Pearson Education Limited Edinburgh Gate Harlow Essex CM20 2JE England and Associated Companies throughout the world

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2 Pearson Education Limited Edinburgh Gate Harlow Esse CM0 JE England and Associated Companies throughout the world Visit us on the World Wide Web at: Pearson Education Limited 04 All rights reserved. No part of this publication ma be reproduced, stored in a retrieval sstem, or transmitted in an form or b an means, electronic, mechanical, photocoping, recording or otherwise, without either the prior written permission of the publisher or a licence permitting restricted coping in the United Kingdom issued b the Copright Licensing Agenc Ltd, Saffron House, 60 Kirb Street, London ECN 8TS. All trademarks used herein are the propert of their respective owners. The use of an trademark in this tet does not vest in the author or publisher an trademark ownership rights in such trademarks, nor does the use of such trademarks impl an affiliation with or endorsement of this book b such owners. ISBN 0: ISBN 0: ISBN 3: ISBN 3: British Librar Cataloguing-in-Publication Data A catalogue record for this book is available from the British Librar Printed in the United States of America

3 CONCEPT EXPLANATION EXAMPLES Solution Set The set of all solutions Usuall a region in the -plane The solution set to + 7 is all points above the line + =. + > Sstem of Linear Inequalities in Two Variables Solutions to sstems must satisf both inequalities. The solution set usuall includes infinitel man solutions. The ordered pair (0, 0) is a solution to , because both inequalities are true when = 0 and = 0. > (0, 0) Eercises CONCEPTS AND VOCABULARY. When the equals sign in A + B = C is replaced with 6, 7,, or Ú, a linear in two variables results.. A solution to a linear inequalit in two variables is a(n) that makes the inequalit a true statement. 3. Describe the graph of the solution set to k for some number k. 4. Describe the graph of the solution set to 7 k for some number k. 5. Describe the graph of the solution set to Ú. 6. When graphing the solution set to a linear inequalit, one wa to determine which region to shade is to use a(n) point. 7. When graphing a linear inequalit containing either 6 or 7, use a(n) line. 8. When graphing a linear inequalit containing either or Ú, use a(n) line. 9. When graphing the linear inequalit A + B 6 C, a first step is to graph the line. 0. A solution to a sstem of two inequalities must make ( both inequalities/one inequalit ) true. 30

4 . If two shaded regions represent the solution sets for two inequalities in a sstem, then the solution set for the sstem is where these two shaded regions.. If a test point is found that satisfies both inequalities in a sstem, do other test points still need to be checked? SOLUTIONS TO LINEAR INEQUALITIES Eercises 34: Determine whether the test point is a solution to the linear inequalit. 3. (3, ), 7 4. ( -3, 4), (0, 0), Ú 6. (0, 0), (5, 4), Ú 8. ( -, ), 6 9. (3, 0), 6-0. (0, 5), ( -, 6), + 4. (, -4), - Ú 7 3. ( -, -), + Ú - 4. (0, ), Ú - Eercises 5: Write a linear inequalit that describes the shaded region = = 3 3 = 3 = = + = Eercises 33 44: Shade the solution set to the inequalit Ú Ú Ú Eercises 45 50: Determine if the test point is a solution to the sstem of linear inequalities. 45. (3, ) ( -, 3) 3 - Ú ( 4, -) - Ú (0, 0) (, ) ( -, -) Eercises 5 54: The graphs of two equations are shown with four test points labeled. Use these points to decide which region should be shaded to solve the given sstem of inequalities Ú 5. Ú - Ú = = + = (, ) = (3, ) (, ) (3, ) 3 (, ) (3, ) = (0, 0) 3 = 3 (, ) 303

