College of Marin Study Guide for Math 103 X,Y

Transcription

1 College of Marin Study Guide for Math 0 X,Y Revised Edition Ted Broomas, Ira G. Lansing, Jeannette Woods Revision Editor Dan Ayer To Accompany Intermediate Algebra, Tenth Edition By Marvin L. Bittinger

2 Copyright 006, 00 by College of Marin Department of Mathematics Copyright 999 by Addison Wesley, Inc. All rights reserved. Permission in writing must be obtained from the publisher before any part of this work may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying and recording, or by any information storage or retrieval system. All trademarks, service marks, registered trademarks, and registered service marks are the property of their respective owners and are used herein for identification purposes only. Printed in the United States of America ISBN SB Please visit our web site at PEARSON CUSTOM PUBLISHING 7 Arlington Street, Suite 00, Boston, MA 06 A Pearson Education Company

3 Contents Math 0X Intermediate Algebra Lesson Page Title 7 Review of Elementary Algebra Linear Equations 4 Inequalities 4 0 Polynomials 4 Factoring 6 8 Review 7 Fractional Expressions 8 Solving Fractional Equations 9 40 Radicals, Part I 0 44 Radicals, Part II 48 Review Final Final Exam for Math 0X Math 0Y Intermediate Algebra Lesson Page Title 9 Quadratic Equations 6 Equations of Quadratic Form 4 66 Graphing Linear Equations 69 Graphs of Quadratic Equations and Nonlinear Inequalities 6 7 More About Functions 7 80 Review 8 8 Exponential and Logarithmic Functions 9 8 Systems of Equations

4 0 88 Matrices 6 Linear Programming Review Comprehensive Review Final 9 Final Exam for Math 0Y 49 Answers to Sample Tests 7 Answers to Homework Problems in the Study Guide

5 Videotape Index for Intermediate Algebra, 0th Ed. Videotape Index for Intermediate Algebra, 9th Ed. Marvin Bittinger There are 9 videotapes that introduce the material in each section and work the following text exercises. VHS Tape # Run Time 9e Sect # 6:4 R. The Set of Real Numbers 9:09 R. Operations with Real Numbers : R. Exponential Notation and Order of Operations 07:4 R. 4 Introduction to Algebraic Expressions :06 R. Equivalent Algebraic Expressions :7 R.6 Simplifying Algebraic Expressions :8 R.7 Properties of Exponents and Scientific Notation Section Title Presenter Exercises Used Examples Used C. Vance,,, 4, 4,,, 7, 9, 6 C. Vance, 7,,,, 4,, 6, 8, 09 J. Penna,,,,, 4, 4, 7, 0 B. Johnson 4, 7 P. Schwarzkopf,, 9, 7, 7,, 9 J. Penna, 9, 7,, 4, 49, 7 J. Penna,, 7, 4, 77, 79, 0 :44. Solving Equations J. Penna, 7,,, 7, 4, 7 06:. Formulas and Applications :0. Applications and Problem Solving 4 :49.4 Sets, Inequalities, and Interval Notation 4 9:4. Intersections, Unions, and Compound Inequalities 4 0:4.6 Absolute-Value Equations and Inequalities,, 4, 7, 9, 4,,,, 6, 7, 8,,, 6, 9, 4, 8, 9, 46,, none, 7,, 8,,, 6 C. Vance, 9, 6 P. Schwarzkopf, 6 P. Schwarzkopf 7,,, 7,, 9, 9, 69 none C. Vance 9,, 9,, 4, 6, 0 J. Penna 7, 9,,,,, 9, 77 4:. Graphs of Equations P. Schwarzkopf,,, 47, 7, 7, 8, 7, 8 :0. Functions and Graphs P. Schwarzkopf,,, 7, 8, 9 0:49. Finding Domain and Range 6 0:0.4 Linear Functions: Graphs and Slope 6 8:4. More on Graphing Linear Equations 6 0:8.6 Finding Equations of Lines; Applications 7 6:. Systems of Equations in Two Variables 7 7:. Solving by Substitution 7 0:7. Solving by Elimination 7 8:4.4 Solving Applied Problems: Two Equations 8 4:. Systems of Equations in Three Variables 8 6:09.6 Solving Applied Problems: Three Equations B. Johnson,, 4 P. Schwarzkopf 7,, 7, 8 P. Schwarzkopf,, 7, 4, none C. Vance,, 9,, 9 4, 6 J. Penna,, J. Penna,, 4 J. Penna,, 7, 9, B. Johnson, 7 6 J. Penna, 7, none C. Vance, 7

