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College of Marin Study Guide for Math 101 X,Y

Revised Edition

Ted Broomas, Ira G. Lansing, Jeannette Woods

Revision Editor—Dan Ayer

To Accompany Introductory Algebra, Eleventh Edition By Marvin L. Bittinger

Copyright © 2006, 2003, 2002 by College of Marin Department of Mathematics Copyright © 1999 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

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ISBN 0-536-26680-8

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Contents

Math 101X Elementary Algebra

Lesson Page Title

1 7 Prealgebra Review

2 16 Introduction to the Real Numbers

3 20 More on Real Numbers

4 23 Solving Equations

5 28 Inequalities and Exponents

6 35 Linear Equations and Linear Graphs

7 40 Systems of Equations

8 43 Sample Final Exam for Math 101X

Math 101Y Elementary Algebra

Lesson Page Title

9 51 Operations with Polynomials and Special Products

10 55 Factoring Polynomials

11 59 Fractional Expressions and Equations

12 65 More Fractional Expressions and Equations

13 76 Radicals

14 83 Quadratic Equations

15 86 Applied Problems Using Quadratic Equations

16 91 Final Exam for Math 101Y

100 Answers to Sample Tests

Study Guide for Math 101 X, Y

The purpose of this Study Guide is to help you in learning the basic skills and concepts that are taught in Math 101.

This Study Guide consists of 14 lessons and two finals. Use it with your textbook and be sure to do the margin exercises the textbook tells you to do. Each lesson of this Study Guide tells you what pages to read in the textbook and which problems to do for homework. After you have completed each lesson, you will be ready to take a test on that lesson. Show your homework and sample test to the instructor or tutor. At that point you will be given a test for a grade on that lesson.

The textbook is Introductory Algebra, 11th Edition, by Marvin Bittinger. It is a work-text, which means that you can do the problems in the textbook if you wish. When you read the text, be sure to have a pencil ready. It is important that you do the Margin Exercises. They accompany the reading. The Margin Exercises are a check to ensure you understand the concepts that are explained, and also to help you gain the necessary skills one step at a time.

The answers to the Margin Exercises are in the back of the textbook. Also, on these pages you will find the answers to the odd-numbered problems in the exercise sets and all the problems in the Chapter Summary and Review Exercises and the Chapter Tests. The homework and sample tests given in this study guide have the answers in the back of this study guide. The homework problems should be checked one at a time. The complete sample test should be worked before being checked. There are 19 videotapes and 9 DVDs available that introduce each section of the textbook and work some of the exercises. On the next page is a list of these videotapes by section number, content description, and which exercises are worked.

Videotape Index for Introductory Algebra, 9th Ed. Marvin Bittinger

There are 19 videotapes that introduce the material in each section and work the following text exercises.

VHS Tape #

Run Time 9e Sect # Section Title Presenter Exercises Used

Examples Used

1 18:48 R.1 Factoring and LCMs

D. Ellenbogen 3, 17, 29 4

1 20:46 R.2 Fraction Notation J. Penna 7, 15, 37, 41 2, 6, 7, 16, 17

1 20:08 R.3 Decimal Notation D. Ellenbogen N/A 4, 7

2 13:32 R.4 Percent Notation P. Schwarzkopf 3, 23, 35, 41, 45, 55

2, 3

2 14:16 R.5 Exponential Notation and Order of Operations

D. Ellenbogen 31, 61 3, 8, 11

2 21:36 R.6 Geometry B. Johnson 3, 7, 23, 33, 41, 43, 45

3, 14

3 7:08 1.1 Introduction to Algebra

B. Johnson 3, 35 5, 9

3 20:37 1.2 The Real Numbers B. Johnson 4, 17, 45, 53 5, 8, 10, 12, 13, 17

3 18:22 1.3 Addition of Real Numbers

C. Vance 9, 21, 23, 31, 41, 51, 53, 65

1, 2, 3, 4, 13, 19

3 10:12 1.4 Subtraction of Real Numbers

C. Vance 5, 21, 23, 25, 59, 75, 87

1, 11, 12

4 13:37 1.5 Multiplication of Real Numbers

J. Penna 11, 3, 35, 37, 41, 73

1, 6, 7, 16

4 17:12 1.6 Division of Real Numbers

P. Schwarzkopf 3, 13, 19, 25, 29, 35, 51, 61

24

4 30:14 1.7 Properties of Real Numbers

B. Johnson 1, 7, 13, 17, 49, 75, 81, 83

1, 2, 10, 13, 19, 26, 29

4 15:35 1.8 Simplifying Expressions; Order of Operations

J. Penna 13, 19, 29, 37, 41

2, 4

5 13:27 2.1 Solving Equations: The Addition Principle

D. Ellenbogen 33 5, 9

5 9:36 2.2 Solving Equations: The Multiplication Principle

D. Ellenbogen 3, 25 3

5 22:20 2.3 Using the Principles J. Penna 1, 29, 47, 2, 6

Together 59, 61, 83

5 10:14 2.4 Formulas J. Penna 7, 17, 29, 31, 47

1, 3

6 17:13 2.5 Applications of Percent

D. Ellenbogen 1, 43a 5

6 16:08 2.6 Applications and Problem Solving

J. Penna 15 2

6 16:41 2.7 Solving Inequalities C. Vance 7, 9, 11, 17, 23, 37, 45, 61, 71

1, 2, 3, 4

6 13:30 2.8 Applications and Problem Solving with Inequalities

C. Vance 11, 15, 17, 19, 27

none

7 13:50 3.1 Graphs and Applications

B. Johnson 15, 19, 21, 43

1, 2

7 22:02 3.2 Graphing Linear Equations

D. Ellenbogen 1, 19 2, 5

7 7:38 3.3 More with Graphing and Intercepts

C. Vance 1, 5, 18, 26, 57

none

7 16:17 3.4 Slope and Applications

B. Johnson 31, 39, 51 5, 6, 11

8 16:44 4.1 Integers as Exponents

D. Ellenbogen 15, 19, 29, 43, 75, 91

4, 10, 16

8 16:57 4.2 Exponents and Scientific Notation

J. Penna 5, 49, 53, 63, 89

2, 11, 18

8 16:55 4.3 Introduction to Polynomials

J. Penna 9, 35, 83, 93, 95, 97, 99

9, 21

8 15:30 4.4 Addition and Subtraction of Polynomials

J. Penna 7, 31, 53 1, 4, 10, 13

9 14:45 4.5 Multiplication of Polynomials

D. Ellenbogen 5, 35, 63 2, 4, 7, 9

9 26:53 4.6 Special Products B. Johnson 17, 23, 57, 61, 65, 71, 77, 83, 85,

1, 3, 6, 11, 14, 18, 23, 28

89

9 21:38 4.7 Operations with Polynomials in Several Variables

J. Penna 3, 9, 29, 37, 59, 71, 73

3, 6, 12

9 17:40 4.8 Division of Polynomials

B. Johnson 9, 23, 29, 35 5, 7

10 13:40 5.1 Introduction to Factoring

B. Johnson 3, 13, 29, 41 2, 6, 10

10 18:25 5.2 Factoring Trinomials of the Type x2 + bx + c

P. Schwarzkopf 1, 23, 31 2, 3

10 18:48 5.3 Factoring ax2 + bx + c, a ? 1, FOIL Method

B. Johnson 31 1

10 13:30 5.4 Factoring ax2 + bx + c, a ? 1, ac-Method

C. Vance 1, 7, 19, 69 1

11 17:26 5.5 Factoring Trinomial Squares and Differences of Squares

J. Penna 1, 3, 5, 7, 9, 17, 19, 39, 41, 51, 61, 63, 73

11, 12

11 13:28 5.6 Factoring: A General Strategy

J. Penna 3, 17, 65 none

11 15:47 5.7 Solving Quadratic Equations by Factoring

P. Schwarzkopf 13, 17, 37 1, 5

11 27:58 5.8 Applications of Quadratic Equations

P. Schwarzkopf 3, 19 2, 6

12 15:33 6.1 Multiplying and Simplifying Rational Expressions

J. Penna 9, 25, 29, 47 3, 11

12 8:08 6.2 Division and Reciprocals

C. Vance 3, 5, 13, 17, 33

1, 2

12 10:38 6.3 Least Common Multiples and Denominators

B. Johnson 13, 19, 37 1, 6

12 19:00 6.4 Adding Rational Expressions

C. Vance 5, 9, 59 1, 6, 11

13 22:40 6.5 Subtracting Rational Expressions

J. Penna 13, 29, 35, 37, 51

2

13 13:57 6.6 Solving Rational Equations

C. Vance 11, 27, 39 1, 4

13 20:34 6.7 Applications Using Rational Equations and Proportions

C. Vance 1, 23, 25, 27, 43

2

13 15:00 6.8 Complex Rational Expressions

C. Vance 7, 9, 13, 15 1, 4

14 9:07 7.1 Slope-Intercept Equation

B. Johnson 9, 31 2

14 13:00 7.2 Graphing Using the Slope and the y-Intercept

P. Schwarzkopf 5, 9, 27 4

14 13:10 7.3 Parallel and Perpendicular Lines

J. Penna 1, 11, 13 none

14 13:58 7.4 Graphing Inequalities in Two Variables

C. Vance 1, 23, 25 1, 3

14 20:03 7.5 Direct and Inverse Variation

J. Penna 1, 9, 39 4

15 17:39 8.1 Systems of Equations in Two Variables

J. Penna 5 none

15 16:00 8.2 The Substitution Method

B. Johnson 3, 19 2

15 12:19 8.3 The Elimination Method

J. Penna 9, 19 6


B. Johnson 2 3, 5

15 15:08 8.5 Applications with Motion

C. Vance 1, 5 2

16 13:31 9.1 Introduction to Radical Expressions

B. Johnson 7, 27, 41, 43, 45, 61

2, 3, 5, 7, 10, 15

16 16:38 9.2 Multiplying and Simplifying with Radical Expressions

L. Vrionis N/A 13, 14

16 9:28 9.3 Quotients Involving Radical Expressions

C. Vance 1, 15, 27, 29, 49

2

17 16:38 9.4 Addition, Subtraction, and More Multiplication

J. Penna 1, 23, 49, 57, 59

2, 6, 10

17 14:39 9.5 Radical Equations J. Penna 3, 27, 39 none

17 16:15 9.6 Applications with Right Triangles

B. Johnson 23 1

18 21:23 10.1 Introduction to Quadratic Equations

B. Johnson 3, 19, 57 3, 4, 7, 9

18 21:26 10.2 Solving Quadratic Equations by Completing the Square

C. Vance 7, 29, 47 1, 2, 5, 11

18 18:49 10.3 The Quadratic Formula

J. Penna 21 1

18 8:50 10.4 Formulas C. Vance 27, 29, 41, 45, 47

none


B. Johnson 1 2, 3

19 19:56 10.6 Graphs of Quadratic Equations

C. Vance 9, 11, 31, 35 1, 4

19 18:54 10.7 Functions B. Johnson 5, 15a-c, 43 6, 7c

Lesson 1: Prealgebra Review In this lesson, you will have a chance to review the arithmetic of fractions, decimals and percents. You will also learn how to use exponential notation to express numbers. Knowing how to use the rules for order of operations is crucial in algebra.

