of positive real numbers { x|x ? , x > 0 } the set of complex numbers, { a + bi|a, b ? } the set with elements x 1, x 2 … the number of elements in the ? nite set A the set of all x such that… is an element of is not an element of the empty (null) set the universal set union intersection is a proper subset of is a proper subset of the complement of set A the Cartesian product of sets A & B, ( A ? B = { ( a, b )|a ? A, b ? B } ) a divides b a to the power a to the power 1 -n 1 -2 C { x 1, x 2 … } n(A) { x| } ? ? O U ? ? ? ? A’ A? B a|b a 1 / n, n a a 1 / 2, a x ? ? or the nth root of a or the square root of a? 0 ? x, x ? 0 the modulus or absolute value of x ? – x, x ;lt; 0 identity is approximately equal to xiii MATHEMATICS Standard Level ;gt; ? ;lt; ? ;gt; ;lt; [ a, b ] ]a, b [ un d r Sn S? is greater than is greater than or equal to is less than is less than or equal to is not greater than is not less than the closed interval a ? x ? b the open interval a ;lt; x ;lt; b the nth term of a sequence or series the common difference of an arithmetic sequence the common ratio of an geometric sequence the sum of the ? rst n terms of a sequence u 1 + u 2 + u 3 + … + u n the sum to in? nity of a sequence u 1 + u 2 + u 3 + … u1 + u2 + … + un ? ui i=1 n ? ui i=1 n u1 ? u2 ? … ? un n! ———————- the rth binimial coef? cient, r = 0, 1, 2, . . . n the expansion of ( a + b ) n r! ( n – r )! f is a function under which each element of set A has an image in set B f is a function under which x is mapped to y the image of x under the function f the inverse function of the function f the composite function of f and g the limit of f ( x ) as x tends to a the derivative of y with respect to x the derivative of f ( x ) with respect to x the second derivative of y with respect to x the second derivative of f ( x ) with respect to x the nth derivative of y with respect to x ? n? ? r? f :A > B f 😡 > y f ( x) f ( x) f og x>a –1 lim f ( x ) dy —-dx f ‘( x) d y ——2 dx f ” ( x ) d y ——n dx n 2 xiv f (n) ( x) he nth derivative of f ( x ) with respect to x the inde? nite integral of y with respect to x the inde? nite integral of y with respect to x between the limits x = a and x = b the exponential function of x logarithm to the base a of x the natural logarithm of x, log ex the circular functions the reciprocal circular functions the point A in the plane with Cartesian coordinates x and y the line segment with endpoints A and B the length of [AB] the line containing points A and B the angle at A the angle between the lines [CA] and [AB] the triangle whose vertices are A, B and C the vector v the vector represented in magnitude and direction by the directed line egment from A to B the position vector OA unit vectors in the directions of the Cartesian coordinate axes the magnitude of a the magnitude of AB the scalar product of v and w the inverse of the non-singular matrix A the transpose of the matrix A the determinant of the square matrix A the identity matrix probability of event A probability of the event “not A” probability of the event A given B observations frequencies with which the observations x 1, x 2, … occur probability distribution function P(X = x) of the discrete random variable X xv ? y dx ? a y dx ex log ax ln x sin,cos,tan csc,sec,cot A ( x, y ) [AB] AB (AB) ? A ? CA B ? ABC v AB b a i,j,k |a| AB v? A –1 AT detA I P(A) P(A’) P(A|B) x 1, x 2, … f 1, f 2, … Px MATHEMATICS Standard Level E( X ) Var ( X ) µ the expected value of the random variable X the variance of the random variable X population mean ?2 ? x =1 population variance, ? 2 = i———————————- where n = n ? k f i ( xi – µ )2 ? i=1 k fi population standard deviation sample mean s2 n sn sample variance, ? 2 = ——————————— where n = n i=1 ? k f i ( xi – x )2 ? i=1 k fi standard deviation of the sample s2 – 1 n B(n,p) N ( µ, ? 2 ) X~B(n,p) unbiased estimate of the population variance s 2 – 1 n binomial distribution with parameters n and p normal distribution with mean µ and variance ? 2 f i ( xi – x )2 n 2 =1 = ———– s n or i——————————–n–1 n–1 k the random variable X has a binomial distribution with parameters n and p X~ N ( µ, ? 2 ) the random variable X has a normal distribution with mean µ and variance ? 2 ? cumulative distribution function of the standardised normal variable: N(0,1) xvi Theory of Knowledge – CHAPTER 1 1. 1 PURE AND APPLIED MATHEMATICS Mathematics has clearly played a signi? cant part in the development of many past and present civilisations. There is good evidence that mathematical, and probably astronomical techniques, were used to build the many stone circles of Europe which are thought to be at least three thousand years old (Thom).It is likely that the Egyptian pyramids and constructions on Aztec and Mayan sites in South America were also built by mathematically sophisticated architects. Similarly, cultures in China, India and throughout the Middle East developed mathematics a very long time ago. It is also the case that there have been very successful cultures that have found little use for mathematics. Ancient Rome, handicapped, as it was, by a non-place value number system did not develop a mathematical tradition at anything like the same level as that of Ancient Greece. Also, the Australian Aborigines who have one of the most long lasting and successful cultures in human history did not ? nd much need for mathematical methods. The same is true of the many aboriginal cultures of Africa, Asia and the Americas.This may well be because these aboriginal cultures did not value ownership in the way that western culture does and had no need to count their possessions. Instead, to aboriginal cultures, a responsible and sustainable relationship with the environment is more important than acquisition and exploitation. Maybe we should learn from this before it is too late! Mathematics has developed two distinct branches. Pure mathematics, which is studied for its own sake, and applied mathematics which is studied for its usefulness. This is not to say that the two branches have not cross-fertilised each other, for there have been many examples in which they have.The pure mathematician Pierre de Fermat (1601-1665) guessed that the equation x n + y n = z n has whole numbered solutions for n = 2 only. To the pure mathematician, this type of problem is interesting for its own sake. To study it is to look for an essential truth, the ‘majestic clockwork’ of the universe. Pure mathematicians see ‘beauty’ and ‘elegance’ in a neat proof. To pure mathematicians, their subject is an art. Applied mathematics seeks to develop mathematical objects such as equations and computer algorithms that can be used to predict what will happen if we follow a particular course of action. This is a very valuable capability. We no longer build bridges without making careful calculations as to whether or not they will stand.Airline pilots are able to experience serious failures in commercial jets without either risking lives or the airline’s valuable aeroplanes or, indeed, without even leaving the ground. CHAPTER 1 1 MATHEMATICS Standard Level 1. 