6 4 Study Guide for Math 0 X, Y VHS Tape # Run Time 9e Sect # 8 :9.7 Systems of Inequalities in Two Variables 8 :.8 Business and Economics Applications 9 8: 4. Introduction to Polynomials and Polynomial Functions 9 :8 4. Multiplication of Polynomials 9 6: 4. Introduction to Factoring 9 :0 4.4 Factoring Trinomials: x bx c 0 6:4 4. Factoring Trinomials: ax bx c, a C. Vance,, 4, 4, C. Vance none, D. Ellenbogen, 67 4a, 0 J. Penna 7,, 4, 8,, 6, C. Vance 7,, 7,, 4, 0 J. Penna, 7,,,, 6, 8 P. Schwarzkopf, 9, 0 7:8 4.6 Special Factoring J. Penna,,, 9, 67,, 7, :6 4.7 Factoring: A General J. Penna 7, 7, 4, Strategy 0 0:8 4.8 Applications of Polynomial Equations and Functions C. Vance,,,, 9 6, 7 :6. Rational Expressions and Functions: Multiplying, Dividing, and Simplifying 7:4. LCMs, LCDs, Addition, and Subtraction 8:8. Division of Polynomials :48.4 Complex Rational Expressions :4. Solving Rational Equations :04.6 Applications and Proportions 0:8.7 Formulas and Applications 0:.8 Variation and Applications :8 6. Radical Expressions and Functions :4 6. Rational Numbers as Exponents 6:0 6. Simplifying Radical Expressions :0 6.4 Addition, Subtraction, and More Multiplication 4 :8 6. More on Division of Radical Expressions 4 4:9 6.6 Solving Radical Equations 4 : Applications Involving Powers and Roots 4 0: 6.8 The Complex Numbers 8:0 7. The Basics of Solving Quadratic Equations Section Title Presenter Exercises Used Examples Used J. Penna 7,,, 7 J. Penna,, 6, 7, 4 P. Schwarzkopf,, 9, none J. Penna 7, J. Penna, 7, 4 4 B. Johnson, 7 D. Ellenbogen none, P. Schwarzkopf, 9,, 9, 8 C. Vance, 7, 9,, 4, 47, 49, B. Johnson,,, 4,, 79 P. Schwarzkopf, 9, 49, 7, 69, 8 9, 4, 6, 4, 7,, 9, 7, 4 J. Penna,, 4, 9 7, C. Vance, 9, 7,, 7, 9 J. Penna, 7,, 6, 8 P. Schwarzkopf 9, 7,, none J. Penna 7, 9,, 9,,, 9,,, 7, 87, 9 J. Penna,, 4a, 7, 8

7 Videotape Index for Intermediate Algebra, 9th Ed. VHS Tape # Run Time 9e Sect # 9:9 7. The Quadratic Formula 6:0 7. Applications Involving Quadratic Equations :0 7.4 More on Quadratic Equations 6 0:40 7. Graphing f(x) a(x h) k 6 9:9 7.6 Graphing f(x) ax bx c 6 4: 7.7 Mathematical Modeling with Quadratic Functions 6 0:4 7.8 Polynomial and Rational Inequalities B. Johnson, 9 C. Vance,, 4 4 P. Schwarzkopf, 7, 9,,,, 9, 47 P. Schwarzkopf,,,,, none C. Vance 7, 4 C. Vance, 7 J. Penna,, 9, 7 7:9 8. Exponential Functions P. Schwarzkopf 7, 9,,, 7 :9 8. Inverse and Composite Functions B. Johnson 7,, 4,, 7 :0 8. Logarithmic Functions C. Vance,, 9, 4,, 6 8 0: 8.4 Properties of Logarithmic Functions 8 07:09 8. Natural Logarithmic Functions 8 0: 8.6 Solving Exponential and Logarithmic Equations 8 :4 8.7 Mathematical Modeling with Exponential and Logarithmic Functions,, 4,, 6, 7 J. Penna 7, 7, 7, 49,, 4, 6, 8, C. Vance 7 8, 0,, 4 C. Vance,,,,, 7 C. Vance,, b 9 :4 9. Parabolas and Circles B. Johnson 7,, 49,,, 6 9 4: 9. Ellipses P. Schwarzkopf 7, 9, 9 7:7 9. Hyperbolas P. Schwarzkopf,,, 9 none 9 8: 9.4 Nonlinear Systems of Equations Section Title Presenter Exercises Used Examples Used J. Penna, 4, 6

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9 Lesson : Review of Elementary Algebra This lesson reviews some of the basic concepts from elementary algebra that you will need to prepare for this course. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson you will be able to:. Write numbers in decimal, fractional, or percent notation.. Do arithmetic using signed numbers.. Simplify algebraic expressions using the properties of arithmetic. 4. Perform basic operations with exponentials that have integer powers. Procedures. Read section R., pp. 9. Be sure to do all the margin exercises when so instructed. The answers to these margin exercises are in the back of the text. Work one problem at a time and check each answer. Homework: Sect. R., pp. 0, 67 All Odd Problems. (Answers to the odd-numbered problems are in the back of your textbook.). Read section R., pp. 0 Note: The rule at the bottom of page 8 can be used for subtracting any combination of positive and negative numbers. Homework: Sect. R., pp. 4, 4 All Odd Problems.. Read section R., pp. 0 Homework: Sect. R., pp. 4, All Odd Problems. 4. Read section R.4, pp. 9 Homework: Sect. R.4, pp. 40 4, All Odd Problems.. Read section R., pp Homework: Sect. R., pp , 67 All Odd Problems. 6. Read section R.6, pp. 0 Homework: Sect. R.6, pp. 4 7, 8 All Odd Problems. 7. Read section R.7, pp. 8 6 Many of the answers in the text are not given in simplified form. When the instruction is to Simplify, the answer cannot have any negative exponents or powers that have not been expanded. For example, x must be written as ----, must be expanded to be written as 8, and (x) 4 must first be changed to 4 x 4 x 7

10 8 Study Guide for Math 0 X, Y Examples and then to 8x 4. You will be expected to give the answers in the correct form for the tests, so ask a staff member to help you if this is not clear.. (X Y )(X 4 Y) = ()(X X 4 )(Y Y) = ()X 4 Y + = X Y 4 = Y X. 6X 7 Y X Y X 7 Y 8 4X 4 Y 6 9 X 7 X Y Y 8 4X Y 6. ( x y 4 ) ( ) ( ) x ( ) y 4( ) y y ( ) x x 6 Recall that () = ()()() = 8 and ( ) 6 x y (x y ) 4 = () (4) x (4) y (4) = () 4 x 8 y 8x 8 = Remember () 4 = ()()()() = (9)(9) = 8, but 4 = () 4 = ()()()() = (9)(9) = 8. Homework: Sect. R.7, pp , 7 All Odd Problems.. Do the following as a sample test for Lesson : The sample test is Lesson : Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson. 6. Show your homework and sample test to a staff member and ask for Lesson Test. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in. y