Objectives

When you have completed this lesson you will be able to:

1. Find the prime factorization of a composite number.

2. Find the LCM of several numbers.

3. Do arithmetic with fractions and decimals.

4. Convert between any two of these three forms: fractions, decimals, and percents.

5. Write exponential notation for a product.

6. Evaluate expressions by using the rules for order of operations.

Procedures

As you read the explanatory material in the text be sure to follow directions with regard to the margin exercises. It is important that you do these problems so you can test your understanding of the explanations. The answers to the margin exercises are in the back of the textbook (see page A1). Also, these margin exercises are worked out, showing all the steps, in the Solutions Manual, which is available from the bookstore. The answers to the odd-numbered problems of the exercise sets are also provided in the back of your textbook. The answers for the assigned even-numbered problems are on page 113 of this study guide. The answers for all the even-numbered problems are in the Teacher’s edition of the textbook which is available in the math office. There are 17 videotapes available that introduce each section and work some of the problems from the exercise sets. A list of the tapes is given after the table of contents in this study guide. After each homework assignment, the number of the videotape and exercises is listed.

Omit all Calculator Corner readings.

1. Read R.1a, pp. 2–3

Note: In this lesson the word factor is used. Factors are numbers that are multiplied to give a result called the product.

For example,

3 × 5 × 1 × 7 = 105

The numbers 3, 5, 1 and 7 are factors. The result, which is 105 is the product. Sometimes we say “factor the number 12.” In this case we mean to write 12 as the product of two or more factors. For example,

12 = 2 × 6

12 = 1 × 12

12 = 1 × 2 × 3 × 2

12 = 3 × 4

12 = 2 × 2 × 3

All of these are different factorizations of 12.

A convenient notation to use if a factor appears more than once is called exponential notation. For example,

2 × 2 × 2 × 2 is written as 24 meaning that 2 is used as a factor 4 times. It is read as: “2 to the 4th power.”

On page 3 of the textbook, a “factor tree” is used to find the prime factorization of a composite number.

It is sometimes helpful to use another procedure for factoring larger numbers down to prime factors. One such procedure is to divide the given number by each of the primes in increasing order. Use the Table of Primes at the top left of page 4.

Example Factor 90 down to prime factors.

45

2 90

Solution Is 2 factor of 90? Yes!

So, 90 = 2 × 45.

But, 45 is not a prime number. Is 2 a factor of 45? No.

Is 3 (the next higher prime) a factor of 45? Yes!

15

3 45

Now, 15 is not prime, however, we know that 15 = 3 × 5, so we end up with

90 = 2 × 3 × 3 × 5 or 90 = 2 · 32 · 5

The · means exactly that same thing as 3.

Another way of showing this factorization is:

2 90

3 45

3 15

5

So, 90 = 2 · 32 · 5

Another example: Factor 525 down to prime factors.

3 525

5 175

5 35

7

So, 525 = 3 · 52 · 7

We say that 3 · 52 · 7 is the prime factorization of 525.

Show the prime factorization for each of these: (Answers are at the end of this lesson.) Do all of your work in this lesson without a calculator.

a. 42

b. 72

c. 168

d. 605

Homework: Set R.1, p. 6, 1–23 Odd Numbered Problems.

2. Read R.1b, pp. 4–5

3. There is another way to find the LCM of several numbers. If each of 2 or more numbers has a prime factorization written in exponential notation, then to find the LCM of the numbers we use each prime factor to the highest power it appears in any one factorization.

For example, consider the numbers 9 and 24. Their prime factorizations are:

9 = 32 24 = 23 · 3

The LCM is: 23 · 32 = 8 · 9 = 72

Examples

a. Find the LCM of 12 and 30.

12 = 22 · 3 30 = 2 · 3 · 5

LCM = 22 · 3 · 5 = 4 · 15 = 60

b. Find the LCM of 12 and 18.

12 = 23 · 3 18 = 2 · 32

LCM = 22 · 32 = 4 · 9 = 36

Homework: Set R.1, pp. 6–7, 25–55 Odd Numbered Problems.

4. Read R.2a, pp. 8–9

Note: Recall the rule for multiplying fractions.

a c a c

b d b d

That is, we multiply the numerators of the fractions to obtain the numerator of the product, and multiply the denominators of the fractions to obtain the denominator of the product.

Example

2 5 2 5 10

3 7 3 7 21

Also, from page 10, Identity Property of 1, any number multiplied by 1 has a product equal to the original number. That is, if a is any number,

a × 1 = 1 × a = a

Every number of arithmetic has many fractional numerals or fractional names. To find different names we can use the notion of multiplying by 1. That is, we can use any fractional name for 1 by using the property given on page 10, such as:

2 3 4etc.2 3 4,

Examples

a. Let’s arbitrarily choose 1515 to find another name for 2

3 .

2 2 15 30 30 21 so is another name for .

3 3 15 45 45 3

b. This time let’s use 33 to find another name for 2

3 .

2 2 3 6 6 21 so is another name for .

3 3 3 9 9 3


5. Read R.2c, pp. 11–12

Note: In multiplying fractions it is easier to simplify the product before the factors are multiplied out.

Example

Multiply and simplify

21 18 21 18 3 7 2 9

45 35 45 35 9 5 7 52 3 7 9 6 2 3 6

or 15 5 7 9 25 5 5 25


6. To rename a fraction so that we obtain a specific denominator, we must choose the appropriate name for 1.

Examples

a. Rename 23 to a fraction whose denominator is 21.

2 2 7 2 7 141

3 3 7 3 7 21

c. Rename 625 to a fraction whose denominator is 150.

6 6 6 6 6 361

25 25 6 25 6 150

The fraction 66 was chosen because 25 · 6 = 150.

Do These Problems: (Answers are at end of this lesson.)

a. 5 ?

7 6

3

b. 4 ?

9 4

5

c. 5 ?

12 60

7. If two fractions have the same denominators, then to add or subtract the fractions we simply add or subtract the numerators and keep the same denominator. The rules are:

and a b a b a b a b

c c c c c c

Examples

a. 2 3 2 3

7 7 7 7

5

b. 9 7 9 7

11 11 11 11

2

If the denominators are not the same, then the fractions need to be renamed so that the denominators are equal before they can be added or subtracted. But, how do we decide on which common denominator to use? One common denominator that can be used is the product of the two denominators. For example,

5 7?

9 24

A common denominator is 9 · 24 = 216, so we rename both fractions to fractions whose denominators are 216.

5 7 5 24 7 9 120 63 183

9 24 9 24 24 9 216 216 216

Now, simplify the result,

183 3 61 61

216 3 72 72

In the above example a smaller common denominator we could have used is 72. That is,

5 7 5 8 7 3 40 21 61

9 24 9 8 24 3 72 72 72

8. Read R.2c, pp. 13–14 (to just before Reciprocals)

Here are two more examples:

a. Add: 5 1

12 18

1

Solution

The LCM of 12 and 18 is 36, so rename to fractions whose denominators are 36.

5 3 11 2 15 22 37

12 3 18 2 36 36 36

b. Subtract: 11 7

12 30

Since the LCM of 12 and 30 is 60,

11 5 7 2 55 14 41

12 5 30 2 60 60 60

Homework: Set R.2b, pp. 16, 13–29 Odd Numbered Problems.

9. Read R.2c, pp. 14–15

Homework: Set R.2c, p. 16–17, 31–71 Odd Numbered Problems.

10. Read R.3, pp. 18–23

(Don’t forget to do margin exercises when instructed by the textbook!)

Homework: Set R.3, pp. 24-–25, 1–73 Every Other Odd Numbered Problem.

11. Read R.4, pp. 26–28

Homework: Set R.4, pp. 29–31, 1–77 Every Other Odd Numbered Problem.

12. Read R.5, pp. 32–34

Homework: Set R.5, pp. 35–36, 1–23, 27-77 Odd Numbered Problems.

13. Read R.6, pp. 37–42

Homework: Set R.6, pp. 44–46, 1–49 Odd Numbered Problems.

14. The sample test is Lesson 1: Sample Test of page 14 of this Study Guide. The answers are given at the end of this Study Guide. Be sure to time this test and allow at least this amount of time when you take the real test. Do this test without a calculator since you won’t be able to use a calculator for the real lesson test.

15. Show your homework and sample test to a staff member and ask for Lesson Test 1. If you use scratch paper number your problems neatly and staple your scratch paper to your tests before you turn it in. Be sure you are ready for this test since every test you take is averaged into your score for this lesson.

Answers for Procedure 1 Problems

a. 2 · 3 · 7

b. 23 · 32

c. 23 · 3 · 7

d. 5 · 112

Answers for Procedure 6 Problems

1. 45

2. 20

3. 25

Lesson 1: Sample Test

1. Find the prime factorization of 300.

2. Find the LCM of 15, 24, and 60.

3. Write an expression equivalent to 37 using 7

7 as a name for 1.

4. Write an equivalent expression with the denominator of 48 for 1116 .

Simplify for 5-8 without using a calculator. (Show all your steps.)

5. 16

24

6. 925

1525

7. 9 5

810

8. 10

27

8

3

9. Convert to fractional notation (do not simplify): 6.78.

10. Convert to decimal notation: 1895

1000

For 11−16, do not use a calculator and show all your steps.

11. Multiply: 123.6

3.52

12. 7.2 11.52

13. Add: 7.14 + 89 + 2.8787

14. Subtract: 1800 − 3.42

15. Convert to decimal notation: 23

11

16. Convert to percent notation: 11

25

17. Round 234.7284 to the nearest tenth.

18. Round 234.7284 to the nearest thousandth.

19. Convert to decimal notation: 0.7%.

20. Convert to fractional notation: 91%.

Evaluate without using a calculator and show all your steps.

21. 54

22. (1.2)2

23. 144 ÷ 9 · 2

24. 200 − 23 + 5 + 10

25. 5(6 7)

(3 1)9

5 4

3

Lesson 2: Introduction to the Real Numbers

In this lesson, you will be concerned with the notion of positive and negative. The number line is again useful here as a geometrical representation of all the real number. Historically, the concept of a negative number did not appear until hundreds of years after positive numbers were easily accepted.

Objectives

As a result of your work in this lesson you will be able to:

1. Evaluate algebraic expressions by substitution.

2. Translate phrases to algebraic expressions.

3. Graph real numbers on a number line.

4. Write a number in the form of a fraction, decimal, or percent.

5. Locate numbers on a number line and use the symbols of inequality.

6. Add and subtract real numbers.

Procedures


1. Read 1.1, pp. 56−59

Homework: Set 1.1, pp. 60−62, 1−67 Odd Numbered Problems.

2. Read 1.2abc, pp. 63−67


3. Read 1.2d, pp. 68−70

Homework: Set 1.2d, pp. 74−75, 29−63 Odd Numbered Problems.

4. Read 1.2e, pp. 70−71

Homework: Set 1.2e, p. 74, 65−79 Odd Numbered Problems.