2 AXIOMS Mathematics is based on axioms. These are ‘facts’ that are assumed to be true. An axiom is a statement that is accepted without proof. Early sets of axioms contained statements that appeared to be obviously true. Euclid postulated a number of these ‘obvious’ axioms. Example: That is, ‘Things equal to the same thing are equal to each other’; if y = a and x = a then y = x. Euclid was mainly interested in geometry and we still call plane geometry ‘Euclidean’. In Euclidean space, the shortest distance between two points is a straight line.We will see later that it is possible to develop a useful, consistent mathematics that does not accept this axiom. Most axiom systems have been based on the notion of a ‘set’, meaning a collection of objects. An example of a set axiom is the ‘axiom of speci? cation’. In crude terms, this says that if we have a set of objects and are looking at placing some condition or speci? cation on this set, then the set thus speci? ed must exist. We consider some examples of this axiom. Example: Assume that the set of citizens of China is de? ned. If we impose the condition that the members of this set must be female, then this new set (of Chinese females) is de? ned.As a more mathematical example, if we assume that the set of whole numbers exists, then the set of even numbers (multiples of 2) must also exist. A second example of a set axiom is the ‘axiom of powers’: Example: For each set, there exists a collection of sets that contains amongst its elements all the subsets of the original set. If we look at the set of cats in Bogota, then there must be a set that contains all the female cats in Bogota, another that contains all the cats with green eyes in Bogota, another that contains all the Bogota cats with black tails, etc. A good, but theoretical, account of axiomatic set theory can be found in Halmos, 1960. Mathematics has, in some sense, been a search for the smallest possible set of consistent axioms.In the section on paradox, we will look further at the notion of axioms and the search for a set of assumptions that does not lead to contradictions. There is a very strong sense in which mathematics is an unusual pursuit in this respect. Pure mathematics is concerned with absolute truth only in the sense of creating a self-consistent structure of thinking. As an example of some axioms that may not seem to be sensible, consider a geometry in which the shortest path between two points is the arc of a circle and all parallel lines meet. These ‘axioms’ do not seem to make sense in ‘normal’ geometry. The ? rst mathematicians to investigate non-Euclidean geometry were the Russian, Nicolai Lobachevsky (1793-1856) and the Hungarian, Janos Bolyai (1802-1860).Independently, they developed self consistent geometries that did not include the so called parallel postulate which states that for every line AB and point C outside AB there is only one line through C that does not meet AB. 2 Theory of Knowledge – CHAPTER 1 C A B Since both lines extend to in? nity in both directions, this seems to be ‘obvious’. Non-Euclidean geometries do not include this postulate and assume either that there are no lines through C that do not meet AB or that there is more than one such line. It was the great achievement of Lobachevsky and Bolyai that they proved that these assumptions lead to geometries that are self consistent and thus acceptable as ‘true’ to pure mathematicians.In case you are thinking that this sort of activity is completely useless, one of the two non-Euclidean geometries discussed above has actually proved to be useful; the geometry of shapes drawn on a sphere. This is useful because it is the geometry used by the navigators of aeroplanes and ships. Rome Djakarta The ? rst point about this geometry is that it is impossible to travel in straight lines. On the surface of a sphere, the shortest distance between two points is an arc of a circle centred at the centre of the sphere (a great circle). The shortest path from Rome to Djakarta is circular. If you want to see this path on a geographer’s globe, take a length of sewing cotton and stretch it tightly between the two cities.The cotton will follow the approximate great circle route between the two cities. If we now think of the arcs of great circles as our ‘straight lines’, what kind of geometry will we get? You can see some of these results without going into any complex calculations. For example, what would a triangle look like? The ? rst point is that the angles of this triangle add up to more than 180?. There are many other ‘odd’ features of this geometry. However, fortunately for the international airline trade, the geometry is self consistent and allows us to navigate safely around the surface of the globe. Thus non-Euclidean geometry is an acceptable pure mathematical structure.While you are thinking about unusual geometries, what are the main features of the geometry of shapes drawn on the ‘saddle surface’? 3 MATHEMATICS Standard Level One ? nal point on the subject of non-Euclidean geometries; it seems to be the case that our three dimensional universe is also curved. This was one of the great insights of Albert Einstein (18791955). We do not yet know if our universe is bent back on itself rather like a sphere or whether another model is appropriate. A short account of non-Euclidean Geometries can be found in Cameron (pp31-40). By contrast, applied mathematics is judged more by its ability to predict the future, than by its self-consistency.Applied mathematics is also based on axioms, but these are judged more on their ability to lead to calculations that can predict eclipses, cyclones, whether or not a suspension bridge will be able to support traf? c loads, etc. In some cases such mathematical models can be very complex and may not give very accurate predictions. Applied mathematics is about getting a prediction, evaluating it (seeing how well it predicts the future) and then improving the model. In summary, both branches of mathematics are based on axioms. These may or may not be designed to be ‘realistic’. What matters to the pure mathematician is that an axiom set should not lead to contradictions.The applied mathematician is looking for an axiom set and a mathematical structure built on these axioms that can be used to model the phenomena that we observe in nature. As we have seen, useful axiom sets need not start out being ‘sensible’. The system of deduction that we use to build the other truths of mathematics is known as proof. B A R V A’ S Q T P C B’ C’ ? P, Q, R are collinear 1. 