11 Lesson : Sample Test x. Evaluate when x = 0 and y =. y. Find the area of a triangle when the height h is 0 feet and the base b is 6 feet. Be sure to give the appropriate units.. Write an algebraic expression: Nine less than some number. 4. Write another inequality with the same meaning as x. Simplify.. a. 7 b Add, subtract, multiply, or divide (6)() Multiply. Factor. 9. y(6 x) 0. (y ). x xy. 7x + 4y Simplify.. 0[4 ( + 8)] 4. 6 ( ) ( 8) 9

12 0 Study Guide for Math 0 X, Y. 4x [6 (x )] 6. 7 {[(y ) + 9] (y + 8)} 7. Graph on a number line: x Simplify the following. x x ( ) a b (x ) (x ) 9a b 4. (x ) (x ). (4x y 6 )

13 Lesson : Linear Equations In applying mathematics to solve problems it is very important that you gain ability to express concepts as equations and to understand the methods of solving them. In this lesson you will review methods of solving equations. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson you will be able to:. Tell what an equation means and what is meant by a solution to an equation.. Solve equations using the addition and multiplication principles.. Solve certain equations containing parentheses. 4. Determine when an equation has no solution or when any real number is a solution.. Solve word problems by translating them to equations. 6. Solve formulas for a specified variable. Procedures. Read section., pp Homework: Sect.., pp , 99 All Odd Problems.. There are equations with no solution and there are equations for which any real number is a solution. The latter type is called an identity. Homework: Solve the following equations. (Answers are provided at the back of this Study Guide beginning on page 7.). (x + ) = (0 x). x + = 6x (x ). 8x 4(x ) = 4. 9 (y + 4) = 0 y. (m ) = 8m (m + ). Read section., pp Homework: Sect.., pp. 9 98, 4 All Odd Problems.

14 Study Guide for Math 0 X, Y 4. Read section., pp Homework: Sect.., pp. 07, 4 All Odd Problems.. Do the following as a sample test for Lesson : The sample test is Lesson : Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson. 6. Show your homework and sample test to a staff member and ask for Lesson Test. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in.

15 Lesson : Sample Test Solve each of these equations:. 7 a = 9. ( x 4) = (n 6) x 0. =. 0.x. 4t (t + 0) = 4(t ) 6. 8 (y 6) = y 7. (x 4) (x 4) = 8 For Exercises 8 and 9, translate to an equation and solve. 8. Find the dimensions of a rectangle whose length is twice its width and whose perimeter is 4 inches. 9. Money is borrowed at 9% simple interest. After one year, $87 pays off the loan. How much was originally borrowed? rr ( h) 0. Solve A for h.

16 Lesson : Inequalities A comparison of any two real numbers A and B gives rise to one and only one conclusion. Either A and B are equal, A is less than B, or B is less than A. thus, we say that the set of real numbers is linearly ordered. In this lesson, we use the properties of the relations less than and greater than to solve sentences involving them. You will also learn a method for converting units of measure from one unit to another within a system, or converting from one system to another. This method uses a well known property of fractions. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson, you will be able to:. Solve and graph simple inequalities in one variable.. Perform a given change of dimension symbols by the method of multiplying by one.. Write set notation for solution sets of equations and inequalities. 4. Find the union or intersection of two given sets.. Solve and graph compound inequalities in one variable. 6. Solve applied problems that involve inequalities. 7. Solve and graph absolute value inequalities. Procedures 4. Read Section.4ab, pp. 4 Note: If a compound sentence uses the conjunction or, the sentence is considered true if either part is true. The sentence 8 is an abbreviated form of 8 or = 8. To determine the truth or falseness of the sentence, we examine each part separately. If either part is true, then the sentence is true. Since 8 is true, the sentence 8 is also true. Homework: Set.4, p.,.. Read section.4c, pp. 9 Homework: Set.4, pp. 6, 69 All Odd Problems.. Read section.4d, pp. 9 Homework: Set.4, pp. 6 8, 7 99 All Odd Problems. 4. Read section., pp. 9 6 Note: An easy way of remembering the meaning of intersection is to think of the intersection of two streets as being the portion of pavement that belongs to both streets.

17 Lesson : Inequalities On p. 9, the size of the oval shapes is immaterial. It is merely an abstract representation. Note: If a compound sentence uses the conjunction and, the sentence is considered true only if both parts are true. Example and 8 is true because is true and 8 is also true. Example x and x 7 is true if x is replaced by any number between and 7 (not including or 7) because each of those numbers make both parts true. The sentence may be abbreviated to x 7. In general, the sentence a x b may be abbreviated to a x b provided that a b. Similarly, a x and x b may be abbreviated to a x b provided that a b. Note: There is no abbreviation for a compound sentence that uses the conjunction or. Note: In the example in the middle of p. 0, if you imagine placing the second number line on top of the first, then the third graph shows the points that are in both graphs and. In Example 7, p. 4, the third graph is the final graph that shows the points that belong to either graph or graph. The following are additional examples of unions and intersections of sets. In these examples, set-builder notation is used. In each of these problems, graph the solution set, and also indicate the solution set by using set-builder notation.. {x x 4} {x x } x 4 x 4 4 x 4 4 The third graph shows all the points that belong to either graph; therefore, it is the graph of the union of the two given sets. Using set-builder notation, we get: {x x 4} {x x } = {x x 4}

18 6 Study Guide for Math 0 X, Y. {x x } {x x 7} x 7 x 7 7 x 7 7 The third graph represents the union of the two given sets. In set builder notation, we get: {x x } {x x 7} = {x x 7}. {x x } {x x 7} x 7 x x 7 {x x } {x x 7} = {x 7 x } 4. {x x 4} {x x } x 4 4 x 4 4 x or x {x x 4} {x x } = {x x or x 4}

