homework 2.2 college algebra

Georgia College Ina Dillard Russell Library

MATH 1111 - College Algebra: 2.2 Intro to Functions

1.1 Sets and Set Operations
1.2 Linear Equations and Inequalities
1.3 Systems of Linear Equations
1.4 Polynomials; Operations with Polynomials
1.5 Factoring Polynomials
1.6 Quadratic Equations
1.7 Rational Expressions and Equations
1.8 Complex Numbers
2.1 Cartesian Coordinates/Relations

2.2 Intro to Functions

2.3 Operations with Functions
2.4 Graph of Functions
3.1 Linear Functions
3.2 Quadratic Functions and Quadratic Inequalities
4.1 Finding Zeros of Polynomial Functions
4.2 Graphing Polynomial Functions
4.3 Rational Functions
4.4 Rational Inequalities
5.1 Composition of Functions
5.2 Inverse Functions
5.3 Introduction to Exponential and Logarithmic Functions

At the end of this section students will be able to:

Determine whether a relation is a function
Find the domain and range of a function
Evaluate functions

Required Reading

1.3 Introduction to Functions

Stitz-Zeager College Algebra - pages 43-47

1.4 Function Notation

Stitz-Zeager College Algebra - pages 55-59

Practice Exercises

Introduction to Functions

Stitz-Zeager College Algebra - pages 49-54

Answers to practice exercises can be found on pages 53-54.

Function Notation

Stitz-Zeager College Algebra - pages 63-65

Answers to practice exercises can be found on pages 69-74.

Supplemental Resources

Introduction to Functions (tutorial): West Texas A&M University Virtual Math Lab (College Algebra Tutorial 30)

Finding the Domain of a Function:

Evaluating Functions:

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Next: 2.3 Operations with Functions >>
Last Updated: Apr 2, 2024 2:52 PM
URL: https://libguides.gcsu.edu/math1111

College Algebra 2e - 2e

(19 reviews)

Jay Abramson, Arizona State University

ISBN 13: 9781951693411

Publisher: OpenStax

Language: English

Formats Available

Conditions of use.

Learn more about reviews.

Reviewed by Sarah Cordell, Professor, Northeastern Illinois University on 4/12/24

Comprehensiveness rating: 5 see less

I gave this textbook a rating of 5 for the comprehensiveness category. The textbook includes all important content typically included in college algebra textbooks. This textbook also includes additional topics that may not be addressed in a typical college algebra course such as probability which provides students with an opportunity to potentially use this resource to study additional topics not covered in their college algebra class.

Each chapter also includes chapter reviews (which includes key terms and key concepts) and exercises (which includes practice exercises and practice tests). There is also a helpful answer key located at the end of the textbook.

The textbook also incorporates "Try It" sections, "Media" sections that provide links to videos containing examples, and technology sections that provide students with support for using a graphing calculator to understand topics.

The index included in this textbook is also very effective. For each item listed in the index, there is a link or links directly next to the item that go to the section or sections in which that particular item appears in the textbook.

Content Accuracy rating: 5

I gave this textbook a rating of 5 for the accuracy category. I did not notice any errors in the textbook. The content is accurate and unbiased.

Relevance/Longevity rating: 5

I gave this textbook a rating of 5 for the relevance category. The content is up to date as it includes real world examples and applications. The real world examples and application problems could be easily updated if interested.

Clarity rating: 4

I gave this textbook a rating of 4 for the clarity category. Overall the textbook is accessible to students and appropriate mathematical terminology is used. However, I believe that some mathematical solutions in the textbook could be presented more clearly. In some cases there could be additional steps of the problem solving process included to support student understanding.

Consistency rating: 5

I gave this textbook a rating of 5 for the consistency category. The overall format of the book is very consistent throughout each chapter and section. For example each section contains learning objectives, key topics and terminology highlighted in bold, and examples, "how to" sections, and definitions highlighted in grey boxes, etc.