5. Read 1.3a, pp. 75−77


6. Read 1.3bc, pp. 77−79


7. Read 1.4a, pp. 83−85

Note: The definition of subtraction, that is, a − b = a + (−b), is an extremely important one int hat it occurs consistently throughout algebra. Be sure to do many subtraction problems practicing this rule so that you can gain facility with it.

Homework: Set 1.4, pp. 86−88, 1-91 Odd Numbered Problems.

8. Read 1.4b, p. 85


9. The sample test is Lesson 2: Sample Test on page 18 of this Study Guide. The answers are given at the end of this Study Guide. Be sure to time this test and allow at least this amount of time when you take the real test. Do this test without a calculator, since you will not be able to use a calculator for the real lesson test.

10. Show your homework and sample test to a staff member and ask for Lesson Test 2. If you use scratch paper number your problems and staple your scratch paper to your test before you turn it in. Be sure you are ready for this test since every test you take is averaged into your score for this lesson.


1. Evaluate 3

2

x

y when x = 10 and y = 5.

2. Find the area of a triangle when the height h is 30 feet and the base b is 16 feet. Be sure to give the appropriate units.

3. Write an algebraic expression: Nine less than some number.

Use either < or > for ? to write a true sentence.

4. a. −4 ? 0

b. −3 ? −8

5. a. −0.78 ? −0.87

b. 1 1

?8 2

6. Write an inequality with the same meaning as x ≤ 22.

Find the absolute value.

7. a. |−7|

b. 9

4

Add and simplify without a calculator. Show all steps.

8. (−15) + 7

9. 5 1

9 9

10. 1 3

5 8

11. −8 + 4 + (−7) + 3

12. Find the opposite

−1.4

13. Find the additive inverse.

2

3

14. Find − x when x = −8.

Add or subtract and simplify without a calculator. Show all steps.

15. 3.1 − (−4.7)

16. 2 − (−8)

17. 3.2 − 5.7

18. 1 3

8 4

19. 6 + 7 − 4 − (−3)

20. June has $143 in her checking account. She writes a check for $25. Then she makes a deposit of $30. How much is now in her checking account.

Lesson 3: More on Real Numbers

This lesson covers multiplication and division of real numbers, the properties of real numbers, and simplifying expressions.

Objectives

When you have mastered this lesson you will be able to:

1. Multiply and divide real numbers.

2. Recognize division by zero as meaningless.

3. Name the reciprocal of any real number.

4. Use the distributive property to do simple factoring and multiplying.

5. Use the rules of order of operations to simplify algebraic expressions.

Procedures


1. Read 1.5, pp. 92–95

Homework: Set 1.5, pp. 96–98, 1–53, 59-91 Odd Numbered Problems.

2. Read 1.6abc, pp. 99–103

Homework: Set 1.6, pp. 105––106, 1–63, 71–79 Odd Numbered Problems.

3. Read 1.7ab, pp. 108–111

Homework: Set 1.7, p. 118, 1–27 Odd Numbered Problems.

4. Read 1.7cd, pp. 114–115

When factoring by using the distributive law it is often helpful to factor each term of the given expression first. The largest common factor should then be pulled out. For example,

12x + 18 = 3 · 4 · x + 3 · 6 = 3(4x + 6)

3(4x + 6) is in factored form, however, 3 is not the largest common factor. Actually, 6 is the largest common factor. So,

12x + 18 = 6 · 2 · x + 6 · 3 = 6(2x + 3)

is the correct factorization.

Examples

a. 16x + 8y + 32 = 8 · 2 · x + 8 · y + 8 · 4 = 8(2x + y + 4)

b. 15a + 5 + 10b = 5 · 3 · a + 5 · 1 + 5 · 2 · b = 5(3a + 1 + 2b)

c. 14xy + 21xy + 7x = 7 · 2 · x · y + 7 · 3 · x · y + 7 · x · 1 = 7x(2y + 3y + 1)

Homework: Set 1.7c, pp. 117–120, 37–91 Odd Numbered Problems.

5. Read 1.7e, p. 116

Homework: Set 1.7e, pp. 119–120, 97–125 Odd Numbered Problems.

6. Read 1.8ab, pp. 121–123


7. Read 1.8cd, pp. 123–125

Additional Examples Using Grouping Symbols

[5( 2) 3 ] {4[3( 2) 4( 2)] 3}

[5 10 3 ] {4[3 6 4 8] 3}

[2 10] {4[2 14] 3}

[2 10] { 4 56 3}

[2 10] { 4 59}

2 4 69

x x y y

x x y y

x y

x y

x y

x y

a.

2{ [2 7 3(2 4 )]}

2{ [2 7 6 12 ]}

2{ [14 13]}

2{ 14 13}

2{ 13 13} 26 26

x x x

x x x

x x

x x

x x

a.

Homework: Set 1.8cd, pp. 128–130, 27–87 Odd Numbered Problems.

8. Do the following as a sample test for Lesson 3:

The sample test is Lesson 3: Sample Test on page 22 of this Study Guide. The answers are given at the end of this Study Guide. Be sure to time this test and allow at least this amount of time when you take the real test. Do this test without a calculator, since you will not be able to use a calculator for the real lesson test three.

9. Show your homework and sample test to a staff member and ask for Lesson Test 3. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in. Be sure you are ready for this test since every test your take is averaged into your score for this lesson.


Multiple or divide and simplify without using a calculator.

1. 4 · (−12)

2. (−2)(−7)(5)(−4)

3. −45 ÷ 5

4. 1

2

·

3

8

5. 3 4

5 5

6. 4.864 ÷ (−0.5)

7. Find the reciprocal.

a. −2

b. 4

7

Multiply.

8. 3(6 − x)

9. −5(y − 1)

Factor completely.

10. 12 − 22x

11. 7x + 21 + 14y

Simplify without using a calculator and show all steps.

12. 5x − (3x − 7)

13. 4(2a − 3b) + a − 7

14. 23 − 10[4 − (−2 + 18)3]

15. − 2(16) − |2(− 8) − 53|

16. 256 ÷ (− 16) · 4

17. 4{3[5(y − 3) + 9] + 2(y + 8)}

Lesson 4: Solving Equations

In this lesson, you will learn to solve algebraic equations by using certain principles of equations. To solve an algebraic equation means to find all possible replacements for the variable that make the equation true. Many verbal problems can be solved by translating them into algebraic equations.

Objectives

When you have finished this lesson you will be able to:

1. Use the addition principle to solve equations.

2. Use the multiplication principle to solve equations.

3. Solve applied problems using the above principles.

Procedures


1. Read 2.1, pp. 140−143


2. Read 2.2, pp. 146−149


3. Read 2.3ab, pp. 152−154, up to “Clearing Fractions and Decimals.”


4. Read 2.3b, pp. 154−156 “Clearing Fractions and Decimals.”

Homework: Set 2.3c, p. 160-, 45−55 Odd Numbered Problems.

5. Read 2.3c, pp. 156−158

Homework: Set 2.3c, pp. 161−162, 59−87 Odd Numbered Problems.

6. Read 2.4, pp. 163−167

Homework: Set 2.4, pp. 167−170, 1,3,7,11−61 Odd Numbered Problems

7. Read 2.5, pp. 173−176

Homework: Set 2.5, pp. 177−180, 1−31, 37-43, 49-53 Odd Numbered Problems

8. Read 2.6, pp. 181−191

Homework: Set 2.6, pp. 193−197, 1−3, 7-23, 27-33, 37-47 Odd Numbered Problems.

.

.


The sample test is Lesson 4: Sample Test on page 25 of this Study Guide. The answers are given at the end of this Study Guide. For the rest of this course all word problems must be solved using algebra. No credit will be given for only the answers. You must label all the letters that you use giving correct units. Then translate the words into algebraic equations and show your solution to the equations. Be sure to time this test and allow at least this amount of time when you take the real test. Do this test without a calculator, since you will not be able to use a calculator for the real lesson test.

10. Show your homework and sample test to a staff member and ask for Lesson Test 4. If you use scratch paper, number your problems neatly and staple your scratch paper to your test before you turn it in. Be sure you are ready for this test since every test you take is averaged into your score for this lesson.


Solve:

1. x + 7 = 15

2. t − 9 = 17

3. 8 − y = 16

4. 2 3

5 4x

5. 3x = −18

6. 4

287

x

7. 1 3

2 5x

2

5

8. 3t + 7 = 2t − 5

9. 3(x + 2) = 27

10. 0.4p + 0.2 = 4.2p − 7.8 − 0.6p

11. −3x − 6(x − 4) = 9

12. 2 is what percent of 40? (Translate using algebra, then solve.)

13. Bill spent $224 for food one month. If this was 28% of his monthly salary, what was his monthly salary? (Translate using algebra, then solve.)

14. The perimeter of a rectangle is 36 cm. The length is 4 cm greater than the width. Find the width and the length. (Translate using algebra, then solve.)

15. If you triple a number and then subtract 14, you get two-thirds of the original number. What is the original number? (Translate using algebra, then solve.)

16. The sum of three consecutive odd integers is 249. Find the integers. (Translate using algebra, then solve.)

17. Money was invested in a savings account at 5% simple interest. After one year, there is $924 in the account. How much was originally invested? (Translate using algebra, then solve.)

18. Solve for r.

A = 2πrh

19. Solve for a.

13 (

2A h a c )

20. Solve for l.

2w + 2l = P

Lesson 5: Inequalities and Exponents In this lesson, you will learn to solve algebraic inequalities by using the addition and multiplication principles that apply to inequalities. You will graph an inequality in one variable on a number line. Also you will use the product, quotient, and power rules to simplify expressions involving exponents. Scientific notation will be defined and used to solve problems.

Objectives

When you have finished this lesson you will be able to:

1. Use the addition principle to solve inequalities.

2. Use the multiplication principle to solve inequalities.

3. Use both the addition and multiplication principles to solve inequalities.

4. Graph inequalities in one variable on the number line.

5. Use the product, quotient, and power rules to simplify expressions involving exponents.

6. Convert between scientific and decimal notation.

7. Solve problems with scientific notation.

Procedures


1. Read 2.7abc, pp. 198−201


2. Read 2.7de, pp. 201−205

Homework: Set 2.7, pp. 204−206, 33−79, 85−95 Odd Numbered Problems.

3. Read 2.8, pp. 210−212

Homework: Set 2.8, pp. 213−216, 1−21, 25−39, 47−57 Odd Problems.

4. Read 4.1abc, pp. 308−310

Homework: Set 4.1, p. 314, 1−39 Odd Numbered Problems.

5. Read 4.1def, pp. 310−313

Homework: Set 4.1, p. 314−315, 43−119 Odd Numbered Problems.

6. Read 4.2ab, pp. 318−320


7. Read 4.2c, pp. 320−321

Homework: Set 4.2c, p. 326, 53−73 Odd Numbered Problems.

8. Read 4.2d, pp. 322−323

Homework: Set 4.2d, pp. 326−327, 75−85 Odd Numbered Problems.