19 Lesson : Inequalities 7 Homework: Set., pp. 7 40, 77 All Odd Problems.. Read section.6abcd, pp Homework: Set..6, pp , 69 All Odd Problems. 6. Read section.6e, pp Homework: Set.6, pp 0, 7 All Odd Problems. 7. Read section.6e, pp again. Note: x = 7 is false for all real x since the absolute value is always greater than or equal to zero. For the same reason, x is impossible also. Therefore, the answer to each of these is that the solution set is the empty set. Whereas, x is true for all real x, and x 0 is true only when x = 0. Homework: Solve the following equations and inequalities. Answers are on page 7 of this Study Guide.. x = 4. 6x 8 0. x. x 0 0. x 8. Do the following as a sample test for Lesson. The sample test is Lesson : Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson. 9. Show your homework and sample test to a staff member and ask for Lesson Test. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in.

20 Lesson : Sample Test Solve and write set notation for the answer.. x +. x 60. 4x 4. x 0 9x. (x ) (x + ) 6. 8(x + ) + 6(4 x) ( 7x) 4(4 + 6x) 7. You can rent a car for either $40 per day with unlimited mileage, or $0 per day with an extra charge of cents per mile. For what number of miles, traveled in one day, would the unlimited mileage plan save you money? (Translate to an inequality using algebra and solve.) 8. Jean is taking a math course. There will be tests, each worth 00 points. She has scores of 8, 8, 9, and 9. What score on the last test will give her an A in the course if she needs an average of at least 90 on the five tests? (Translate to an inequality using algebra and solve.) 8

21 Lesson : Sample Test 9 9. Graph: x or x 6 0. Graph: x 4 Solve and write set notation for the answer.. {, } {, }. {, } {, }. + x and x x 7 or x 7. x x 7. x = x = 0 9. x = 4 x 0. x. 6 x x x x. x

22 Lesson 4: Polynomials This lesson will review the vocabulary associated with polynomials. You will also learn to do arithmetic operations with polynomial expressions. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson, you will be able to:. Identify the terms and degrees of a polynomial.. Identify the coefficient and degree of a term.. Add, subtract, multiply, or divide polynomials. Procedures. Read section 4.abc, pp. 6 Note: Remember that any expression (except 0) with a 0 exponent is equal to. If necessary, a number like can be thought of as = = y 0 (or x 0, etc.). This means that remains the same no matter what values the variable assumes, so is referred to as the constant term. Homework: Sect. 4., pp. 6, 9 All Odd Problems.. Read section 4.d, pp. Homework: Sect. 4., pp. 6 7, 6 9 All Odd Problems.. Read section 4.abc, pp. 8 4 Note: Don t forget that you can use the FOIL method for multiplying: I L ( x + )(y 7) = xy 7x + 6y 4 F O This process can be extended when either polynomial has more than two terms (as per Product of Two Polynomials, just before Ex. 7 on p. 0): ( x + )(y 7 + 8z ) = xy 7x + 8xz + 6y 4 + 6z Homework: Sect. 4., pp. 4 46, 6 All Odd Problems. 0

23 Lesson 4: Polynomials 4. Read section 4.d, pp Note: In this homework be sure to use the special product rules to multiply. Becoming familiar with these will help you in future work. Homework: Sect. 4., pp , 6 99 All Odd Problems.. Read section.ab, pp Note: Remember that any number (except 0) divided by itself is : 7 6. x (if x 0) 7 6. x Also, remember the names of a division problem: divisor quotient dividend remainder (may be zero) When dividing by polynomials, the terms of the quotient are found by multiplying only the first term of the divisor by the correct number. That is, x x x 8 x x x x 8 x x 8x 8 x x x x 8 x x 8x 8 x 8 x x x 8 x x 8x 8 8x 4 to get this term, ask yourself what times x (from x + ) is x? Then, Multiply x and (x + ) = x + x. Then, subtract (x + x) from (x x + 8). i.e., x x + 8 (x + x) = x x + 8 x x = 8x + 8. to get this term, ask yourself what times x (from x + ) is 8x? Then, multiply 8(x + ) = 8x 4, and subtract 8x 4 from 8x + 8 as ( 8x + 8) ( 8x 4) for a remainder of. (Watch your signs!) In arithmetic, -- means + -- where the + is understood. But in algebra, x means times x, so the + is necessary in our answer. The answer in mixed form is x x Homework: Sect., p. 44, 9 All Odd Problems.

24 Study Guide for Math 0 X, Y 6. Do the following as a sample test for Lesson 4. The sample test is Lesson 4: Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson. 7. Show your homework and sample test to a staff member and ask for Lesson Test 4. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in.

25 Lesson 4: Sample Test. What is the degree of the polynomial 7x y + x 9y?. What is the coefficient of the third degree term of the polynomial in Exercise?. Collect like terms: abc + a bc 4abc 4. Add: 4x y + 6x and x + x y + 7y +. Subtract: (x + 6x ) (x + 4x ) Multiply the following: 6. (y ) and (y + ) 7. (4z + 8) 6x x x 8. Divide: x 9. Divide: (6y 4y 4) (y + 4) 0. Multiply and collect like terms: (x + y )(x 7x + )

26 Lesson : Factoring In this lesson, you will review factoring of polynomials that you learned in elementary algebra. Also, you will study the factorization of the sum and difference of two cubes. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson you will be able to:. Factor certain polynomials.. Factor differences of two squares having more than two terms.. Factor sums and differences of two cubes. 4. Solve equations by factoring. Procedures. Read section 4., pp. 49 Homework: Set 4., pp., 6 All Odd Problems.. Read section 4.4, pp. 6 9 Homework: Set 4.4, pp. 60 6, All Odd Problems.. Read section 4., pp Homework: Set 4., pp , 6 All Odd Problems. 4. Read section 4.6abc, pp Homework: Set 4., pp , 67 All Odd Problems. Answers for 9, 49, and could be as follows: ( x ) ( 6z) ( 6z) ( x y) ( x y). Read section 4.6d, pp Homework: Set 4.6, pp. 80 8, 69 9 All Odd Problems. Answers for 8 and 9 could be as follows: ( a ) ( 4a a ) ( 0x ) ( 00x 0x ) 4