3 PROOF Proof has a very special meaning in mathematics. We use the word generally to mean ‘proof beyond reasonable doubt’ in situations such as law courts when we accept some doubt in a verdict. For mathematicians, proof is an argument that has no doubt at all.When a new proof is published, it is scrutinised and criticised by other mathematicians and is accepted when it is established that every step in the argument is legitimate. Only when this has happened does a proof become accepted. Technically, every step in a proof rests on the axioms of the mathematics that is being used. As we have seen, there is more than one set of axioms that could be chosen. The statements that we prove from the axioms are known as theorems. Once we have a theorem, it becomes a statement that we accept as true and which can be used in the proof of other theorems. In this way we build up a structure that constitutes a ‘mathematics’.The axioms are the foundations and the theorems are the superstructure. In the previous section we made use of the idea of consistency. This means that it must not be possible to use our axiom set to prove two theorems that are contradictory. There are a variety of methods of proof available. This section will look at three of these in detail. We will mention others. 4 Theory of Knowledge – CHAPTER 1 1. 3. 1 RULES OF INFERENCE All proofs depend on rules of inference. Fundamental to these rules is the idea of ‘implication’. As an example, we can say that 2x = 4 (which is known as a proposition) implies that x = 2 (provided that x is a normal real number and that we are talking about normal arithmetic).In mathematical shorthand we would write this statement as 2x = 4 ? x = 2 . This implication works both ways because x = 2 implies that 2x = 4 also. This is written as x = 2 ? 2x = 4 or the fact that the implication is both ways can be written as x = 2 ? 2x = 4 . The ? symbol is read as ‘If and only if’ or simply as ‘Iff’, i. e. , If with two fs. Not every implication works both ways in this manner: If x = 2 then we can conclude that x 2 = 4 . However, we cannot conclude the reverse: i. e. , x 2 = 4 implies that x = 2 is false because x might be –2. Sothat x = 2 ? x 2 = 4 is all that can be said in this case. There are four main rules of inference: 1.The rule of detachment: from a is true and a ? b is true we can infer that b is true. a and b are propositions. If the following propositions are true: It is raining. If it is raining, I will take an umbrella. We can infer that I will take an umbrella. 2. The rule of syllogism: from a ? b is true and b ? c is true, we can conclude that a ? c is true. a, b & c are propositions. If we accept as true that: if x is an odd number then x is not divisible by 4 ( a ? b )and, if x is not divisible by 4 then x is not divisible by 16 ( b ? c ) Example: Example: We can infer that the proposition; if x is an odd number then x is not divisible by 16 ( a ? c ) is true. 5 MATHEMATICS Standard Level 3.The rule of equivalence: at any stage in an argument we can replace any statement by an equivalent statement. If x is a whole number, the statement x is even could be replaced by the statement x is divisible by 2. Example: 4. The rule of substitution: If we have a true statement about all the elements of a set, then that statement is true about any individual member of the set. If we accept that all lions have sharp teeth then Benji, who is a lion, must have sharp teeth. Example: Now that we have our rules of inference, we can look at some of the most commonly used methods of proof 1. 3. 2 PROOF BY EXHAUSTION This method can be, as its name implies, exhausting! It depends on testing every possible case of a theorem.Example: Consider the theorem: Every year must contain at least one ‘Friday the thirteenth’. There are a limited number of possibilities as the ? rst day of every year must be a Monday or a Tuesday or a Wednesday…. or a Sunday (7 possibilities). Taking the fact that the year might or might not be a leap year (with 366 days) means that there are going to be fourteen possibilities. Once we have established all the possibilities, we would look at the calendar associated with each and establish whether or not it has a ‘Friday the thirteenth’. If, for example, we are looking at a non-leap year in which January 1st is a Saturday, there will be a ‘Friday the thirteenth’ in May.Take a look at all the possibilities (an electronic organiser helps! ). Is the theorem true? 6 Theory of Knowledge – CHAPTER 1 1. 3. 3 DIRECT PROOF The following diagrams represent a proof of the theorem of Pythagoras described in ‘The Ascent of Man’ (Bronowski pp 158-161). The theorem states that the area of a square drawn on the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares drawn on the two shorter sides. The method is direct in the sense that it makes no assumptions at the start. Can you follow the steps of this proof and draw the appropriate conclusion? 7 MATHEMATICS Standard Level 1. 3. 4 PROOF BY CONTRADICTIONThis method works by assuming that the proposition is false and then proving that this assumption leads to a contradiction. Example: The number 2 greatly interested classical Greek mathematicians who were unable to ? nd a number that, when it was squared, gave exactly 2. Modern students are often fooled into thinking that their calculators give an exact square root for 2 as when 2 is entered and the square root button is pressed, a result (depending on the model of calculator) of 1. 414213562 is produced. When this is squared, exactly 2 results. This is not because we have an exact square root. It results from the way in which the calculator is designed to calculate with more ? gures than it actually displays. The ? rst answer is stored to more ? ures than are shown, the result is rounded and then displayed. The same is true of the second result which only rounds to 2. Try squaring 1. 414213562, the answer is not 2. The theorem we shall prove is that there is no fraction that when squared gives 2. This also implies that there is no terminating or recurring decimal that, when squared, gives exactly 2, but this further theorem requires more argument. p The method begins by assuming that there is a fraction — (p & q are integers) which has been q p cancelled to its lowest terms, such that — = 2 . From the assumption, the argument proceeds: q p — = q p2 2 ? —– = 2 ? p 2 = 2q 2 ? p 2 is even ? p is even q2As with most mathematical proofs, we have used simple axioms and theorems of arithmetic. The most complex theorem used is that if p 2 is even, then p is even. Can you prove this? The main proof continues with the deduction that if p is even there must be another integer, r, that is half p. p = 2r ? p 2 = 4r 2 ? 