9. Read 4.2e, pp. 323−324

Homework: Set 4.2e, pp. 327−328, 87−93 Odd Numbered Problems.

10. Some of the answers in the text are not given in simplified form so do the worksheet on the next page to see the proper way to give your answers. When the instruction is to “simplify,” the answer cannot have any negative exponents or powers that have not been expanded. For example, x−2 must be written as

3, 22

1

x must be expended to be written as 8, and (3x)4 must first be changed to

34x4, and then to 81x4. 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.


The sample test is Lesson 5: Sample Test on page 33 of this Study Guide. The answers are given at the end of this Study Guide. Be sure to time this test and allow at least this amount of time when you take the real test. Do this test without a calculator, since you will not be able to use a calculator for the real lesson test.

12. Show your homework and sample test to a staff member and ask for Lesson Test 5. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in. Be sure you are ready for this test since every test you take is averaged into your score for this lesson.

Worksheet on Operations with Exponentials

Rules

1. xm · xn 5m1n (To multiply exponentials with the same base, keep the same base and add the exponents.)

2. 2m

mn

x nxx

(To divide exponentials with the same base, keep the same base

and subtract the exponents.)

3. (xm)n = xmn

4 1nx

nx

Examples Simplify:

2 3 4 2 4 3 2 4 3 1

42 4

2

( )( ) ( 1)( )( ) ( 1)X Y X Y X X Y Y X Y

YX Y

X

1.

2 2 7 2 47 3 2 8 4 6

3 8 3 8 6

36 36 44 4

9 9

X Y X Y XX Y X Y

X Y X Y Y

2.

2 4 3 1( 3) 2( 3) 4( 3) 3 6 12

12 12

3 6 6

( 2 ) ( 2) ( 2)

( 2) 8

x y x y

y y

x x

3.

x y

Recall that 3 1 1 1 1( 2) ( 2)( 2)( 2) 28 and

8 8 1 8

82 3 4 1(4) 2(4) 3(4) 4 8 12

12

81( 3 ) ( 3) ( 3)

xx y x y x y

y 4.

Remember (−3)4 = (−3)(−3)(−3)(−3) = (9)(9) = 81,

but −34 = −(3)4 = − (3)(3)(3)(3) = −(9)(9) = −81.

Simplify These: (Answers can be found on page 32.)

1. (X2Y3)4 =

2. 33

4

X

Y

3 7 2

5 3

9

3

X Y

X Y

4 5

7

6

8

X

X

5. −2(X2Y)4 =

6. (−2S2 T4)4 =

7. 43

2

MN

P Q

8. 3 3

2 4

( 4 )

(2 )

U V

UV

9. 6(XY3)5 =

10. (2X3 Y2)2 · (3XY4)3 =

11. (3x3y−4)−2 =

Answers to Worksheet on Exponentials

1. X8Y12

2.9

12

X

Y

3. 23X

Y

4. 2

3

4X

5. −2X8Y4

6. 16S8T16

7. 4 1

8 4

2M N

P O

8. 5

5

4U

V

9. 6X5Y15

10. 108X9Y16

11. 8

69

y

x

Lesson 5: Sample Test Graph on a number line.

1. y ≤ 9

2. −2 ≤ x < 2

3. 6x − 3 < x + 2

Solve and write set notation for the answer.

4. x + 6 ≤ 2

5. 12x ≤ 60

6. − 2y ≥ 16

7. −4x ≤ −32

8. −5x ≥ 14

9. 4 − 6x > 40

10. 14x + 9 > 13x − 4

11. 5 − 9x ≥ 19 + 5x

12. Find all numbers such that six times the number is greater than the number plus 30. (Translate to an inequality using algebra, then solve.)

13. The width of a rectangle is 96 yards. Find all possible lengths so that the perimeter of the rectangle will be at least 540 yards. (Translate to an inequality using algebra, then solve.)

Multiply, but do not simplify:

14. 6−2 · 6−3

5. x6 · x2 · x

16. (4a)3(4a)8

Divide, but do not simplify:

17.

5

2

3

3

18.

3

8

x x

19.

5

5

)

)

(2

(2

x

x

20. Express using a positive exponent: 5−3.

21. Express using a negative exponent: 8

1 y

Simplify the following:

22. (x3)2

23. (−3y2)3

24. (2a3b2)4

25. 3

a

c 2

26. (3x2)3 (−2x5)3

27. 3(x2)3 (−2x5)3

28. (4x−3y6)−3

29. Convert to scientific notation: 3,900,000,000.

30. Convert to decimal notation: 5 × 10−8.

Multiply or divide and write scientific notation for the answer.

31. 6

11

10

0

5.6

3.2 1

32. (2.4 × 105)(5.4 × 1016)

Lesson 6: Linear Equations and Linear Graphs

In Lesson 6 we are going to use the Cartesian Coordinate System (named after the French mathematician René Descartes, 1596−1650) to graph ordered pairs of numbers on a plane. The point (2, 4) is an ordered pair of numbers since it makes a difference which number comes first. In other words, the pair (2, 4) is different from the pair (4, 2).

Objectives


1. Plot an ordered pair of numbers (coordinates) and indicate in which quadrant it lies, if any.

2. Recognize a linear equation.

3. Determine if a given ordered pair is a solution of a given system of equations.

4. Solve for one of two variables in a given linear equation.

5. Graph a linear equation.

6. Graph an equation of the form y = mx + b by using the slope and y-intercept.

7. Find the x- and y-intercepts of a given linear equation.

8. Find the equation of a line given two points on the line.

9. Find the equation of a line given one point and the slope.

Procedures


1. Read 3.1abc, pp. 228−231

Homework: Set 3.1a, pp. 239−240, 1−19 Odd Numbered Problems.

2. Read 3.1de, pp. 232−237

Note: A linear equation is one that can be rewritten in the form ax + by = c where a, b, and c are real numbers. No term of a linear equation can contain the product or quotient of two variables. Also, a variable cannot appear in the denominator of any term.

These are examples of linear equations:

3x − 2y = 4

y = 7x + 6

28

3 5

x y

7 1

4 2y x

These are examples of equations that are not linear:

x + 2y2 = 5

13y

x

3 + 2xy − 5y = 0

2 8x

xy


3. Read 3.2 pp. 245−250


4. Read 3.3, pp. 257−262


5. Read 3.4, pp. 267−270


6. Read 3.5, pp. 275−277


7. Read 3.6, pp. 281−283


8. Read 3.7, pp. 286−290


9. Read 6.9ab, pp. 552−555


10. Read 6.9cd, pp. 555−557


11. The sample test is Lesson 6: Sample Test on page 37 of this Study Guide. The answers are given at the end of this Study Guide. Be sure to time this test and allow at least this amount of time when you take the real test. Do this test without a calculator, since you will not be able to use a calculator for the real lesson test.



1. In which quadrant is (−2, −7)?

2. Determine whether (−2, −1) is a solution of 2x − 3y = −7. Graph each of the following:

3. 5

4y x

4. 3y − 6 = 9x

5. x = − 2

6. Find the slope of the line containing (−7, −6) and (−4, −2).

7. Find the slope, if it exists, of the line x = −2.

8. Find the slope and the y-intercept of −2x − 8y = 16.

9. Find an equation of the line that passes through (4, −2) with m = 6.

10. Find an equation of the line containing (2, − 1) and (−2, −4).

11. Determine whether the graphs of the equations are parallel, perpendicular, or neither. 2x − 5y = −3 and 2y + 5x = 6.

12. Find an equation of variation where y varies directly as x and y = 0.8 when x = 0.5.

13. Find an equation of variation where y varies inversely as x and y = 2 when x = 45.

14. The number of servings S varies directly as the weight W. From 9 kg, one can get 70 servings. How many servings can one get from 12 kg? (Translate using algebra, then solve.)

15. It takes 4 hours for 9 workers to complete a certain job. The number of hours varies inversely as the number of workers. How long would it take 8 workers to do the job? (Translate using algebra, then solve.)

16. Is (−2, 3) a solution of the inequality 4y − 5x < −4?

17. Graph 5x + 4y ≥ 20.

Lesson 7: Systems of Equations

In this lesson, we are going to find the solution that a system of linear equations might have. This is the point of intersection of the two graphs. Applied problems will be solved using systems of equations.

Objectives


1. Determine if an ordered pair is a solution of a system of equations.

2. Solve a linear system of equations in two variables by graphing.

3. Solve a linear system of equations in two variables by the elimination method (also known as the addition method).

4. Solve applied problems by translating them to a system of linear equations in two variables and then solve the system.

Procedures


1. Read 7.1, pp. 574–578

Homework: Set 7.1, pp. 579–580, 1–25 Odd Numbered Problems.

2. Read 7.2ab, pp. 581–583


3. Read 7.2c, pp. 583–584


4. Read 7.3, pp. 588 – 593


5. Read 7.4, pp. 598–604

Homework: Set 7.4, pp. 605–609, 1–19, 23–37 Odd Numbered Problems.

6. Read 7.5, pp. 611–615




8. 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. Be sure you are ready for this test since every test you take is averaged into your score for this lesson.


1. Determine whether (− 1, − 5) is a solution of the system of equations: y − 4x= − 1, and 2x = − y − 7.

2. Solve by graphing: x + y = 8 and 2x − y = 1.

Solve using the substitution method:

3. x = 1 − y

2x + y = 4

4. y = 2x − 5

3y − x = 5

5. x + 2y = 10

3x + 4y = 8

Solve using the elimination method, i.e., the addition method:

6. 5x − y = 5

3x + y = 11

1 18

2 32 1

53 2

x y

x y

7.

8. 3x + 4y = − 5

7x + 3y = 1

9. 3x − 4y = 16

5x + 6y = 14

10. The sum of two numbers is 26. One number is 12 more than the other. Find the numbers. (Translate to a system of equations using algebra, then solve.)

11. A chemist has one solution that is 50% alkaline and a second that is 30% alkaline. How many liters of each mixed together to get 200 liters that is 42% alkaline? (Translate to a system of equations using algebra, then solve.)

12. A boat took 3 hours to travel 18 miles against the current. The return trip with the same current took 1

21 hours. Find the speed of the boat in still water. (Translate to a

system of equations using algebra, then solve.)

Lesson 8: Final Exam for Math 101X Congratulations, you have completed all your lesson tests for the first module Math 101X of the Elementary 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 seven lessons of this study guide.

Objectives

When you have completed the final exam, you will be able to:

1. Say you have mastered the material in Lessons 1−7.

2. Receive a grade in Math 101X.

Procedures

1. 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.

2. Redo your sample tests from the first seven lessons.

3. Do the following as a sample test for the final exam:

The sample test is Lesson 8: Sample Final for Math 101X given on page 43 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 3 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 101X. If you use scratch paper, number your problems neatly 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 35% 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.

Lesson 8: Sample Final for Math 101X

1. Find the prime factorization of 300.

2. Find the LCM for 15, 24, and 60.

3. Simplify: −102 ÷ 5 · 2.

4. Evaluate 4 2

14

x y when x = −2 and y = −3.

5. Translate to an algebraic expression: A number less than ten.

6. Simplify: −9 − (−2).

7. Simplify: 7 1

3 2 .