27 Lesson : Factoring 6. Read section 4.7, pp. 8 8 Homework: Set 4.7, pp , 7 All Odd Problems. Answers for 7 and could be as follows: ( x 4y ) (x )(9x + x + ) Read section 4.8a, pp Note: When solving an equation by factoring and the principle of zero products, it is essential that zero be on one side of the equation. Example: Solve x + x 6 = 6 Although the left side can be factored, there must be a zero on the right before you factor. So, x + x = 0 (x + 4)(x ) = 0 The solutions are 4 and. Homework: Set 4.8, pp , 6 All Odd Problems. 8. Read section 4.8b, pp Homework: Set 4.8, pp , 6 97 All Odd Problems. 9. Do the following as a sample test for Lesson : The sample test is Lesson : Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson. 0. Show your homework and sample test to a staff member and ask for Lesson Test. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in. 7

28 Lesson : Sample Test Factor each of the following completely.. 6x y x y + x y. ab + ac + db + dc. 9t 0t x 0x. x 40x x + 8x y 7. 8x + 0x 6 8. x 64 6

29 Lesson : Sample Test a 7 b + 4ab 7 0. x + 6x 7x 4. Solve: 0x + 6 = 6x. Solve: 47x = x Translate Exercises and 4 into an equation, then solve:. A photograph is cm longer 4. The shorter leg is 7 feet less than it is wide. Its area is 40 cm. than the longer leg of a right Find the dimensions of the triangle. If the hypotenuse is photograph. one foot more than the longer leg, find the longer leg.

30 Lesson 6: Review We now pause to reflect upon and review the material you have learned in the first five lessons. When you have completed this lesson you will have firm foundation of concepts needed for the succeeding lessons. NO CALCULATORS MAY BE USED ON TESTS. Procedures. Homework: Test Chapter R. pp. 7 74, 6 All Odd Problems.. Homework: Test Chapter, pp. 6, 4 All Odd Problems.. Homework: Test Chapter 4, pp , 9 All Odd Problems. 4. Review sample test.. Do the following as a sample test for Lesson 6: The sample test is Lesson 6: Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson 6. Show your homework and sample test to a staff member and ask for Lesson Test 6. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in. 8

31 Lesson 6: Sample Test. Remove grouping symbols and simplify: a [( 7a) (4a + 6)]. Simplify: 8b 4 c b 9 c. Solve for T: R -- T 6 4. Solve and graph: x. Solve and write the solution set in set notation: x Subtract and collect like terms: (4x 6x + x ) (x + 8x x ) 7. Divide 9x 4 x 6x by x 8. Factor completely: 4x + x 9. Factor completely: 8x 7y 6 9

32 0 Study Guide for Math 0 X, Y 0. Translate to an equation, then solve: A hardware store drops the price of a lawn mower % to a sale price of $0. What was the former price?

33 Lesson 7: Fractional Expressions The properties of rational numbers and operations within that set are now extended to algebraic fractions. In order to solve fractional equations that arise in the solution of certain types of applied problems, you must first learn how to manipulate fractional expressions. A necessary prerequisite skill is factoring polynomial expressions. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson you will be able to:. Rename a fractional expression using multiplication by a factor of one.. Simplify fractional expressions.. Multiply and divide fractional expressions. 4. Find the LCM of several algebraic expressions.. Simplify complex fractional expressions. Procedures. Read Section. abcd, pp Homework: Set., pp. 4 4, 9 All Odd Problems.. Read section.e, pp Review the definition of reciprocal and examples 4 44 on page 9 of your textbook. Homework: Set., pp. 4 44, 4 7 All Odd Problems. 4. Read section.a, pp to the end of Example 4. Note: It is very important that you use this method of addition of fractions because it is precisely the method used in algebraic fractions. Notice that the denominators are kept in factored form until the very last step. This makes it easier to check if there are any common factors in the numerator and denominator. If there are, the fraction should then be reduced by removing the factor(s) of. Homework: Set., p. 4, 4.. Read section.a, pp Homework: Set., p. 4, 9 All Odd Problems. 6. Read section.b, pp Note: In Example 6 why is 0 not a sensible replacement? In Example 7, notice that the numerator and denominator of the sum on the first line are factored in

34 Study Guide for Math 0 X, Y order to reduce the fraction to the lowest terms. Fractional answers should be reduced to lowest terms whenever possible. Homework: Set., pp. 4 44, 6 All Odd Problems. 7. Read section.c, p. Homework: Set., pp. 44 4, 6 8 All Odd Problems. 8. Read section.4, pp Homework: Set.4, pp , 4 All Odd Problems. 9. Do the following as a sample test for Lesson 7: The sample test is Lesson 7: Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson. 0. Show your homework and sample test to a staff member and ask for Lesson Test 7. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in.