2q 2 = 4r 2 ? q 2 = 2r 2 ? q 2 is even ? q is even p We now have our contradiction as we assumed that — was in its lowest terms so p & q cannot q both be even. This proves the result, because we have a contradiction. This theorem is a very strong statement of impossibility. There are very few other areas of knowledge in which we can make similar statements.We might be virtually certain that we will never travel faster than the speed of light but it would be a brave physicist who would state with certainty that it is impossible. Other methods of proof include proof by induction which is mainly used to prove theorems involving sequences of statements. Whilst on the subject of proof, it is worth noting that it is much easier to disprove a statement than 8 Theory of Knowledge – CHAPTER 1 to prove it. When we succeed in disproving a statement, we have succeeded in proving its negation or reverse. To disprove a statement, all we need is a single example of a case in which the theorem does not hold. Such a case is known as a counter-example. Example: The theorem ‘all prime numbers are odd’ is false.This can be established by noting that 2 is an even prime and, therefore, is the only counter-example we need to give. By this method we have proved the theorem that ‘not every prime number is odd’. This is another example of the way in which pure mathematicians think in a slightly different way from other disciplines. Zoo-keepers (and indeed the rest of us) might be happy with the statement that ‘all giraffes have long necks’ and would not be very impressed with a pure mathematician who said that the statement was false because there was one giraffe (with a birth defect) who has a very short neck. This goes back to the slightly different standards of proof that are required in mathematics. Counter-examples and proofs in mathematics may be dif? cult to ? nd.Consider the theorem that every odd positive integer is the sum of a prime number and twice the square of an integer. Examples of this theorem that do work are: 5 = 3 + 2 ? 1 2, 15 = 13 + 2 ? 1 2, 35 = 17 + 2 ? 3 2 . The theorem remains true for a very large number of cases and we do not arrive at a counterexample until 5777. Another similar ‘theorem’ is known as the Goldbach Conjecture. Christian Goldbach (16901764) stated that every even number larger than 2 can be written as the sum of two primes. For example, 4 = 2 + 2, 10 = 3 + 7, 48 = 19 + 29 etc. No-one has every found a counter-example to this ‘simple’ conjecture and yet no accepted proof has ever been produced, despite the fact that the conjecture is not exactly recent!Finally, whilst considering proof, it would be a mistake to think that mathematics is a complete set of truths that has nothing which needs to be added. We have already seen that there are unproved theorems that we suspect to be true. It is also the case that new branches of mathematics are emerging with a fair degree of regularity. During this course you will study linear programming which was developed in the 1940s to help solve the problems associated with the distribution of limited resources. Recently, both pure and applied mathematics have been enriched by the development of ‘Chaos Theory’. This has produced items of beauty such as the Mandelbrot set and insights into the workings of nature.It seems, for example, that the results of Chaos Theory indicate that accurate long term weather forecasts will never be possible (Mandelbrot). 9 MATHEMATICS Standard Level 1. 4 PARADOX 1. 4. 1 WHAT IS A PARADOX? Pure mathematics is a quest for a structure that does not contain internal contradictions. A satisfactory mathematics will contain no ‘nonsense’. Consider the following proof: Let x = 1 Then x 2 – 1 = x – 1 ( x + 1)( x – 1) = x – 1 x+1 = 1 2=1 Try substituting x = 1 to check this line. Factorising using the difference of two squares. Dividing both sides by x – 1. Substituting x = 1. There is obviously something wrong here as this is the sort of inconsistency that we have discussed earlier in this chapter, but what is wrong?To discover this, we must check each line of the argument for errors or faulty reasoning. Line 1 Line 2 Line 3 Line 4 Line 5 must be acceptable as we are entitled to assign a numerical value to a pronumeral. is true because the left hand and right hand sides are the same if we substitute the given value of the pronumeral. is a simple factorisation of the left hand side. is obtained from line 3 by dividing both sides of the equation by x – 1 and should be acceptable as we have ‘done the same thing’ to both sides of the equation. is obtained from line 4 by substituting x = 1 and so should give the correct answer. Obviously we have an unacceptable conclusion from a seemingly watertight argument.There must be something there that needs to be removed as an acceptable operation in mathematics. The unacceptable operation is dividing both sides by x – 1 and then using a value of 1 for x. What we have effectively done is divide by a quantity that is zero. It is this operation that has allowed us to prove that 2 = 1, an unacceptable result. When a paradox of this sort arises, we need to look at the steps of the proof to see if there is a faulty step. If there is, then the faulty step must be removed. In this case, we must add this rule to the allowed operations of mathematics: Never divide by a quantity that is, or will become, zero. This rule, often ignored by students, has important implications for Algebra and Calculus.Some paradoxes are arguments that seem to be sound but contain a hidden error and thus do not contain serious implications for the structure of mathematical logic. An amusing compilation of simple paradoxes can be found in Gardner (1982). An example is the ‘elevator paradox’. Why does it always seem that when we are waiting for an elevator near the bottom of a tall building and wanting to go up, the ? rst elevator to arrive is always going down? Also, when we want to go back down, why is the ? rst elevator to arrive always going up? Is this a real phenomenon or is it just a subjective result of our impatience for the elevator to arrive? Or is it another example of Murphy’s Law; whatever can go wrong will go wrong? 10Theory of Knowledge – CHAPTER This is quite a complex question, but a simple explanation might run as follows: If we are waiting near the bottom of a tall building, there are a small number of ? oors below us from which elevators that are going up might come and then pass our ? oor. By contrast, there are more ? oors above us from which elevators might come and then pass our ? oor going down. On the basis of this and assuming that the elevators are randomly distributed amongst the ? oors, it is more likely that the next elevator to pass will come from above and will, therefore, be going down. 1 By contrast, if we are waiting near the top of a tall building, there are a small number of ? oors above us from which elevators that are going down might come and then pass our ? oor. Also, there are more ? ors below us from which elevators might come and then pass our ? oor going up. It is more likely that the next elevator to pass will come from below and will, therefore, be going up. A fuller analysis of this paradox can be found in Gardner (pp96-97). The elevator paradox does not contain serious implication for the structure of mathematics like our ? rst example. We will conclude this section with a look at a modern paradox that did cause a re-evaluation of one of the basic ideas of mathematics, the set. 1. 