8. Simplify: 3 4

5 5

9. Simplify: − 2(4x − 5)

10. Factor: 6x − 6.

11. Simplify: 5{[6(x − 1) + 7] − [3(3x − 4) + 8]}

12. Solve: 3x − 2 = −38.

13. Solve: 3 2

5 2

x

5

14. Solve: − 3x − 6(x − 4) = 9

15. Solve for x: 19 − (2x + 3) = 2(x + 3) + x

16. Solve and graph: Use set notation. 5 − 9x ≥ 19 + 5x.

17. The sum of two consecutive even integers is 106. Find the integers. (Translate using algebra, then solve.)

18. The perimeter of a rectangle is 36 cm. The length is 4 cm greater than the width. Find the dimensions. (Translate using algebra, then solve.)

19. Money is invested in a savings account at 5% simple interest. After one year, there is $924 in the account. How much was originally invested? (Translate using algebra, then solve.)

20. Solve: 5

45

Ay

x for x.

21. Find, in set notation, all numbers such that three times the number is less than six more than the number. (Translate using algebra, then solve.)

22. Simplify: x−7 x−6

23. Simplify: 8

3

x

x

24. Simplify: (2a−4b7)−3

25. Multiply and write scientific notation for the answer: (7.1 × 10−7)(8.6 × 10−5).

26. Find an equation of the line containing the point (−2, −5) and having a slope of 2

3 .

27. Find an equation of the line containing the points (−2, −4) and (2, − 1).

28. Graph the equation 4x + 3y = 12.

29. It takes 5 hours for 2 washing machines to wash a fixed amount. How long would it take 10 washing machines? The number of hours, H, varies inversely as the number of washing machines, W. (Translate using algebra, then solve.)

30. Solve by graphing: x + y = 8

2x − y = 1

31. Solve this system: 5x + 5y = 30

(Use substitution) y = −2x + 3

32. Solve this system: 2x + 3y = 8

(Use elimination) 5x + 2y = −2

33. Solution A is 30% acid and solution B is 60% acid. How many liters of each should be mixed to make 80 liters of 45% acid? (Translate to a system of equations, then solve.)

Lesson 9: Operations with Polynomials and Special Products

In this lesson, you will learn to identify a polynomial and will do arithmetic with these expressions. The concepts you learned in Lessons 2, 3, and 5, such as inverses and arithmetic of real numbers, will form part of the basis for this lesson. You will also learn some “special products” of polynomials that will be very helpful in Lesson 10, Factoring Polynomials.

Objectives

As a result of your work on this lesson, you will be able to:

1. Recognize a polynomial, find its degree and name its coefficients and terms.

2. Evaluate a polynomial for a given value of its variable.

3. Add and subtract polynomials.

4. Multiple and divide polynomials.

5. Recognize special product forms and find their products.

6. Do all of the above with polynomials of more than one variable.

Procedures

Omit all Calculator readings.

1. Read 4.3abc, pp. 330−333


2. Read 4.3defghi, pp. 333−337

Note: On page 336 in Example 24, the polynomial 8x5 − 2x3 + 5x2 + 7x + 8 is given, and it is pointed out that the x4 term is missing. The last term, 8, is the x0 term (since x0 = 1) or the constant term. For example, which terms are missing in the polynomial 2x5 + x4 − 4x? The missing terms are the x3, x2, and x0 terms. Another way of stating this is to say that the 3rd degree, 2nd degree, and constant terms are missing. The polynomial could be rewritten as:

2x5 + x4 + 0 · x3+ 0 · x2− 4 x + 0 · x0


3. Read 4.4a, p. 343


4. Read 4.4bcd pp. 344−346


5. Read 4.5abc, pp. 353−355

Homework: Set 4.5, p. 357, 1− 45, 51, 53 Odd Numbered Problems.

6. Read 4.5d, pp. 355− 356

Homework: Set 4.5d, pp. 358, 61−75, 87, 89 Odd Numbered Problems.

7. Read 4.6a, pp. 360−361


8. Read 4.6bcd, pp. 361−366


9. Read 4.7, pp. 371−374

Homework: Set 4.7, pp. 375−379, 1−11, 17−39 Odd Numbered Problems, 41−81 Every Other Odd Numbered Problem.

10. Read 4.8, pp. 380−383



The sample test is Lesson 10: Sample Test on page 57 of this Study Guide. The answers are given at the end of this Study Guide. Be sure to time this test and allow at least this amount of time when you take the real test. Do this test

without a calculator, since you will not be able to use a calculator for the real lesson test.



1. Give the coefficients of each term and the degree of the polynomial.

x4 + 2x3 − 17 x2 + 4x − 1

2. Evaluate the polynomial xy − x2 + y2 for x = − 2 and y = 5.

3. Add: (x2 − 2x + 1) + (3x − 5) + (2x2 + 5x + 3).

4. Subtract (4x − y − 6z3) from (3x + 2y − 14z3).

5. Subtract: (6x4 − 5x3 + 3x − 2) − (7x5 − x3 + 4x − 6).

Multiply and simplify:

6. −3x3(5x3 − 3x2 − 2)

7. (2x + 3)(x − 7)

8. (5x − 1)(7x2 − x − 9)

9. (9x − 7)(9x + 7)

10. (5x6 + 8y4)(5x6 − 8y4)

11. (6x − 11)2

12. 2

12

3x

13. Divide:

14. Divide: (Give the answer in mixed form.) (4x2 − 105x + 27) ÷ (x − 5)

15. Evaluate the polynomial A = P(1 + i)2 to find the amount to which $3500 will grow at 5% interest for two years.

Lesson 10: Factoring Polynomials

Factoring is the rewriting of a polynomial as a product of lower degree polynomials. Factoring is an essential skill for much of your work in algebra; therefore, be sure to do many problems in order to gain proficiency.

Objectives

When you have finished this lesson, you will be able to:

1. Factor a polynomial into the product of lower degree polynomials.

2. Factor polynomials of several variables.

3. Solve quadratic equations by factoring.

4. Solve applied problems involving quadratic equations that can be solved by factoring.

Procedures


1. Read 5.1ab, pp. 398–402


2. Read 5.1c, pp. 403–404

Homework: Set 5.1c, pp. 405–406, 35–65 Odd Numbered Problems.

3. Read 5.2, pp. 407–412


4. Read 5.3, pp. 417–421


5. Read 5.4, pp. 425–426

Homework: Set 5.4, pp. 427–430, 1–99 Every Other Odd Number Problem.

6. Read 5.5ab, pp. 433–435

Homework: Set 5.5, pp. 439–440, 1–41 Odd Numbered Problems

7. Read 5.5cd, pp. 435–438

Homework: Set 5.5cd, pp. 440–442, 45–91 Odd Numbered Problems.

8. Read 5.6, pp. 443–446


9. Read 5.7, pp. 451–455


10. Read 5.8, pp. 460–466

Homework: Set 5.8, pp. 467–471, 1–27, 33, 35 Odd Numbered Problems.





In Problems 1–15, factor completely:

1. 10x − 20

2. x2 − 5x

3. 13w2 − 52 + 26w

4. x2 − 8x + 15

5. x2 + 36 − 12x

6. 56x − 96 + 20x2

7. a2 − 3a + ay − 3y

8. 9x2 − 4y2

9. x2 − x − 20

10. 6x3 + 15x2 + 9x

11. x2 y6 + 4xy5 − 32y4

12. 2x2 − 50

13. x4 + 2x3 − 5x − 10

14. 6x3 + 9x2 − 15x

15. 16x4 − 1

Solve each of the following in Problems 16 − 18.

16. (x − 9)(2x + 3) = 0

17. 3x2 − 7x = 20

18. x(x − 2) = 35

For Problems 19–20, translate to an algebraic equation, then solve.

19. The product of two consecutive even integers is 168. Find the integers.

20. The height of a triangle is 3 centimeters greater than the base. The area is 44 square centimeters. Find the height and base.

Lesson 11: Fractional Expressions and Equations

In Lessons 9 and 10 you added, subtracted, multiplied, divided, and even factored polynomials. So, what now? Well, in the next two lessons you will review fractions with numerical numerators and denominators. You will also learn how to operate with fractions whose numerators and denominators are polynomials—that is, fractional expressions.

Objectives

When you have mastered Lesson 11, you will be able to:

1. Simplify or reduce a fractional expression to lowest terms.

2. Multiply fractional expressions.

3. Add and subtract fractional expressions when the denominators are the same.

4. Find the least common multiple (LCM) of a given group of polynomials.

Procedures


1. Read 6.1abc, pp. 484−488. Also, go over the following additional examples.

Example 1

1x y

y x

If you are wondering how the result was obtained, consider the following explanation. It is always possible to factor any number or expression so that one of the factors is −1.

For example,

2 2 2

6 ( 1)( 6)

2 ( 1)(2)

( 1)( )

( 1)( )

( 1)[ ( )] ( 1)( )

( 1)[ ( )] ( 1)( ) ( 1)( )

3 2 ( 1)[ ( 3 2)] ( 1)( 3 2)

x x

a a

x y x y x y

x y x y x y y

x x x x x x

x

You can check each of the above equations by multiplying the right side. Notice that if an expression is written as a product with 21 as one factor, then the other factor is just the additive inverse of the original expression. Now, to show how the result of Example 1 was obtained:

( 1)( ) ( 1)( ) ( 1) ( )

(1)( ) (1) ( )

( 1)(1) 1

x y y x y x y x

y x y x y x y

x

Example 2

( )( ) ( ) ( )( 1)( )

( ) 1

x y a b x y a ba b

y x y x

a b b a

Now, here is another interesting result:

−3 is the additive inverse of 3, but if you square both of these numbers, the results are equal. That is,

(−3)2 = (3)2

Also, (a − b) is the additive inverse of (b − a), but,

(a − b)2 = (b − a)2

So, this fact can be used to reduce certain fractions. For example,

2 2( ) ( ) ( )( ) ( ) ( )

( ) ( ) 1 ( ) 1 ( )

( )(1)

x y y x y x y x y x y x

y x y x y x y

y x y x

x

Note that x y

x y

cannot be reduced since (x + y) is not the additive inverse of (x − y). See

the steps below:

( 1)( )

( )

x y x

x y x y

y

and since (− x − y) is not equal to (x − y), the fraction cannot be

reduced.


2. Read 6.1d, pp. 489


3. Read 6.2, pp. 494−496


4. Read 6.3, pp. 488−489

Note: Recall from Lesson 1 that if exponential notation is used then the LCM rule is:

To find the LCM, we use each factor to the highest power it appears in any one factorization.

For example, 24 = 23 · 3 and 36 = 22 · 32 so the LCM = 23 · 32 or 72.

Another example:

24x3y2z = 23 · 3 · x3 · y2 · z

90x2yz2 = 2 · 32 · 5 · x2 · y · z2

LCM = 23 · 32 · 5 · x3 · y2 · z2 or 360x3y2z2


6. Read 6.5ab, pp. 511−514

Homework: Set 6.5ab, pp. 515−517, 1−35, 41−53 Odd Numbered Problems.