35 Lesson 7: Sample Test. Simplify: y 6y y y. Multiply and simplify: x x x x x. Divide and simplify: x 6x x x 6 x x x x 4. Subtract and simplify: x y x. Find the LCM: x 4, x + 4x Add and simplify: x x -- x 7. Simplify: x 9 -- x

36 4 Study Guide for Math 0 X, Y 8. Add and simplify: y y y 4y 9. Calculate and simplify: x x x x Calculate and simplify: x x y x y y x

37 Lesson 8: Solving Fractional Equations The solutions of applied problems in algebra oftentimes involve fractional equations. The technique used to solve these equations involves the use of the LCM. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson you will be able to:. Solve fraction equations.. Solve applied problems involving work, motion, or variation.. Solve a formula containing fractional expressions for a given letter. 4. Given a description of direct or inverse variation find the variation constant and the equation of variation. Procedures. Read section., pp Note: You must check your values of x into the original equation and throw out any value of x that makes any denominator equal to zero. Homework: Set., pp , All Odd Problems.. Read section.6a, pp Homework: Set.6, pp , 9 All Odd Problems.. Read the first paragraph of page 466 of section.c. d Homework: Given r --, solve for: t a. t b. d 4. Read section.6c, pp Note: Since systems of equations have been deferred to Lesson 9, an adjustment is needed in the solution of the problem in Example 6. The table must be filled using only one variable. It is common practice, in this case, to fill in two columns from the given information, then filling in the third column using the other two entries in the same row. The sample is redone as follows: Example 6 Instead of using the two variables w and t, we choose to use only one. Either one will do, but usually we choose the one asked for in the problem. So if we use w,

38 6 Study Guide for Math 0 X, Y two columns of the table may be filled in as follows: Let w = speed of the wind in mph then, 00 + w = speed of plane traveling with the wind in mph and 00 w = speed of plane traveling against the wind in mph No matter which one is the unknown, it is always the speed of the plane in still air minus the speed of the wind for the speed of the plane against the wind. Distance (miles) Rate (mph) Time (hours) With Wind Against Wind w w Now we will fill in the remaining column using the fact that time distance rate Distance (miles) Rate (mph) Time (hours) With Wind Against Wind w w w w Since the two times are the same, we write: Time with wind = Time against wind, which is w 00 w The rest of the solution is the same as that on the top half of p. 48. Study the following additional example. Example 7 A car leaves a town traveling east at 40 mph. Two hours later, a truck leaves the town in the same direction at mph. How far from town will the truck catch the car?

39 Lesson 8: Solving Fractional Equations 7 Let d = the distance from the town in miles, then, d = the distance of the car in miles, and d = the distance of the truck in miles. Distance (miles) Rate (mph) Time (hours) Car d 40 Truck d Now we will fill in the remaining column using the fact that time distance rate Distance (miles) Rate (mph) Time (hours) Car d 40 Truck d d d We then compare the quantities in the column last filled (in this case the two values in the Time column). Since the time for the car is two hours more than the time for the truck, we write: Time for the car = Time for the truck + hrs or, d d = and =, so the LCD = 880 so, clearing of fractions, we have d = 8d and d or 9 miles. -- The truck will catch the car 9 -- miles from town. Homework: Set.6, pp , 9 All Odd Problems.. Read section.7, pp Homework: Set.7, pp , All Odd Problems. 6. Read section.8ab, pp Homework: Set.7, p. 486,

40 8 Study Guide for Math 0 X, Y 7. Read section.8cd, pp Homework: Set.7, p. 487, 7 (odds) 9 All Odd Problems. 8. Do the following as a sample test for Lesson 8: The sample test is Lesson 8: Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson. 9. Show your homework and sample test to a staff member and ask for Lesson Test 8. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in.

41 Lesson 8: Sample Test. Solve for x: -- x -- 4 x x. Solve for y: y y. Solve for t: t t t 4. Solve for x: x x x. Solve for t : Q R st ( t ) 6. Find the variation constant and an equation of variation where y varies directly as x, and y = 8 when x Find the value of y when x --. The description is: y varies inversely as x, and when y = 0, x = Joe can paint a room in hours. Cathy can paint the same room in hours. Working together, how long would it take them to paint the room? 9. A fishing boat takes twice as long to go miles up a river than to return. If the boat cruises at 9 mph in still water, what is the rate of the current? 0. The volume V of a gas varies inversely as the pressure p upon it. The volume of a gas is 00 cubic cm under a pressure of km per square cm. What will be its volume under a pressure of 40 km per square cm? 9

42 Lesson 9: Radicals, Part I In this lesson, you will review the properties of, and operations with, square roots. These will be extended to include other roots as well. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson, you will be able to:. Simplify expressions with integer exponents.. Convert to and from scientific notation.. Multiply and divide using scientific notation. 4. Simplify expressions involving odd and even roots.. Add, subtract, multiply, divide, and simplify radical expressions. 6. Identify rational and irrational numbers. Procedures 40. Read section R.7, pp. 8 6 Note: Since scientific notation is of the form a 0 b, where a 0, to convert to scientific notation move the decimal point in the original number to a place where the result is a number between and 0. The b is an integer, positive or negative. For example, 4,87,00 would become would become 9. (zeroes at the beginning or end may be dropped). This new number is now a. The exponent b is how many places the decimal moves to give back the original number; positive if to the right, negative if to the left. So, is 6 places right and b is is places left and b is Thus, 4,87, Remember that while a number like.9 0 looks like scientific notation, it is not because.9 is not between and ( )

43 Lesson 9: Radicals, Part I 4 Homework: Set R.7, pp , (odds), 07, 09, 7 All Odd Problems.. Read section 6.a, pp Note: Remember that a b = a b, but a + b a + b. Homework: Set 6., pp. 04 0, 4 All Odd Problems.. Read section 6.cd, pp Note: When an index is, this is the square root. Also observe you need to use only absolute value for even roots. Never use absolute values for odd roots. Homework: Set 6., pp , 9 All Odd Problems. 4. Read section 6.a, pp. 8 Note: Examples 9 on pp. 7 8 do not require absolute value only because the problem says assume that all expressions under radicals represent non-negative numbers. This is not usually the case! Be sure you can use absolute value appropriately in these problems. For example, the answers for each would be the following: 9. x 0. x y. 6x y 6xy Do not use absolute values for, 4, or. Homework: Set 6., pp. 0, 49 All Odd Problems.. Read section 6.b, pp. 8 9 Note: The Power-Root Rule allows you to do either roots or powers first, whichever is easier. For example, find to mention its cube root Here, 8 6 is rather complicated, not But 8 is. So, 8 6 ( 8) 6 6. On the other hand, for the expression ( 4 9) the radical part does not simplify. But squaring the radicand first gives ( 4 9) Homework: Set 6., pp., 9 All Odd Problems. 6. Read section 6.4a, p. 4 Homework: Set 6.4, pp. 6 7, All Odd Problems. 7. Do the following as a sample test for Lesson 9: The sample test is Lesson 9: Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson 8. Show your homework and sample test to a staff member and ask for Lesson Test 9. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in.