5 = 3 (, 0) (, 0) = (, 3) (, ) 54. Ú - = (, 0) (, 0) = (0, ) (0, ) Eercises 55 7: Shade the solution set to the sstem of inequalities Ú Ú 5 APPLICATIONS Ú Ú Ú Ú 3 - Ú Ú 3 - Ú Ú Ú Ú Ú 6 - Ú Manufacturing (Refer to Eample 5.) A business manufactures at least two MP3 plaers for each digital video plaer. The total number of MP3 plaers and digital video plaers must be less than 90. Shade the region that represents the number of MP3 plaers M and digital video plaers V that can be produced within these restrictions. Put V on the horizontal ais. 74. Working on Two Projects An emploee is required to spend more time on project X than on project Y. The emploee can work at most 40 hours on these two projects. Shade the region in the -plane that represents the number of hours that the emploee can spend on each project. 75. Maimum Heart Rate When eercising, people often tr to maintain target heart rates that are a percentage of their maimum heart rate. Maimum heart rate R i s R = 0 - A, where A is the person s age and R is the heart rate in beats per minute. (a) Find R for a person 0 ears old; 70 ears old. (b) Sketch a graph of R 0 - A. Assume that A is between 0 and 70 and put A on the horizontal ais of our graph. (c) Interpret our graph. 76. Target Heart Rate (Refer to the preceding eercise.) A target heart rate T that is half a person s maimum heart rate is given b T = 0 - A, where A is a person s age. (a) What is T for a person 30 ears old? 50 ears old? (b) Sketch a graph of the sstem of inequalities. T Ú 0 - A T 0 - A Assume that A is between 0 and 60. (c) Interpret our graph. 77. Height and Weight Use Figure 0 in Eample 6 to determine the range of recommended weights for a person who is 74 inches tall. 78. Height and Weight Use Figure 0 in Eample 6 to determine the range of recommended heights for a person who weighs 50 pounds. WRITING ABOUT MATHEMATICS 79. What is the solution set to the following sstem of inequalities? Eplain our reasoning Write down a sstem of linear inequalities whose solution set is the entire -plane. Eplain our reasoning. 304

6 SECTIONS 3 and 4 Checking Basic Concepts. Use elimination to solve the sstem of equations. + 3 = 5-7 = -6. Use elimination to solve each sstem of equations. How man solutions are there in each case? (a) + = - - = (b) 5-6 = = (c) - 3 = 0-6 = 0 3. Solve the sstem of equations smbolicall, graphicall, and numericall. How man solutions are there? - + = 0 = 4. Shade the solution set to each inequalit. (a) 6 - (b) Shade the solution set to the following sstem of inequalities Large U.S. Cities The combined population of New York and Chicago was million people in 009. The population of New York eceeded the population of Chicago b 5 million people. (a) Let be the population of New York and be the population of Chicago. Write a sstem of equations whose solution gives the population of each cit in 009. (b) Solve the sstem of equations. Summar SECTION. SOLVING SYSTEMS OF LINEAR EQUATIONS GRAPHICALLY AND NUMERICALLY Sstem of Linear Equations Solution Solution Set Graphical Solution Numerical Solution An ordered pair (, ) that satisfies both equations The set of all solutions Eample: The ordered pair (3, ) is the solution to the following sstem. + = 4 - = Graph each equation. A point of intersection is a solution. (Sometimes determining the eact answer when estimating from a graph ma be difficult.) Solve each equation for and make a table for each equation. A solution occurs when two -values are equal for a given -va l ue. 3 + = 4 is a true statement. 3 - = is a true statement. A Graphical Solution The point of intersection, (3, ), is the solution to the sstem of equations. 3 + = 4 3 = (3, ) A Numerical Solution The ordered pair ( 3, ) is the solution. When = 3, both -values equal. 3 3 = =

7 Tpes of Sstems of Linear Equations A sstem of linear equations can have no solutions, one solution, or infinitel man solutions. No Solutions One Solution Infinitel Man Solutions Inconsistent Sstem Consistent Sstem Consistent Sstem Independent Equations Dependent Equations SECTION. SOLVING SYSTEMS OF LINEAR EQUATIONS BY SUBSTITUTION Method of Substitution This method can be used to solve a sstem of equations smbolicall and alwas gives the eact solution, provided one eists. Eample: - + = -3 + = 3 STEP : Solve one of the equations for a convenient variable. + = 3 becomes = 3 -. STEP : Substitute this result in the other equation and then solve. - + (3 - ) = -3 Substitute (3 - ) for. -3 = -6 Combine like terms; subtract 3. = Divide each side b -3. STEP 3: Find the value of the other variable. Because = 3 - a nd =, it follows that = 3 - =. STEP 4: Check to determine that (, ) is the solution. -() + () -3 A true statement + 3 A true statement The solution (, ) checks. Recognizing Tpes of Sstems No solutions The final equation is alwas false, such as 0 =. One solution The final equation has one solution, such as =. Infinitel man solutions The final equation is alwas true, such as 0 = 0. SECTION 3. SOLVING SYSTEMS OF LINEAR EQUATIONS BY ELIMINATION Method of Elimination This method can be used to solve a sstem of linear equations smbolicall and alwas gives the eact solution, provided one eists. Eample: + 3 = - + = 3 4 = 4, or = Add and solve for. 306