1. Simplify:

8 24

6 2

a

a

2. Simplify:

2

2

3 4

2 10

x x

8x x

3. Multiply. Simplify, if possible.

2

2

24 3 6

23 12 12

a a

aa a

4. Add. Simplify, if possible.

3 2 5

3 4 4 3

x x

x x

5. Divide. Simplify, if possible.

2 1 1

1

x x

x x

6. Divide. Simplify, if possible.

2

2

4 5 1

39

x x x

xx

7. Subtract. Simplify, if possible.

2

2 4 2 4

x x

x x


2 2

1 2 5

4 4

x x

x x


2 25

5 5

x

x x


2

2 2

2

1 1

x x

x x

11. Find the LCM of 6, 9, 21.

12. Find the LCM of 12xy2, 15x3y, 9y4.

13. Find the LCM of x2 − y2, 2x + 2y, x2 + 2xy + y2.

14. Perform the indicated operations. Simplify the result.

2 2 3 3x y x y x y

x y y x x y

15. Perform the indicated operations. Simplify the result.

2 4 6 12 3 1

6 6 2

x x x

x x x

5

2

Lesson 12: More Fractional Expressions and Equations In this lesson, you will continue to use LCMs. You will also use all the previous techniques for simplifying expressions in order to solve fractional equations. Finally, you will solve applied problems that involve fractional expressions.

Objectives

When you have mastered this lesson you will be able to:

1. Add and subtract fractional expressions with different denominators.

2. Solve and check fractional equations.

3. Solve applied problems that involve fractional expressions.

Procedures


1. 1. Read 6.4, pp. 503−506

Note: In this example, the denominator of the 2nd fraction is the LCM of the 2 denominators, so the 2nd fraction does not have to be renamed.

2

3 2

2 3 103 2

LCM ( 5)( 2)2 ( 5)( 2)

3 5 2

2 5 ( 5)( 2)

3 15 2

( 2)( 5) ( 2)( 5)

3 15 2 4 17

( 2)( 5) ( 2)( 5)

x

x x xx

x xx x x

x x

x x x x

x x

x x x x

x x x

x x x x


2. Read 6.5b, pp. 514

Homework: Set 6.5b, pp. 517, 41−53 Odd Numbered Problems.

3. Read 6.6, pp. 521−525

Here is an extra example:

Solve for x: 5 1 3

3 6x x

4 LCM = 6(x − 3)(x − 4)

Multiply both sides of the equation by the LCM.

5 16( 3)( 4) 6( 3)( 4)

3 63

6( 3)( 4)4

x x x xx

x xx

2

2

2

2

30( 3)( 4) 6( 3)( 4) 18( 3)( 4)

( 3) 6 ( 4)

30( 4) ( 3)( 4) 18( 3)

30 120 ( 7 12) 18 54

30 120 7 12 18 54

2 19 78 0

19 78 0 (By multiplying the equation by 1.)

( 6)( 13) 0

x x x x x x

x x

x x x x

x x x x

x x x x

x x

x x

x x

x

6 0 or 13 0

6 or 13

x

x x

Check for x = 6: Check for x = 13(6):

5 1

66 3

3

13 4

5 1

13 3 6

3

13 4

5 1

3 6

3

2

5 1

10 6

3

9

10

1

6 6

1 1

2 6

1

3

9

6

3 1

6 6

3

2

2

6

1

3


4. Some motion problems can be solved by writing a fractional equation and

The verbal problems given in this section deal with uniform motion, which means that an object m nt rate of speed. The

Distance = (Rate) × (Time) or d = rt

at 60 mph. Write an expression for each of d.

Solution: d = rt = (60)(3) = 180 miles

ar to travel 150 m

finding its solution. We will now develop some skills that you will need in order to be able to extract the appropriate equation from the given information in a verbal problem.

oves along a straight line at a constabasic formula is:

where d is the distance traveled, r is the constant rate of speed and t is the time of travel.

Example Suppose a car is traveling these. You may use a variable if neede

c. The distance traveled in 3 hours.

d. The time it takes the c iles.

Solution: (d = rt), so t = 150 1

2d h

60 2rours

− x) mph

d of

e. The rate of a car going x mph slower.

Solution: (60

Now, Do These Problems. (Use same car as above.) (Answers are at the enthis lesson.)

f. The distance traveled in 2 hours =_______________________________

g. The distance traveled in 5.5 hours = _____________________________

h. The distance traveled in x hours = ______________________________

i. The distance traveled in 2 less hours than t hours = _________________

j. The rate of a car going 15 mph slower = __________________________

k. The rate of a car going 20 mph faster = __________________________

l. The rate of a car going x mph faster =____________________________

m. The rate of a car going twice as fast =___________________________

n. The time it takes the car to travel 90 miles = ______________________

o. The time it takes the car to travel 240 miles =______________________

p. The time it takes the car to travel x miles =________________________

to travel 300 miles =________

________

___

iles. How many hours does the trip take if

q. The time it takes a car going x mph faster

Other Practice Problems for You

Translate these into algebraic expressions:

1. The distance a car travels in 2 hours at 35 mph = ___________________

2. The distance a car travels in t hours at 35 mph = ___________________

3. The distance a car travels in (t + 2) hours at 35 mph = _______________

4. The distance a car travels in 3 hours at r mph = ____________________

5. The distance a car travels in 3 hours at (r − 15) mph = _______

6. The distance a car travels in 6 hours at (2r) mph = _______________

7. A boat travels at 20 mph. What is the rate of a boat traveling:

a. twice as fast?____________________________________________

b. 15 mph faster?___________________________________________

c. 3 mph slower?___________________________________________

d. r mph faster?____________________________________________

e. x times as fast?__________________________________________

8. A train travels a distance of 300 mthe train travels at a rate of:

a. 50 mph? __________________

b. 20 mph? __________________

_____________

iles. How fast is the car traveling if the time

____________

_____

pp

Here Are Thr

Example 1 and the other south. The rate of the northbound car is 20 mph faster than the speed of the southbound car. If they are 440

art at the end of 4 hours, find the speed of each car.

c. 100 mph?_________________

d. r mph?___________________

e. (2r) mph? ________________

f. (r + 10) mph?_____________

g. (r − 15) mph?

9. A car travels a distance of 200 mfor the trip is:

a. 4 hours?__________________

b. 10 hours?_________________

c. 6 hours?__________________

d. t hours?__________________

e. (t + 1) hours?_

f. (2t) hours?___________

g. (t − 2) hours?

5. Read 6.7, . 530−538

ee More Examples

At 2 P.M., two cars leave Novato. One car is traveling north

miles ap

Solution

1. First draw a sketch.

2. Organize the given information in chart form.

3. Write an equation using the chosen variable.

4. Solve the equation.

Let r be the rate in sou nd n r + 20 is the rate in mph of th northbound car. trav ou

mph of theEach car

thbouels 4 h

car, thers. e

D = R · T

Distance Rate Time

( miles) (m )ph (hours)

Northbound (r + 20) · 4 r + 20 4

Southbound r · 4 r 4

Since the total distance traveled by the two cars is 440 miles, the equation is:

· 4 = 440

=

r + 20 = 45 + 2

Example 2 rned several days later, a detour increased her

peed returning was 36 mph, how far did she travel going to the construction site?

olution d = dista ce in miles going to the construction site.

(r + 20) · 4 + r

4r + 80 + 4r = 440

8r + 80 = 440

8r = 360 or r 45 mph southbound car

0 = 65 mph northbound car

An engineer drove to a construction site at an average speed of 48 mph. When she retutrip by 8 miles and increased her time by 1 hour. If her s

S Let n

DT

R

Distance R ate Time (miles) (mph) (hours)

48

d d Going 48

Returning d + 8 36 8

36

d

Since the return trip took 1 hour longer, the time returning is 1 hour more than the time going. The equation is:

8+1 LCM = 144, so multiply both sides of the equation by 144.

36 48+8

144 144 144 136 48

4( 8) 3 144

4 32 3 144

32 144d 122 miles

d d

d d

d d

d d

d

Example 3 Two hours later, a car starts from the same depot and travels at

olution Let t = the number of hours the truck travels. Then, t − 2 = the

(distance to the construction site)

At 8 A.M., a truck leaves a depot and travels north at 35 mph.

55 mph until it overtakes the truck. At what clock time does thecar overtake the truck?

Snumber of hours the car travels.

D = R · T

Distance Rate T e im(miles) (mph) (hours)

Truck 35 · d 35 t

Car 55 · (t − 2) 55 t − 2

Since the distance traveled by the truck is the same as the distance traveled by

)

35t = 55t − 110

−20t = −110

the car, the equation is:

35t = 55(t − 2

15 hourst

2

Therefore, the car overtakes the truck at 1:30 P.M.

Homework: Set 6.7, pp. 539−545, 1−21, 25, 27, 33, 43−51, 59, 61 Odd

The

since you will not be able to use a calculator for the real

ore for this lesson.

A mework Problems in This Lesson:

Numbered Problems.


The sample test is Lesson 12: Sample Test on page 72 of this Study Guide.answers are given at the end of this Study Guide. Be sure to time this test and allow at least this amount of time when you take the real test. Do this test without a calculator, lesson test.

7. Show your homework and sample test to a staff member and ask for LessonTest 12. If you use scratch paper, number your problems and staple your scratch paper to your test before you turn it in. Be sure you are ready for this test since every test you take is averaged into your sc

nswers to Ho

f. 120 miles g. 330 miles h. 60x miles

i. 60(t − 2) miles j. 45 mph k. 80 mph

1. (60 1 x) mph m. 120 mph n. 1.5 hours

o. 4 hours hours60

p. x 300

hours60 x

q.

1. 70 miles 2. 35t miles 3. 35(t + 2) miles

4. 3r miles 5. 3(r + 15) mi 6. 12r miles

7a. 40 mph 7b. 35 mph 7c. 17 mph

7d. (20 + r) mph 7e. 20x mph 8a. 6 hours

8b. 15 hours 8c. 3 hours 300hours

r8d.

150hours8e.

r

300hours

10r 8f.

300hours

15r 8g.

9a. 50 mph 9b. 20 mph 133

39c.

200mph

1t 9d.

200 100mph

2t t9f.

200mph

t9d.

200mp

2t 9g. h



3 2

1 3x x


2

2

2 4

3 9

a a

a a


2 1

3x


1 2

4

x x 3

x x

5. Perform the indicated operations. Simplify, if possible.

2 2

2 1b

b c b c b

1

c

6. Perform the indicated operations. Simplify, if possible.

2 2

5

11 30 9 20

x

x x x x

7. Solve tor a: 2 3

3 8

a

a

8. The instructions for mixing a spray for roses read:

Mix 3 tablespoons of liquid fungicide to each gallon of water.

How many tablespoons should be mixed with 1

24

gallons of water? (Translate

using algebra, then solve.)

9. Solve for x: 1 5

2 8

3

x x

10. Solve tor x: 10

xx

3

11. Solve for x: 2

42

1

2 2

x x

x x x

12. It takes Dave 6 hours to paint a room. It takes Sierra and Dave 4 hours to paint the same room working together. How long would it take Sierra to paint the room alone? (Translate using algebra, then solve.)