Modularity rating: 5

I gave this textbook a rating of 5 for the modularity category. This book is well-organized into various chapters and sections that can be assigned at various points throughout the course and that can be easily reorganized to align with the objectives of a particular course.

Organization/Structure/Flow rating: 4

I gave this textbook a rating of 4 for the organization category. Overall I feel that topics are presented in a logical and clear fashion. However, there are some improvements that could be made such as not repeating topics in more than one place in the book. For example, there is some overlap between Chapters 2 and 4.

Interface rating: 5

I gave this textbook a rating of 5 for the interface category. I did not notice any significant interface issues. I thought it was easy to navigate the textbook, images/charts were clear throughout the textbook, and other items were also displayed clearly throughout the textbook. There are links provided for tables and figures within the writing of the sections making it is easy to quickly navigate to these items.

Grammatical Errors rating: 5

I gave this textbook a rating of 5 for the grammatical category because I did not notice any grammatical errors.

Cultural Relevance rating: 5

I gave this textbook a rating of 5 for the cultural category. I did not find this textbook to be culturally insensitive or offensive in any way. There are real world exercises included in the textbook that are inclusive of a variety of backgrounds. A suggestion would be to include additional culturally relevant examples in the textbook in the future.

Reviewed by Kristin Riesgraf, Assistant Teaching Professor, University of Wisconsin - Superior on 2/20/23

The text does a thorough job of covering essential topics needed in a College Algebra course. Topics are explained with lots of details and examples. It also includes opportunities for students "Try It" themselves after each example problem. ... read more

I did not find any errors. However, the textbook website does have a place where suggestions for corrections can be submitted. There is also a listing of all the correction suggestions that have been made, and what the result was after it was reviewed.

The content is up-to-date. The application problems are relevant, and can be easily updated (if needed) in the future.

The text uses appropriate mathematical terminology. It provides many good examples to help show the meaning each concept. The text also incorporates instruction one using the graphing calculator, which is nice. The formatting of some solutions to example problems gets a little clumsy, which can make it hard to follow.

The text is consistent in its language/terminology and format.

Modularity rating: 4

Chapter 2 took be reorganized. I feel there could be a better place for talking about the rectangular coordinate system. Starting chapter 2 just focusing on linear equations and one model with applications and then moving into other types of equations, would maybe nice. Save the graphing pieces for later.

For some parts, there seems to be a logical flow. However, some sections in chapter 2 (like the ones talking about graphing) would be suited well other place. The textbook webpage does provides Instructor Resources. One of the recourses is the ability to down load docx files of the textbook, which makes it easy for the teacher to reorganize the book how the feel it works best for their class.

Interface rating: 4

The only interface issue I can think of is finding a more convenient way for students to access the "Try it" solutions.

I did not notice any grammatical errors.

There are many real world application problems that are current. The website for the textbook provides instructor resources. One resource that it provides is a "Improving Diversity, Equity, and Inclusion in Course Materials" document, which provides suggestions of how to continue to make the document culturally relevant.

Reviewed by Haley Dohrmann, Assistant Professor, Kirkwood Community College on 11/15/21

The text is a very comprehensive study of college algebra. Every topic was covered in depth with plenty of explanation, examples, and exercises. It has an index at the end of the book and a very useful glossary of key terms at the end of each... read more

I did not find any errors or bias in the text.

The content mostly is up to date with relevant real world examples. There were only a couple of examples/exercises that I found that could be updated. It would be very easy to update these as well as any content as the book ages.

Clarity rating: 5

I found this book very easy to read clearly defined new terms as they were introduced. I was very impressed with the ability of the text to be more relatable and simplistic for students without taking away rigor.

Overall the book is very consistent. Sections and chapters consistently follow the same organization. I found terminology to also be consistent throughout the text.

This book is easily reorganized and divisible especially on the level of the chapters. There were some sections that were excessively long in my opinion, but were divided into subtopics that could be rearranged pretty easily.