44 Lesson 9: Sample Test Simplify: x y x 8 y 7 [ x ] a x a Write in scientific notation:. ( )(. 0 7 ) Simplify Exercises 7, assuming the letters represent any allowable real number.. a. ( 8) 6 b. ( ) y a a 4 4

45 Lesson 9: Sample Test 4 Simplify Exercises 8, assuming all expressions are positive real numbers. 8. 7x x 4 8x x 4 6x 7x x 6 y x y

46 Lesson 0: Radicals, Part II This lesson will extend the concept of radicals to more complex expressions and to equations. New ways of expressing and using roots will also be discussed. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson you will be able to:. Rationalize radical expressions.. Use and simplify fractional exponents.. Solve radical equations. 4. Perform arithmetic operations with complex numbers. Procedures. Read section 6.4b, p. Note: Remember that (a + b) a + b. Also, 9 but ( ) 9. Homework: Set 6.4, pp. 7 9, 7 89 All Odd Problems.. Read section 6.a, pp. 0 Homework: Set 6., p., 9 All Odd Problems.. Read section 6.b, p. Homework: Set 6., p. 4, 7 All Odd Problems. 4. Read section 6.abc, pp Homework: Set 6., pp. 4, All Odd Problems.. Read section 6.d, pp. Homework: Set 6., p. 4, 89 All Odd Problems. 6. Read section 6.6, pp. 4 Homework: Set 6.6, pp. 4 46, 7 All Odd Problems. Be sure to check your solutions when you use the principle of powers with an even exponent. 7. Read section 6.7, pp Homework: Set 6.7, pp. 0, 9 All Odd Problems. 8. Read section 6.8ab, pp. 4 Homework: Set 6.8, p. 6, 9 All Odd Problems. 44

47 Lesson 0: Radicals, Part II 4 9. Read section 6.8cdef, pp. 60 Note: On p. 6, Example, the FOIL method of multiplication may be used: ( i) ( i) ( i) 9 6i 6i 4i 9 i 4 i Also, Examples 4 7 lead to the following: i 0 = i = i i = i = i i 4 = i = i, etc. It can be seen that every four times the powers of i start over again. This leads to a shortcut for simplifying powers of i: divide the exponent by 4 and use the remainder as the new exponent. For example, i = i = i since 4 = R. Also, i 48 = i 0 = since 48 4 = R 0. On page 7, Example, the conjugate of 4i is 4i because 4i = 0 + 4i. In general, the conjugate of bi is bi. However, the conjugate of 4 is not 4. Note that 4 = 4 + 0i and the conjugate is 4 0i = 4. In general, the conjugate of a is a. Homework: Set 6.8, pp. 6 64, 4 07 All Odd Problems. 0. Do the following as a sample test for Lesson 0: The sample test is Lesson 0: Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide beginning on page 49. Work the entire test before checking the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson.. Show your homework and sample test to a staff member and ask for Lesson Test 0. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in.

48 Lesson 0: Sample Test 7. Rationalize the Simplify: denominator: 8x x x Simplify, leaving the 4. Use exponents to 6 answer in exponent form: 64x 4 y have a single radical: x Simplify Problems 9:. (9 + i)( i) 6. ( i) (8 + 4i) 7. i 7 8. ( x) ( x) 46

49 Lesson 0: Sample Test i 0. Rationalize the 7 i denominator: y x y x Solve Problems and.. x 6. x x. In a right triangle, find b if c = 0 and a = 4. (Simplify the answer.)

50 Lesson : Review Before going on further, this lesson lets you review the material you have learned on Lessons 7 through 0. When you have completed this lesson, you will have a good grasp of fractional expressions and equations, and you will feel comfortable with exponents, powers, and roots. NO CALCULATORS MAY BE USED ON TESTS. Procedures. Homework: Test: Chapter, pp , All Odd Problems.. Homework: Test: Chapter 6, pp , 4 All Odd Problems.. Review Sample Tests Do the following as a sample test for Lesson : The sample test is Lesson : Sample Test on the next page of this Study Guide. The answers are given at the end of this Study Guide. Text references for all Review Sample Tests are to the left of the problem number in the answers. Be sure to time this test and allow at least this amount of time when you take the real test. Remember that every test you take is averaged into your score for this lesson.. Show your homework and sample test to a staff member and ask for Lesson Test. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in. 48

51 Lesson : Sample Test. Simplify: 4x x x. Solve: x x 8 x x. A train traveling at maximum speed takes -- hours longer to travel a distance of 0 miles than to travel a distance of 0 miles at the same maximum speed. Find the maximum speed of the train. (Translate using algebra and then solve.) 4. The electrical resistance of a certain wire varies directly as its length. If a wire 40 ft long has a resistance of ohm, what is the resistance of a wire of length 00 ft? (Translate using algebra and then solve.). Simplify: x 60x 6 (Assume x is any allowable real number.) 6. Rationalize the numerator. x x x x 7. Simplify: 8a x Simplify: 7 9y x 49