13. Jim took a trip to Paris, Texas at a rate of 48 mph. The return trip, due to a detour, was 8 miles longer and took one extra hour. If the return rate was 36 mph, find the distance going to Paris. (Translate using algebra, then solve.)

Lesson 13: Radicals

In this lesson, you will learn about radicals. The name, radical, is from a Latin word, radicis, which means root. So, radicals are often called roots. They are

symbolized by . You will also learn an important rule about right triangles, called the Pythagorean Theorem. It was proved by a Greek philosopher named Pythagoras who lived in southern Italy (approximately 520 B.C.)

Objectives

As a result of your work in this lesson, you will be able to:

1. Add subtract, multiply and divide with radicals.

2. Factor and simplify radical expressions.

3. Use the Pythagorean Theorem to find the lengths of sides of right triangles.

4. Solve equations with radicals.

Procedures

Omit all Calculator Corner readings..

1. Read 8.1, pp. 628−632


2. Work the following examples:

3. Determine whether 6 is a meaningful replacement in 1 6 .

If we replace y by 6, we get 1 6 5 , which has no meaning as a real number because the radicand is negative.

1. Determine whether 7 is a meaningful replacement in 3 2x .

If we replace x by 7, we get 3 2(7) 17 , which has meaning as a real

number because the radicand is nonnegative.

Homework: Determine whether the given number is a meaningful replacement in the given radical expression. (Answers are at the end of this lesson.)

1. 4; y

2. 8; m

3. 11; t 5

4. 11; 2 x

3. Determine the meaningful replacements in each of the following expressions. Remember the radicand must be nonnegative in order for the radical to be meaningful as a real number.

1. x Any real number greater than or equal to zero is meaningful.

2. 2x We solve the inequality x + 2 ≥ 0. Any number greater than or equal to −2 is meaningful. Set-builder notation is {x | x ≥ − 2}.

3. 2x Squares of real numbers are never negative. All real numbers are meaningful replacements.

4. 2x 1 Since x2 is never negative, x2 + 1 is never negative. All real numbers are meaningful replacements.

5. 2 5x We solve the inequality 2x − 5 ≥ 0. Any number greater than or equal to 5

2 is meaningful. Set-builder notation is 52{ | }x x .

Homework: Determine the meaningful replacements of the following. (Answers are at the end of this lesson.)

1. 5x

2. 3y

3. 5t

4. 8 y

5. 8y

6. 18m

7. 2 7y

8. 3 8x

9. 2t 5

10. 2y 1

4. Read 8.2, pp. 636−639


5. Read 8.3, pp. 644−647

Homework: Set 8.3, pp. 647−650, 1−91 (every other odd)

6. Read 8.4ab, pp. 653−655


7. Read 8.4c, pp. 655−656

Homework: Set 8.4, pp. 659, 53−73 Odd Numbered Problems.

8. Read 8.5a, pp. 661−662 to just before Example 3.


10. Read 8.6, pp. 669−672





Answers to Homework Problems in This Lesson:

Procedure 2

1. Yes

2. No

3. No

4. Yes

Procedure 3

1. {x | x ≥ 0}

2. {y | y ≥ 0}

3. {t | t ≥ 5}

4. {y | y ≤ 8}

5. {y | y ≥ −8}

6. {m | m ≥ 18}

7. 72{ |y y }

8. {x│x ≥ − }

9. All real numbers

10. All real numbers


1. Find the square root of 144.

2. Simplify: 36

3. Simplify: 64

4. Simplify: 125

5. What is the radicand in 5 6 ?x y

6. Which of these expressions are meaningless as a Real number?

a. 64

b. 81

c. 0

d. 2)( 5

7. Determine if − 11 is a meaningful replacement in 2 x .

8. Determine the meaningful replacements in 3 2x .

9. Simplify: 2k

10. Simplify: 2121y

Simplify by factoring:

11. 45

12. 9 9x

13. 5x

Multiply and then simplify by factoring, if possible.

14. 7 3

15. 6 15

16. 22 12xy x y

17. 5 5x x

Simplify:

18. 50

18

19. 2

225

a

Rationalize the denominator.

20. 2

7

21. 3x

y

Divide and simplify.

22. 56

40

23. 2

35

80

x

xy

Add or subtract and simplify

24. 7 45 3 4 5

1

525. 5

Simplify.

26. 2

7 3

27. 3 5 3 5

28. Rationalize the denominator: 12

4 7

29. In a right triangle, a = 5 and c = 13. Find b.

30. Solve for : 3 2x x 14

Lesson 14: Quadratic Equations

Second degree polynomial equations in one variable are called quadratic equations. These equations can always be put in the standard form ax2 + bx + c = 0. The quadratic formula is a powerful tool that can always be used to find the solutions, if any exist, to a quadratic equation. However, it is not always the best method to use. Sometimes, solving equations by factoring is quicker and easier.

Objectives

When you have mastered this lesson, you will be able to:

1. Write any quadratic equation in the form ax2 + bx + c = 0.

2. Solve quadratic equations of the form ax2 = k, if solutions exist.

3. Solve some quadratic equations by factoring.

4. Solve any quadratic equation that has real solutions using the method of completing the square.

5. Solve any quadratic equation that has real solutions using the quadratic formula.

Procedures


1. Read 9.1abc, pp. 686−690


2. Read 9.2ab, pp. 694−695


3. Read 9.2c, pp. 696−698

Homework: Set 9.2c, pp. 700−701, 31−51 Odd Numbered Problems.

4. Read 9.3, pp. 703−706






1. Write the following equation in the form ax2 + bx + c = 0, then, identify a, b and c.

x(x − 3) + 12 = (3x + 7)x

2. Solve for x: 3x2 = 27

3. Solve by factoring: x2 − 6x = −8

4. Solve for r: r2 + 5 = 1

5. Solve by factoring: 8x2 − 16x = 0

6. Solve using completing the square: x2 + 6x − 91=0

7. Solve using the quadratic formula: 4x2 − 8x = − 1

8. Solve using completing the square: 2y2 − 3y = 9

9. Solve for x: x(x − 1) = 12

10. Solve for 2

5 3: 1

3 9x

x x

0

Lesson 15: Applied Problems Using Quadratic Equations

Many other types of problems are solved by using quadratic equations. In this lesson, we will solve applied problems and radical equations using quadratic equations. Also, we will graph quadratic equations.

Objectives

When you have mastered this lesson, you will be able to:

1. Use quadratic equations to solve radical equations.

2. Solve applied problems using quadratic equations.

3. Graph quadratic equations.

4. Find the x-intercepts of quadratic equations.

Procedures


1. Read 8.5a, pp. 662−−663 (From Example 3 to part B).

Homework: Set 8.5a, p. 666, 21−35 Odd Numbered Problems.

2. Read 8.5b, p. 663

If we have two radicals, we may need to square twice. First, we must isolate one of the radicals on one side of the equation and then square both sides. This removes one radical, but when we square the binomial we still have another radical left on the other side from the middle term. Consider the following example.

Solve for x: 3 5x x 4

2 2

3 4 5 (Isolate one radical.)

( 3) (4 5) (Square both sides.)

x x

x x

Using 2( )x = x and (a − b)2 = a2 − 2ab + b2, we have

2 23 (4) 2(4)( 5) ( 5)

3 16 8 5 5 (Simplify.)

3 16 5 8 5 (Isolate the other radical.)

24 8 5 (Simplify.)

x x x

x x x

x x x

x

24 8 5(Divide by the common number.)

8 8

x

Lesson 15: Applied Problems Using Quadratic Equations 87

2 2

3 5 (Simplify.)

(3) ( 5) (Square both sides again.)

9 5(Simplify.)

4

x

x

x

x

Check: (4) 3 (4) 5 4

1 9 4

1 3 4 (Remeber is always 0.)

4 4 (True, so accept 4.)x

Homework: Set 8.5, p.667, 37−43 Odd Numbered Problems.

3. 3. Read 9.5, pp. 717−720


4. 4. Read 9.6, pp. 725−729






1. Solve for x: 6 13 3x x

2. Solve: 1 6x x 1

3. Graph: y = x2 −1

4. Graph y = 5 − x − x2

5. Find the x-intercepts: y = x2 − 4x − 2.

6. The area of a rectangle is 231 square feet. The length is one foot less than twice the width. Find the dimensions of the rectangle. (Translate using algebra, then solve.)

7. The hypotenuse of a right triangle is 5 meters long. One leg is 3 meters longer than the other leg. Find the lengths of the legs. (Translate using algebra, then solve.)

8. The current in a stream moves at a speed of 2 mph. A boat travels 56 miles upstream and 64 miles downstream in a total time of 4 hours. What is the speed of the boat in still water? (Translate using algebra, then solve.)

Lesson 16: Final Exam for Math 101Y Congratulations, you have completed all your lesson tests for the second module Math 101Y of the Elementary 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 last seven lessons of this Study Guide.

Objectives

When you have completed the final exam, you will be able to:

1. Say you have mastered the material in Lessons 9−15.

2. Receive a grade in Math 101Y.

Procedures

1. 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.

2. Redo your sample tests from the last seven lessons.

3. Do the following as a sample test for the final exam:

The sample test is Lesson 16: Sample Final for Math 101Y given on page 105 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 3 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 101Y. If you use scratch paper, number your problems 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 35% 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.

Lesson 16: Sample Final for Math 101Y

This is a very close representation for the actual final exam, so be sure you can work these problems easily.

1. Simplify: 2

2

3 9 1

6 30

x x 2

24x x

2. Multiple and simplify: 2 2

2 3

2 2 4

2 3 4 8 2

x x x x

x x x x

3. Divide and simplify: 2

2 2 2

2

6 7 8 7

x xy x y2x xy y x xy y

4. Add. Simplify, if possible. 7 8 8 1

5 3 3 5

x x

x x


2

2

1

2 22 2

x

xx x

6. Solve for x: 5 3

13x x

7. Morgan can proofread 25 pages in 40 minutes working alone. Shelby can proofread the same 25 pages in 30 minutes. How long would it take them, working together, to proofread the 25 pages? (Translate using algebra, then solve.)

8. Evaluate 2x2 + 5x − 4 when x = −4.

9. Simplify: 2 2 237 5 3x x x x .

10. Subtract: (7x4 − 4x3 + x2 − 3) − (2x3 − 6x2 + 9x + 5).

11. Multiply and simplify: −3x2(2x3 − 6x + 5).

12. Multiply and simplify: (5x − 3)2.

13. Multiply and simplify: (6 + 5y)(8 − y).

14. Divide and give the answer in mixed form: (6x3 − 8x2 − 14x + 13) ÷ (3x + 2).

15. Factor completely: −10x3 + 8x2 + 4x4

16. Factor completely: x2 − x − 42.

17. Factor completely: 18x2 − 9 − 21x.

18. Factor completely: 4x3 − 6x2 − 6x + 9.

19. Factor completely: 128x3 − 50xy2.

20. Solve: x2 − 2x − 24 = 0.

21. Solve: x(x − 5) = 14.

22. The width of a rectangle is 2 cm less than the length. The area is 15 cm2. Find the length and the width. (Translate using algebra, then solve.)