4. 2 RUSSELL’S PARADOX Bertrand Russell (1872-1970) looked in detail at the basic set axioms of mathematics. We do regard the existence of sets as axiomatic in all mathematical structures.Does this mean that we can make a set that contains ‘everything’? There would seem to be no dif? culty with this as we just move around the universe and sweep everything that we meet into our set, numbers, words, whales, motorcycles etc. and the result is the set that contains everything. Russell posed the following question which we will relate in the context of library catalogues. Every library has a catalogue. There are various forms that CATALOGUE NEWEL LIBRARY this catalogue might take; a book, a set of cards, a computer Castle, The. Catherine the Great disc etc. Whatever form the catalogue in your local library F Kafka 231. 72 A Biography J Nelson 217. 2 takes, there is a sense in which this catalogue is a book (or Catalogue At reception publication) owned by the library and, as such, should Catullus The complete works appear as an entry in the catalogue: Edited by F Wills Catcher in the Rye JD Salinger 123. 64 312. 42 Of course, many librarians will decide that it is silly to include the catalogue as an entry in the catalogue because people who are already looking at the catalogue know where to ? nd it in the 11 MATHEMATICS Standard Level library! It follows that library catalogues can be divided into two distinct groups: Catalogues that do contain an entry describing themselves. Catalogues that do not contain an entry describing themselves. Next, let us make a catalogue of all the catalogues of type two, those that do not contain themselves. This gives us a problem. Should we include an entry describing our new catalogue?If we do, then our catalogue ceases to be a catalogue of all those catalogues that do not contain themselves. If we do not, then our catalogue is no longer a complete catalogue of all those catalogues that do not contain themselves. The conclusion is that making such a catalogue is impossible. This does not mean that the library catalogues themselves cannot exist. We have, however, de? ned an impossible catalogue. In set terms, Russell’s paradox says that sets are of two types: Type 1 Type 2 Sets that do contain themselves. Sets that do not contain themselves. The set of all sets of type 2 cannot be properly de? ned without reaching a contradiction.The most commonly accepted result of Russell’s paradox is the conclusion that we have to be very careful when we talk about sets of everything. The most usual way out is to work within a carefully de? ned universal set, chosen to be appropriate to the mathematics that we are undertaking. If we are doing normal arithmetic, the universal set is the set of real numbers. 1. 5 MATHEMATICS AND OTHER DISCIPLINES When writing Theory of Knowledge essays, students are required to develop their arguments in a cross disciplinary way. For more details on this, you are strongly advised to read the task speci? cations and the assessment criteria that accompany the essay title. You are reminded that it is these statements that de? e what is expected of a good essay, not the contents of this Chapter which have been provided as a background resource. A good essay will only result if you develop your own ideas and examples in a clear and connected manner. Part of this process may include comparing the ‘mathematical method’ described earlier with the methods that are appropriate to other systems of knowledge. As we have seen, mathematics rests on sets of axioms. This is true of many other disciplines. There is a sense in which many ethical systems also have their axioms such as ‘Thou shalt not kill’. The Ancient Greeks believed that beauty and harmony are based, almost axiomatically, on mathematical proportions.The golden mean is found by dividing a line in the following ratio: A B C The ratio of the length AB to the length BC is the same as the ratio of the length BC to the whole 12 Theory of Knowledge – CHAPTER 1 1 length AC. The actual ratio is 1: — ( 1 + 5 ) or about 1:1. 618. The Greek idea was that if this line 2 is converted into a rectangle, then the shape produced would be in perfect proportion: A B C Likewise, the correct place to put the centre of interest in a picture is placed at the golden mean position between the sides and also at the golden mean between top and bottom. Take a look at the way in which television pictures are composed to see if we still use this idea: Centre of interestIn a similar way, the Ancient Greeks believed that ratio determined harmony in music. If two similar strings whose lengths bear a simple ratio such as 1:2 or 2:3 are plucked together the resulting sound will be pleasant (harmonious). If the ratio of string lengths is ‘awkward’, such as 17:19, then the notes will be discordant. The same principle of simple ratios is used in tuning musical instruments (in most cultures) today. The most common connection between mathematics and other disciplines is the use of mathematics as a tool. Examples are: the use of statistics by insurance actuaries, probability by quality control of? cers and of almost all branches of mathematics by engineers.Every time mathematics is used in this way, there is an assumption that the calculations will be done using techniques that produce consistent and correct answers. It is here that pure mathematical techniques, applied mathematical modelling and other disciplines interface. In some of these examples, we apply very precise criteria to our calculations and are prepared to accept only very low levels of error. Navigation satellite systems work by measuring the position of a point on or above the Earth relative to the positions of satellites orbiting the Earth. Navigator Satellite Satellite This system will only work if the positions of the satellites are known with very great precision.By contrast, when calculations are made to forecast the weather, whilst they are done with as much precision as necessary, because the data is incomplete and the atmospheric models used are approximate, the results of the calculations are, at best, only an indication of what might happen. Fortunately, most of us expect this and are much more tolerant of errors in weather forecasting than we would be if airlines regularly failed to ? nd their destinations! There are, therefore a large number of ways in which mathematics complements other disciplines. In fact, because computers are essentially mathematical devices and we are increasingly dependent on them, it could be argued that mathematics and its methods underpin the modern world. 