52 0 Study Guide for Math 0 X, Y 9. Solve: x x 0 8 6i 0. Simplify: i

53 Comprehensive Lesson: Final Exam for Math 0X Congratulations, you have completed all your lesson tests for the first module Math 0X of the Intermediate Algebra course. The only remaining task for you to complete is the final examination. This will demonstrate that you have mastered the material contained in the first eleven lessons of this Study Guide. NO CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed the final exam you will be able to:. Say you have mastered the material in Lessons.. Receive a grade in Math 0X. Procedures. Read through your past homework assignments and redo as many problems as needed to make sure you feel that you are comfortable with all the material.. Redo your sample tests from the first eleven lessons.. Do the following as a sample test for the final exam: The sample test is Sample Final for Math 0X given on the next page of this Study Guide. Be sure to time this test and allow at least that amount of time when you take the real test. If you take this test during the designated day of finals week, you will be restricted to a time limit of hours. During the regular class times you can take the entire continuous period for which the math lab is open. 4. Show your sample test to a staff member and ask for the final for Math 0X. If you use scratch paper, number your problems in sequence and staple your scratch paper to your test before you turn it in. Be sure you are ready for this test, since you can take it only once and it counts % of your course grade. Remember, you cannot receive a passing grade without the attendance hours. If you complete the course work for a module, but have not satisfied the attendance requirement, you can request an IP, in-progress grade, and complete your attendance requirement during the following semester. If you are on an IP grade from the previous semester and you complete the course work for a module, but have not satisfied the attendance requirement, you can request an I, incomplete grade, and complete your attendance requirement during the following semester. If you are on an I grade from the previous semester and you complete the course work for a module, but have not satisfied the attendance requirement, you can request an extension of the incomplete grade for another semester by filling out a petition with the admissions office.

54 Sample Final for Math 0X This is a very accurate representation of the actual final, so be certain you can work these problems easily (without a calculator).. Simplify: {4x [(4x ) (x + )]} +. Simplify: + (9 ). Simplify: y 4 x y 6 x 4 4. Divide and write scientific notation for the answer: Solve: x

55 Lesson : Sample Final for Math 0X 6. The perimeter of a rectangle is 96 feet. The length is three feet less than twice the width. Find the dimensions. (Translate using algebra and then solve.) hc ( d) 7. Solve: A for c. 8. Solve and graph: Use set notation. 9 (x + ) (x + ) + x 9. Solve: 7x or 6x Solve: x =. Solve: 4 y 8

56 4 Study Guide for Math 0 X, Y. Solve: 6 x Subtract: (9x 4x + 4) (7x + 4x + 4) 4. Multiply: (a b)(a + b)(a + 4b ). Factor: ax bx ay + by 6. Factor: 6x x x 7. Factor: 4x 6 8. Factor: 0z 4zy 9. Solve: x + x = 8

57 Lesson : Sample Final for Math 0X 0. If 4 times the square of a number is 4 more than 8 times the number, what is the number? (Translate using algebra and then solve.). Divide and give the answer in mixed form: (a 8a 6) (a + 4). Multiply and simplify: x x x x 4x 9x x 49. Simplify: x x x x 4. Add. Simplify, if possible. 4x xy x y xy y x y. Simplify, if possible. x x x x 6. Solve for x: x x

58 6 Study Guide for Math 0 X, Y 7. Solve for x: x 60 x x 8. An old machine does a certain job in 9 hours while a new machine can do the same job in hours. How long would it take to get the job done if both machines worked together? (Translate using algebra and then solve.) 9. A train leaves town and travels north at 7 kph. Two hours later, a second train leaves on a parallel track traveling north at kph. How far from the starting point will the second train overtake the first? gs 0. Solve: R for s. g s. Simplify: x x x. Evaluate: 8 ( 6) 8

59 Lesson : Sample Final for Math 0X 7. Simplify: ( 4x 4 )y 4. Simplify: 8x y. Simplify: x 4 4x 6. Simplify: ( ) ( ) 7. Rationalize the denominator: y x 4 8. Rationalize the denominator: Use rational exponents to write a single radical and simplify: x x 40. In a right triangle, find the length of the side a if b =, and c =. Give an exact answer in simplified form.

60 8 Study Guide for Math 0 X, Y 4. Divide and simplify: i i 4. Solve: x x 4 0

61 Lesson : Quadratic Equations Many applied problems cannot be solved using only linear for first degree equations. For example, problems describing the motion of objects frequently require quadratic or second degree equations. In this lesson, you will review some methods of solving quadratic equations and learn how complex numbers, studied in Lesson 0, come into the theory of quadratic equations. CALCULATORS MAY BE USED ON TESTS. Objectives When you have completed this lesson, you will be able to:. Solve a quadratic equation by root extraction, factoring, completing the square, and the quadratic formula.. Approximate solutions by square root table or calculator.. Solve fractional equations that are quadratic. 4. Solve applied (geometric, motion, rate) problems.. Describe the solutions of a quadratic equation without solving it by using the discriminant. Procedures. Read section 7., pp Note: Remember x = a has two values for x, a and a. Do not multiply or divide both sides of an equation by anything that could be zero. Factor first, then use the principle of zero products. Homework: Set 7., pp , 7 All Odd Problems.. Read section 7., pp Note: When simplifying from the quadratic formula, factor first, then line out (or cancel) like factors in numerator and denominator ( ) ( ) Homework: Set 7., pp. 94 9, All Odd Problems.. Read section 6., p. 0 Calculator Corner. Homework: Set 6., p. 0 Calculator Corner, Exercises Read section 7.a, pp Note: In solving fractional equations you should first find the restrictions on the variable. The values of the variable that make the denominators zero cannot possibly be solutions. The textbook states check possible solutions by substituting 9