23. Simplify:

a. 416x

b. 527x

24. a. Multiply and simplify: 21 7

b. Divide and simplify: 175

7

25. Simplify: 356

12

x

y

26. Simplify: 5 8 15 18

27. Simplify: 20 50

10

28. If one leg of a right triangle is 18 feet and the hypotenuse is 24 feet, find the length of the other leg. (Give the answer in simplified form.) (Translate using algebra, then solve.)

29. Solve using the principle of square roots: 3x2 − 49 = 0

30. Solve by completing the square: 5x2 − 110x + 510 = 0

31. Solve by the quadratic formula: 3x2 − 4x − 2 = 0

32. Solve for x: 2

3 11

2x

2

4x

33. Graph: y = 3 + x − x2

34. The current in a stream moves at a speed of 4 mph. A boat travels 4 miles upstream and 12 miles downstream in a total time of 2 hours. What is the speed of the boat in still water? (Translate using algebra, then solve.)

35. Solve for x: 2 1 2x x

Answers to Sample Tests

Test 1

1. 2 · 2 · 3 ·5 · 5

2. 120

3. 21

49

4. 33

48

5. 2

3

6. 37

61

7. 11

40

8. 5

36

9. 67

10

8

0

10. 1.895

11. 435.072

12. 1.6

13. 99.0187

14. 1796.58

15. 2.09

16. 44%

17. 234.7

18. 234.728

19. 0.007

20. 91

100

21. 625

22. 1.44

23. 32

24. 207

25. 3

Test 2

1. 10

2. 240 ft2

3. x − 9

4a. <

4b. >

5a. >

5b. <

6. 22 ≥ x

7a. 7

7b. 9

4

8. −8

9. 2

3

10. 7

40

11. −8

12. 1.4

13. 2

3

14. 8

15. 7.8

16. 10

17. −2.5

18. 7

8

19. 12

20. $148

Test 3

1. −48

2. −280

3. −9

4. 3

16

5. 3

4

6. −9.728

7a. 1

2

7b. 7

4

8. 18 − 3x

9. −5y + 5

10. 2(6 − 11x)

11. 7(x + 3 + 2y)

12. 2x + 7

13. 9a − 12b − 7

14. 448

15. −173

16. −64

17. 68y − 8

Test 4

1. 8

2. 26

3. −8

4. 7

20

5. −6

6. 49

7. 2

8. −12

9. 7

10. 2.5

11. 5

3

12. 5%

13. $800

14. width is 7 cm; length is 11 cm

15. 6

16. 81; 83; 85

17. $880

18. 2π

Ar

h

19. 6A hc

ah

20. 2

2

P wl

Test 5

4. {x| x ≤ −4}

5. {x| x ≤ 5}

6. {y| y ≤ − 8}

7. {y| y ≥ 8}

8. 1

|20

x x

9. {x| x< −6}

10. {x| x >−13}

11. {x| x ≤ −1}

12. {x| x >6}

13. {l| l ≥ 174 yards}

14. 6−5

15. x9

16. (4a)11

17. 33

18. x−5

19. (2x)0

20. 3

1

5

21. y−8

22. x6

23. −27y6

24. 16a12b8

25. 3

3

8a

c

26. −216x21

27. − 24x21

28. 9

64 18

x

y

29. 3.9 × 109

30. 0.00000005

31. 1.75 × 1017

32. 1.296 × 1022

Test 6

1. Q III

2. No

3.

4.

5.

6. 4

3

7. undefined

8. slope is 14 ; y-intercept is (0, − 2)

9. y = 6x − 26

10. 3 5

4 2y x

11. perpendicular

12. 8

3y x

13. 90

yx

14. 1393

15. 124 hours

16. No

17.

Test 7

1. Yes

2.

3. (3, −2)

4. (4, 3)

5. (−12, 11)

6. (2, 5)

7. (12, −6)

8. (1, −2)

9. (4, −1)

10. 7; 19

11. 80 liters of 30%; 120 liters of 50%

12. 9 mph

Test 8

There are text references in front of each problem on this final. For example, [1.4a] means Chapter 1, Section 4a of your textbook.

[R.1 a] 1. 2 · 2 · 3 · 5 · 5

[R.1b] 2. 120

[R.5c] 3. −40

[1.1a] 4. 1

7

[1.1b] 5. 10 − x

[1.4a] 6. −7

[1.4a] 7. 17

6

[1.6c] 8. 3

4

[1.7c] 9. − 8x + 10

[1.7d] 10. 6(x − 1)

[1.8c] 11. − 15x + 25

[2.3b] 12. −12

[2.3a] 13. 2

[2.3c] 14. 5

3

[2.3c] 15. 2

[2.7e] 16. {x|x ≤ −1}

[2.6a] 17. 52; 54

[2.6a] 18. width is 7 cm, length is 11 cm

[2.5a] 19. $880

[2.4a] 20. 20

5

A yx

or x =

[2.8b] 21. {x| x < 3}

[3.1d, f] 22. 13

1

x

[3.1e, f] 23. x11

[3.2a, b] 24. 12

218

a

b

[3.2d] 25. 6.106 × 10−11

[17.11] 26. 2 11

or 2 3 113 3

y x x y

[7.1] 27. 3 5

or 3 4 104 2

y x x y

[7.2] 28.

[7.5] 29. 1 hour

[8.1] 30.

[8.2] 31. (−3, 9)

[8.3] 32. (−2, 4)

[8.4] 33. 40 L each

Test 9

1. 1, 2, −17, 4, −1 Degree is 4.

2. 11

3. 3x2 + 6x − 1

4. − x + 3y − 8z3

5. − 7x5 + 6x4 − 4x3 − x + 4

6. − 15x6 + 9x5 + 6x3

7. 2x2 − 11x − 21

8. 35x3 − 12x2 − 44x + 9

9. 81x2 − 49

10. 25x12− 64y8

11. 36x2 − 132x + 121

12. 2 4 14

3 9x x

13. 3 23 6x y y 1

2

14. 2 24 20 5

5x x

x

15. $3858.75

Test 10

1. 10(x − 2)

2. x(x − 5)

3. 13(w3 − 4 + 2w)

4. (x − 3)(x − 5)

5. (x −6)2

6. 4(5x − 6)(x + 4)

7. (a + y)(a − 3)

8. (3x + 2y)(3x − 2y)

9. (x − 5)(x + 4)

10. 3x(2x + 3)(x + 1)

11. y4(xy + 8)(xy − 4)

12. 2(x + 5)(x − 5)

13. (x3 − 5)(x + 2)

14. 3x(2x + 5)(x − 1)

15. (2x − l)(2x + l)(4x2 + 1)

16. 3

or2

x x

9

17. 5

or3

x x

4

18. x = −5 or x = 7

19. The two consecutive even integers are 12 and 14, or the two consecutive even integers are −14 and −12.

20. The base of the triangle is 8 cm and the height of the triangle is 11 cm.

Test 11

1. −4

2. 4

2( 4)

x

x

3. 12

a 2

a

4. 4 3

3 4

x

x

5. 21)(x

x

6. 5

3

x

x

7. 1

2

x

x

8. 3

2x

9. x + 5

10. 2

1

x

x

11. 2 · 32 · 7 or 126

12. 22 · 32 · 5x3y4 or 180x3y4

13. 2(x − y)(x + y)2

14. 2( 2 ) 2 4

orx y x y

x y x

y

15. 5 5

or2( 1) 2 2

x x

x x

Test 12

1. 11 2

3 (

x

x x

1)

2. . or (2 1)( 2)

(3 )(3 )

a a

a a

3. x6

3

x

4. 7 13 7 1

or4 4

x x 3

x x

5. 2

b c

6. 6

( 6)(

x

x x

4)

7. a = 5

8. 3

6 T 4

9. x = 4

10. x = −5 or x = 2

11. No solution, since x = −2 makes the denominator zero.

12. 12 hours

13. 112 miles

Test 13

1. −12; 12

2. 6

3. −8

4. 5 5

5. 6 − y

6. b

7. Yes

8. 3

|2

x x

9. k

10. 11y

11. 3 5

12. 3 1 x

13. 2x x

14. 21

15. 3 10

16. 2 6xy y

17. 225 x

18. 5

3

19. 15

a

20. 14

7

21. 3xy

y

22. 35

5

23. 7

4y

24. 12 5

25. 4

5

5

26. 52 14 3

27. −22

28. 4(4 7

3

)

29. 12

30. 48

Test 14

1. 2x2 − 4x − 12 = 0; a = 2, b = 10, c = −12

2. x = 3 or x = −3

3. x = 4 or x = 2

4. No real-number solution.

5. x = 0 or x = 2

6. x = −13 or x = 7

7. 2 3

2x

8. 3

or2

x x

3

9. x = − 3 or x = 4

10. Only x = 2, since x = 3 makes the denominator zero.

Test 15

1. 2; −2

2. −3

3. y = x2 − 1 (0, −1)

4. 1 21

5 ,2 4

y x x

5. (2 6); (2 6 )

6. width is 11 ft; length is 21 ft.

7. x = x + 3 =

8. 30 mph

Test 16

There are text references in front of each problem on this final. For example, [5.1] means Chapter 5, Section 1 of your textbook.

1. 4

2( 4)

x

x

] 2. 2

2 (

x

x x

3)

[ 3. ( )

2

x x y

x y

[6.4] 4. 3

[6.5] 5. 1

2

[6.6] 6. 14

3

[6.7] 7. 1717 min

[3.3a] 8. 8

[3.3e] 9. 2 134

3x x

[3.4c] 10. 7x4 − 6x3 + 7x2 − 9x − 8

[3.5b] 11. − 6x5 + 18x3 − 15x2

[3.6c] 12. 25x2 − 30x + 9

[3.6a] 13. 48 + 34y − 5y2

[3.8b] 14. 2 172 4 2

3 2x x

x

[4.1b] 15. 2x2(2x2 − 5x + 4)

[4.2a] 16. (x − 7)(x + 6)

[4.3a] 17. 3(3x + l)(2x − 3)

[4.1b] 18. (2x2 − 3)(2x − 3)

[4.5d] 19. 2x(8x + 5y)(8x − 5y)

[4.7b] 20. −4; 6

[4.1c] 21. −2; 7

[4.8a] 22. width is 3 cm; length is 5 cm

[9.1] 23a. 4x2

[9.2] 23b. 23 3x x

[9.2] 24a. 7 3

[9.3] 24b. 5

[9.3] 25. 42

3

x xy

y

[9.4] 26. 55 2

[9.4] 27. 4 2

2

[9.5] 35. 5

[9.6] 28. 6 7

[10.2] 29. 7 3

;7 3

3 3

[10.2] 30. 11 19

[10.3] 31. 2 10 2

;3 3

10

[10.1] 32. 1

[10.6] 33. Y = 3 + x − vertex = ( ½, )

34. 8 mph

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