13 MATHEMATICS Standard Level That is not to say that mathematics is ‘everywhere’. Many very successful people have managed to avoid the subject altogether.Great art, music and poetry has been produced by people for whom mathematical ideas held little interest. In using mathematical ideas in essays, remember that you should produce original examples, look at them in a mathematical context and then compare the ways in which the example might appear to a mathematician with the way in which the same example might appear to a thinker from another discipline. As a very simple example, what should we think of gambling? To the mathematician (Pascal was one of the ? rst to look at this activity from the mathematical perspective), a gambling game is a probability event. The outcome of a single spin of a roulette wheel is unknown.If we place a single bet, we can only know the chances of winning, not whether or not we will win. Also, in the long run, we can expect to lose one thirtyseventh of any money that we bet every time we play. To the mathematician, (or at least to this mathematician) this rather removes the interest from the game! Other people look at gambling from a different standpoint. To the politician, a casino is a source of revenue and possibly a focus of some social problems. To a social scientist, the major concern might be problem gamblers and the effect that gambling has on the fabric of society. A theologian might look at the ethical issues as being paramount. Is it ethical to take money for a service such as is provided by a casino?Many of these people might use mathematics in their investigations, but they are all bringing a slightly different view to the discussion. As we can see, there are many sides to this question as there are many sides to most questions. Mathematics can often illuminate these, but will seldom provide all the answers. When you choose an essay title, you do not have to use mathematical ideas or a mathematical method to develop your analysis. However, we hope that if you do choose to do this, you will ? nd the brief sketch of the mathematical method described in this Chapter helpful. We will ? nish with one observation. Mathematics and mathematicians are sometimes viewed as dry and unimaginative. This may be true in some cases, but de? nitely not all.We conclude with some remarks by the mathematician Charles Dodgson (1832-1898), otherwise known as Lewis Carroll: ‘The time has come’, the Walrus said, ‘To talk of many things: Of shoes and ships and sealing wax, Of cabbages and kings, Of why the sea is boiling hot And whether pigs have wings’. Through the Looking Glass References: Megalithic Sites in Britain. Thom, A. (1967). U. K. Oxford University Press. Heritage Mathematics. Cameron, M. (1984). U. K. E. J. Arnold. The Ascent of Man, Bronowski, J. (1973). U. K. BBC. The Fractal Geometry of Nature, Mandelbrot, B (1977), U. S. W. H. Freeman & Co. 1977. Gotcha! , Gardner, M. (1977). U. S. A. W. H. Freeman & Co. 14 Algebra of Linear and Quadratic Expressions – CHAPTER 2 2. 1 THE REAL NUMBER LINE 2. 1. 1 THE REAL NUMBER LINE A visual method to represent the set of real numbers, , is to use a straight line.This geometrical representation is known as the real number line. It is usually drawn as a horizontal straight line extending out indefinitely in two directions with a point of reference known as the origin, usually denoted by the letter O. Corresponding to every real number x there is a point P, on the line, representing this value. If x > 0, the point P lies to the right of O. If x < 0 the point P lies to the left of O. If x = 0, the point P is at the origin, O. O . –6 –5 –4 –3 –2 –1 .. 0 1 2 3 4 5 6 … P x 2. 1. 2 SET BUILDER NOTATION The set of points on the real number line can also be written in an algebraic form: = { x : –? < x < ? This means that any real number set can be expressed algebraically. For example, the set of positive real numbers = += { x : x > 0 } , negative real numbers = = { x : x < 0 } Note that = ? {0 } ? + . CHAPTER 2 Similarly we can construct any subset of the real number line. The great thing about using set notation is that we can quickly identify if a point on the number line is included or excluded in the set. How do we represent these inclusions and exclusions on the real number line? If the number is included in the set, you ‘black-out’ a circle at that point on the number line – this is called a closed circle. For example, the set { x : x ? } has the following representation: If the number is excluded from the set, you place a circle at that point on the number line – this is called an open circle. For example, the set { x : x > 3 } has the following representation: 0 3 x 0 3 x 2. 1. 3 INTERVAL NOTATION Another notation that is used to describe subsets of the real numbers is interval notation. This form of notation makes use of only ‘square brackets’ or ‘square brackets’ and ‘round brackets’ to indicate if a number is included or excluded. For the examples above we have: { x : x ? 3 } = [3, ? ) or [3, ? [ and for { x : x > 3 } = (3, ? ) or ]3, ? [ Notice that ‘? ’ is never included! 15MATHEMATICS Standard Level It should be noted that { x : x ? 3 } can also be expressed as { x : 3 ? x < ? } . Hence the reason for having ‘? ’ in the interval notation representation. A summary of the different possible intervals is given in the table below: Real Number Line a a b b x x Set Notation { x : a ? x ? b} Interval Notation x ? [ a, b ] Example x 3 8 { x : 3 ? x ? 8} or x ? [ 3, 8 ] x 3 8 { x : 3 ? x < 8} or x ? [ 3, 8 ) x ? [ 3, 8[ x 3 8 { x : 3 < x ? 8} or x ? ( 3, 8 ] x ? ]3, 8] x 3 8 { x : 3 < x < 8} or x ? ( 3, 8 ) x ? ]3, 8[ x 3 { x : x ? 3} or x ? [ 3, ? ) x ? [ 3, ? [ x 3 { x : x > 3} or x ? (3, ? ) x ? ]3, ? [ x 8 { x : x ? 8} or x ? ( – ? , 8 ] x ? ]– ? 8] x 8 { x : x < 8} or x ? ( – ? , 8) x ? ]– ? , 8[ a a b b x x { x : a ? x < b} x ? [ a, b ) or x ? [ a, b[ a a b b x x { x : a < x ? b} x ? ( a, b ] or x ? ]a, b] a a b b x x { x : a < x < b} x ? ( a, b ) or x ? ]a, b[ a a b x x { x : x ? a} or { x : a ? x < ? } x ? [ a, ? ) or x ? [ a, ? [ a a b x x { x : x > a} or { x : a < x < ? } x ? (a, ? ) or x ? ]a, ? [ a a x x { x : x ? a} or { x : –? < x ? a } x ? ( – ? , a ] or x ? ]– ? , a] a a x x { x : x < a} or { x : –? < x < a } x ? ( – ? , a) or x ? ]– ? , a[ 16 Algebra of Linear and Quadratic Expressions – CHAPTER 2 We also make the following point in relation to set notation.Rather than using the colon ‘:’ in expressions such as { x : x < 3 } we can also use the separator ‘|’. That is, { x x < 3 }. Either notation can be used. 2. 1. 4 NUMBER SYSTEMS The set of real numbers can be broken down into two subsets, namely, the set of rational numbers and the set of irrational numbers. The set of rational numbers can itself be broken down into two sets, the set of integers and the set of fractions. The set of integers can then be broken down into the set of positive integers, the set of negative integers and the set that includes the number zero. Each of these sets can be represented by a different symbol. Real Numbers Irrational Numbers Rational Numbers Fractions Integers Positive Integers Zero Negative IntegersIn this book we will use the following notation and definitions: Set of positive integers and zero The set of integers The set of positive integers The set of rational numbers Definition: ? = ? x ? = = = {0, 1, 2, 3, . . . }. = {0, ±1, ±2, ±3, . . . } [Also known as Natural numbers] = + = {1, 2, 3, . . . } = ? a x = — , b ? 0 and a and b are integers ? b ? + + The set of positive rational numbers = The set of positive real numbers The empty set = = {x x ? , x > 0} = ? = The set with no members. EXAMPLE 2. 1 i. ii. iii. (a) (b) { x – 1 < x ? 4} Write each of the following using interval notation. Represent the sets on the real number line. { x x ? 3} ? { x x < 6} { x : x < 8 } { 5 } 17 MATHEMATICS Standard Level S i. o l u ii. t i o iii. n a) (b) (a) (b) (a) (b) { x – 1 < x ? 4 } = ]–1, 4] or using round bracket: (–1, 4]. –1 0 4 x { x x ? 3 } ? { x x < 6 } = [3, ? [ ? ]? ,6[ = [3, 6[ or [3, 6) – using round bracket. 0 3 6 x { x : x < 8 } { 5 } = ]–? , 8[ {5} or (– ? , 8){5} – using round brackets. 0 5 8 x 2. 1. 5 IRRATIONAL NUMBERS We provided a definition of rational numbers earlier. The question then remains, what is an irrational number? The obvious answer is ‘Whatever is not a rational number’! So, what do these numbers look like? The best way to answer this is to say that it is a number that cannot be written a in the form — , b ? 0 where a and b are integers.Examples of irrational numbers are 2, 3, ? b and so on. How do we know that 2 is an irrational number? We can show this as follows – a process known as reductio ad absurdum – meaning to prove by contradiction: Assume that 2 is a rational number. Then by the definition of rational numbers there exist integers a and b (where a and b have no common factors) such that a 2 = — , b ? 0 b a2 Upon squaring we have: 2 = —-b2 ? a 2 = 2b 2 Then, a 2 must be even [because 2b 2 is even – i. e. , any number multiplied by 2 is even] and so a must also be even. This means we can express a as 2k. So, setting a = 2k we have: a 2 = 4k 2 But, a 2 = 2b 2 , so that 2b 2 = 4k 2 ? 2 = 2k 2 And so, b 2 must also be even, meaning that b is even. Then, since both a and b are even it follows that a and b have a common factor (i. e. , 2). This is contrary to our original hypothesis. Therefore therefore be an irrational number. 2 is not a rational number and must One subset of irrational numbers is known as the set of surds. Surds can be expressed in the form n a where n ? + and a ? + . A commonly encountered surd is a (i. e. , the square root). 18 Algebra of Linear and Quadratic Expressions – CHAPTER 2 The laws of operations apply to surds in the same way that they apply to real numbers. We summarise some results involving surds: a? = a ——- = b ab a -b a 2 b = a b, a > 0 a b ? c d = ac bd ( a + b)( a – b) = a – b Notice that the last result shows that we obtain a rational number! The surds a – b and a + b are know as conjugate pairs. Whenever conjugate pairs are multiplied together they produce a rational number. EXAMPLE 2. 2 (a) Expand the following (b) ( 5 – 2 3)( 5 – 3) ( 2 + 3)( 6 – 3) ( 2 + 3)( 6 – 3) = = S o (a) l u t i o n (b) 2? 6–3? 2+ 3? 6–3? 3 2 ? ( 2 ? 3) – 3 2 + 3 ? ( 3 ? 2) – 3 3 =2 3–3 2+3 2–3 3 =– 3 ( 5 – 2 3)( 5 – 3) = 5? 5– 5? 3–2 3? 5+2 3? 3 = 5 – 15 – 2 15 + 2 ? 3 = 11 – 3 15 EXAMPLE 2. 3 Rationalise the denominator of (a) 1 —————-2+ 2 (b) + 2 ——————-2 3–1 S o l (a) u t i o n (b) 1 1 2– 2 —————– = —————– ? —————– [multiplying numerator and denominator by conjugate] 2+ 2 2+ 2 2– 2 2– 2 = —————-4–2 1 = 1 – — 2 2 2 3+ 1+ 2 6+ 2 1+ 2 1+ 2 2 3+1 ——————– = ——————– ? ——————— = ——————————————————4? 3–1 2 3–1 2 3–1 2 3+1 2 3+ 1+ 2 6+ 2 = ——————————————————- [cannot be simplified further] 11 19 MATHEMATICS Standard Level 2. 1. 6 THE ABSOLUTE VALUE The absolute value or modulus of a number x, denoted by x , is defined as follows: If x ? 0 ? = x and if x < 0 ? x = – x . This means that the absolute value of any number will always be positive. E. g. , 4 = 4 :– seeing as 4 > 0, we use the value of 4. Whereas, – 2 = – ( – 2 ) = 2:– by taking the negative of a negative number we obtain a positive number. EXAMPLE 2. 4 (a) (b) Find { x : x = 7 } . Use the number line to represent the sets i. { x : x ? 3} ii. { x : x > 1} Express these sets using interval notation. S (a) o l u t i (b) o n We are looking for value(s) of x such that when we take the absolute value of x it is 7. From the definition of the absolute value, we must have that x = 7 or x = – 7. That is, x = 7 ? x = ± 7 .Therefore, the solution set is {7, –7} i This time we want all values of x such that their absolute value is less than or equal to 3. For example, if x = –2. 5 then – 2. 5 = 2. 5 which is less than 3. However, if x = –4 then – 4 = 4 which is greater than 3. So we cannot have x = –4. Working along these lines we must have: –3 0 3 x Using interval notation we have { x : x ? 3 } = {x : –3 ? x ? 3} = [–3, 3] ii. This time we want numbers for which their absolute value is greater than 1. For example, 1. 2 = 1. 2 > 1 and – 3. 2 = 3. 2 > 1 . We then have: –1 0 1 x Using interval notation we have { x : x > 1 } = ]–? , –1[ ? ]1, ? [ EXERCISES 2. 1 1.Show the following sets on the real number line. { x 2 ? x ? 8} { x x > 7} (a) (b) (d) 2. ]2, 7] ? ]4, 8[ (e) (–? , 4) ? [–2, 5) (c) (f) { x – 2 < x ? 6 } {4} { x : x < –6 } Write the following using interval notation. (a) (b) { x – 2 ? x ? 7} (c) (e) { x 0 < x ? 5} { x : x < 8 } ? { x : x > –4 } 20 (d) (f) { x x > 9} { x : x ? 0} { x : x < –1 } ? { x : x > 2 } Algebra of Linear and Quadratic Expressions – CHAPTER 3. Simplify the following. (a) 4. 3 5 + 20 (b) 2 3 – 27 (c) 2+ 3+ 2 8 – 18 Simplify the following. (a) (c) ( 5 + 1)( 5 – 1) (3 2 – 6)( 3 + 3) (b) (d) (2 3 – 2)( 2 + 3) ( 2 + 3 3 )2 5. Rationalise the denominator in each of the following. a) (d) 1 —————-2+ 3 2 5+1 ——————–3–2 If x = (b) (e) 3 —————-7–2 2+ 3 ———————3– 5 i. i. (c) (f) 1 x + -x 1 x – -x 3 —————-5–2 2 3 —————————-2 5–3 2 ii. ii. 1 x 2 + —x2 1 x 2 + —x2 6. (a) (b) 5 + 3 , find the value of If x = 4 + 3 , find the value of 7. Find the value of x if (a) { x x = 3} (d) {x x + 1 = 3} (b) (e) { x x = 10 } { x x + 2 = 10 } (c) (f) { x x = –2 } { x x – 2 = 2} 8. Represent each of the following on the real number line. (a) (b) { x : x ? 5} { x x > 2} (c) { x : 2 ? x < 5} (d) { x : 2 x ? 8} 9. Write the following using interval notation. (a) { x x – 1 > 0} (b) ? 1 ? ? x — x > 2 ? 2 ? ? (c) { x : x > 4 } ? x : 2 x < 12 } 2. 2 LINEAR ALGEBRA 2. 2. 1 REVIEW OF LINEAR EQUATIONS A linear equation in the variable x (say) takes on the form ax + b = c where a, b and c are real constants. The equation is linear because x is raised to the power of one. To solve such equations we use the rules of transposition: ax + b = c ? ax = c – b c–b ? x = ———–a Solving ax + b = c produces a solution that can be represented on the real number line. 21 MATHEMATICS Standard Level EXAMPLE 2. 5 (a) 4x + 5 = 21 Solve the following linear equations (b) 9 – 2x = 7 (c) 3 ( 5x – 2 ) = 12 S o (a) l u (b) t i o (c) n 4x + 5 = 21 ? 4x = 16 ? x = 4 9 – 2x = 7 ? – 2x = – 2 ? = 1 3 ( 5x – 2 ) = 12 ? 15x – 6 = 12 [don’t forget to multiply the 3 and –2] ? 15x = 18 6 ? x = -5 Sometimes equations might not appear to be linear, but with some algebra, they form into linear equations. The following examples shows how this works. EXAMPLE 2. 6 (a) x–3 Solve for x, ———— – 1 = x (b) 2 – ? Find ? x ? ? x 2–x — – ———— = 1 ? . 2 3 ? S o l (a) u t i o n (b) x–3 x–3 ———— – 1 = x ? ———— = x + 1 ? x – 3 = 2 ( x + 1 ) 2 2 ? x – 3 = 2x + 2 ? –x = 5 ? x = – 5 x 2–x 2x 3 ( 2 – x ) — – ———— = 1 ? —– – ——————— = 1 ? 2x – 3 ( 2 – x ) = 6 3 2 66 ? 2x – 6 + 3x = 6 ? 5x = 12 12 ? x = —-5- EXAMPLE 2. 7 a) bx – b 2 = ab Solve the following literal equations for x, where a and b are real constants. (b) bx = a ( b – x ) S o (a) l u t i (b) o n bx – b 2 = ab ? bx = ab + b 2 ? bx = b ( a + b ) [taking b out as a common factor] ? x = a + b [dividing both sides by b] bx = a ( b – x ) ? bx = ab – ax ? bx + ax = ab 22 Algebra of Linear and Quadratic Expressions – CHAPTER ? ( b + a )x = ab ab ? x = b+a ————- 2 Linear Equations involving Absolute values Recall that if x = a (where a ? 0) then, x = a or – a. Using this result we can solve similar linear equations. That is, ax + b = c ? ax + b = c or ax + b = – c ? ax = c – b or ax = – c – b c–b (b + c) ? = ———— or x = – —————–a a Notice that this time we have two solutions! EXAMPLE 2. 8 (a) 2x = 6 Solve the following. (b) x–1 = 5 (c) 1 3 – — x = 2 2 S o (a) l u t i (b) o n (c) 2x = 6 ? 2x = 6 or 2x = – 6 ? x = 3 or x = – 3 x – 1 = 5 ? x – 1 = 5 or x – 1 = – 5 ? x = 6 or x = – 4 1 1 1 3 – — x = 2 ? 3 – — x = 2 or 3 – — x = – 2 2 2 2 1 1 ? – — x = – 1 or – — x = – 5 2 2 ? x = 2 or x = 10 The next example illustrates a demanding algebraic solution. EXAMPLE 2. 9 S o l u t i o n (a) Solve for x where x = 2x + 1 By definition, x = x if x ? 0 and –x if x < 0. Therefore we have two separate equations to solve, one for x ? 0 and one for x < 0. Case 1 ( x ? 0): x = 2x + 1, x ? ? – x = 1, x ? 0 ? x = – 1, x ? 0 23 MATHEMATICS Standard Level Now, our solution is that x = – 1, however, we must also satisfy the condition that x ? 0. As both statements do not agree with each other, we conclude that for x ? 0 there is no solution. Case 2 ( x < 0): – x = 2x + 1, x < 0 ? – 3x = 1, x < 0 1 ? x = – — , x < 0 3 1 This time our solution is that x = – — , and we must also satisfy the condition that x < 0. 3 As both statements agree with each other, we conclude that for x < 0 there is a solution. 1 1 Namely, x = – — . Therefore, x = 2x + 1 has only one solution, x = – –. 3 3 Solving Equations with the TI-83Equations such as the we have just looked at can also be solved using the solve( option on the TI–83. We do this by calling up the Catalogue and then 1. locating the solve( option 2. enter the relevant equation [The equation must be entered in the form Equation = 0. So, to solve the equation 2x + 6 = 15, we must rewrite it as 2 x + 6 –15 = 0 so that the equation that is entered into the TI–83 is 2x + 6 – 15 = 0] 3. indicate the variable we are solving for 4. provide a reasonable guess (for the answer) To obtain the solve( option we use the following sequence: 2nd LN then use the arrow key to reach solve( and then press ENTER : 0 We look at some of the problems we have aleady solved: EXAMPLE 2. 10 (a) 4x + 5 = 21Solve the following linear equations (b) 3 ( 5x – 2 ) = 12 (c) x–3 ———— – 1 = x (d) 2 x = 2x + 1 S o l u t i o n (a) (b) (c) 24 Algebra of Linear and Quadratic Expressions – CHAPTER (d) 2 Notice that in each case we have used a guess of 5. Another method is to use the Equation solver facility. The expression must still be entered in the form Equation = 0. To call up the Equation solver screen 1. 2. 3. press MATH 0 Enter the equation in the form Equation = 0 Move the cursor over the variable for which you want to solve and then press ALPHA ENTER . It is important that you become familiar with both modes of solving equations, although eventually you will prefer one method over the other.EXERCISES 2. 2. 1 1. Solve the following linear equations. (a) (d) 2. 2x = 8 3 – 2x ————— = 2 7 (b) (e) 5x – 3 = 12 5x 1 2 —– + — = -3- 2 3 (c) (f) 1 2 – — x = 4 3 1 2x + — = 1 4 Solve the following equations. (a) (d) 5 ( x – 1 ) = 12 2x – 1 = 3 – x (b) (e) 1 3 ? 2 – — x? = 4 ? 2 ? (c) – 2 ( 2x + 1 ) = 1 1 5 ( 2x – 3 ) = 8 – — x 4 x–2 1–x ———— + 1 = ———–3 4 3(u + 1) (u + 1) ———————- – 2 = —————–5 5 (c) (f) (i) ax = b ( a – x ) 1 1 1 — + — = — a x b a b ———— = ———–b–x a–x 1 3 ? 5 – — x? = x – 20 (f) ? 3 ? u–1 u ———— – 3 = -4 3 1 2 ———— + 1 = ———–y–1 y–1 x. ( x – b) = b + a x x — – a = a – b -b 1 – ax 1 – bx ————— + ————— = 0 b a – 3. Solve the following equations. 2–u ———— + 8 = 1 – u (b) (a) 6 5 1 ———— + 2 = ———— (e) (d) x+1 x+1 Solve the following equations for (a) x–b = b–2 (b) x — – a = b (d) (e) a b–x b+x ———— = ———–(g) (h) a+x a–x (c) (f) 4. 25 MATHEMATICS Standard Level 5. Solve for x. (a) (d) (g) (j) 2x = 8 3 – 2x = 2 ————–7 5 ( x – 1 ) = 12 a–2 x = b (b) (e) (h) (k) 5x – 3 = 12 5x 1 2 —– + — = -3 3- 2 1 3 ? 2 – — x? = 4 ? 2 ? 1 a – — x = b b (c) (f) (i) (l) 1 2 – — x = 4 3 1 2x + — = 1 4 ? 2 x + 1? = 1 ? -? 3 2ax – b = 3b 2. 2. 2 LINEAR INEQUATIONSInequalities are solved in the same way as equalities, with the exception that when both sides are mulitplied or divided by a negative number, the direction of the inequality sign reverses. EXAMPLE 2. 11 S (a) o l u (b) t i o n Find (a) {x : x + 1 < 4} (b) { x 2x – 5 < 1 } x+1 3 ? 3x > 1 1 ? x > –. 3 ? 1? Therefore, s. s. = ? x : x > — ? 3? ? (b) 3 – 2x 4x – 3 ? 3 – 2x? ? 4x – 3? ————— ? ————— ? 14 ? ————— ? ? 14 ? ————— ? [multiply both sides by 14] 7 2 7 2 ? 2 ( 3 – 2x ) ? 7 ( 4x – 3 ) ? 6 – 4x ? 28x – 21 ? – 32x ? – 27 27 ? x ? —– [notice the reversal of the inequality – as we divided 32 by a negative number] 26 Algebra of Linear and Quadratic Expressions – CHAPTER ? 27 ? Therefore, s. s. is ? x : x ? —– ?. 32 ? ? When dealing with inequalities that involve absolute values, we need to keep in mind the following: 1. 2. x < a ? –a < x < a x > a ? x < – a or x > a EXAMPLE 2. 13 S (a) o l u t (b) i o n Find (a) {x : x + 1 < 4} (b) { x 2x – 5 ? 1 } x + 1 < 4 ? –4 < x + 1 < 4 ? – 5 < x < 3 [subtracting 1 from both inequalities] Therefore, s. s. is { x : –5 < x < 3 }. 2x – 5 ? 1 ? – 1 ? 2x – 5 ? 1 ? 4 ? 2x ? 6 [adding 5 to both sides of inequality] ? 2 ? x ? 3 [dividing both sides by 2] EXAMPLE 2. 14 Find (a) ? ? 1 ? x : 1 – — x > 3 ? 2 ? ? (b) { x 3x – 2 – 1 ? 5 } S o l (a) u t i o n (b) 1 1 1 1 – — x > 3 ? – — x > 3 or 1 – — x < – 3 2 2 2 1 1 ? – — x > 2or – — x < – 4 2 2 ? x < – 4 or x > 8 [Note the reversal of inequality sign, i. e. , ? by –2] Therefore, s. s. is { x : x < – 4 } ? { x : x > 8 }. 3x – 2 – 1 ? 5 ? 3x – 2 ? 6 ? 3x – 2 ? 6 or 3x – 2 ? – 6 ? 3x ? 8 or 3x ? – 4 4 8 ? x ? –or x ? – -3 3 4? ? ? 8? Therefore, s. s. is ? x : x ? – — ? ? ? x : x ? — ?. 3 3? ? ? ? 27 MATHEMATICS Standard Level EXERCISES 2. 2. 2 1. Solve the following inequalities. (a) (d) 2. 2x + 1 < x – 3 x ? 3( x + 4) (b) (e) x–4 ———— ? 2x – 1 3 x–4 2–x ———— > ———–5 2 (c) (f) x+3 x + 1 > ———–2 1 – 3x < 5x – 2