Organization/Structure/Flow rating: 3

I believe this text lacks a good flow mostly because of Chapter 2. For example, in 2.2 Linear Equations in One Variable the text discusses linear equations in two variables and even goes as far as to discuss parallel and perpendicular lines. This is all covered again in 4.1 Linear Functions. Most of Chapter 2 seemed redundant or out of place to me. However, with some rearranging of these topics or even omission, the rest of the text is organized logically.

I did not have any issue navigating through the book online or in the pdf. Everything displayed clearly. My note here is that the online text overuses gray boxes. It uses them for examples, definitions, Q&A, and more. This is confusing as the reader as it seems like all the gray boxes should be highlighting the same type of information. The pdf does a much better job a visually separating these types of information.

I did not find any grammatical errors.

I did not find the text to be culturally insensitive or offensive in any way. At the beginning of the section the book relates to the reader with a real world motivation for the material being covered in the section and finishes most sections with real world exercises relating to a variety of backgrounds.

Overall I think this is a great text for a College Algebra course and I will be adopting it in my class!

Reviewed by Scott Hornby, Mathematics Department Course instructor, Bridgewater State University on 6/30/21

A comprehensive text I will consider using in Fall 2021 read more

A comprehensive text I will consider using in Fall 2021

The accuracy of all the terms I reviewed in several of the chapters appears accurate.

At no time did I feel there was any problem with the relevance of any of the examples in the book.

The text was clear and easy to follow with several color examples.

The framework and consistency of the text is very much in alignment with what I currently use.

The section size of each part appears to be appropriate in my view.

Organization/Structure/Flow rating: 5

Organization of this text was similar to the 3 that I have used in the past and is easy to follow.

The interface was easy to follow using the online and PDF version of the text.

No grammatical errors were noted.

There were no culturally in appropriate issues that I detected.

Answers to all questions at the end of the text is my only suggestion.

Reviewed by Doug Joseph, Mathematics Instructor, Allen Community College on 5/29/21

This textbook covers the outcomes and objectives for a typical college algebra course. The main reason I am giving this text such a high rating is the index and key terms included after each chapter, I really like how terms are included in the index and linked back to the text. I feel linking back to each section is a better approach for students than having definitions listed out at the end of the text, the ability to quickly link back to the appropriate section is also a great benefit for students. Listing of key terms and definitions after each chapter is also helpful as students see a short list of terms used for each chapter.

Since this text is on that is considered for adoption at my institution it has went through a very in-depth review. All figures in this text are clearly and correctly labeled, and all problems have been worked out correctly.

Relevance/Longevity rating: 4

Mathematics textbooks typically do not change much from edition to edition. What I look for is how old are the examples in the text, if they have data then that data should be current and up to date. This text did a great job of insuring that data was current and up to date and most examples were relevant. I found only one application that was out of date, this was cell phone plans and pay per minute. Since almost no cell phone plan are no longer pay per minute, I feel this very slightly dates the text.

This text is not much different than most other college algebra texts. One place where this text excels is repetition of key terms and concepts, I feel having definitions listed out in each section and again at the end of each chapter is helpful to students.

After a careful review I found this text to be very consistent with symbols and terminology. Another important place where algebra texts must be consistent is working out example problems, all instructors work problems out differently and this can cause confusion for students. Reviewing this text I felt examples were presented in a consistent manner and style throughout.

The sections and chapters were of appropriate length, and only a few chapters directly linked to others. One example where one chapter linked to another is section 4.1linking back to 3.1, I do not find this a bad thing, however if you were to break up and reorganize this text then you would need to make sure chapter linking was preserved or removed.

This text is very well organized and presented in a logical order. Once place that could be an issue for some instructors is the placement of graphing sections. I personally like having graphing of linear functions along with solving linear functions, however some instructors like having graphing combined into a single stand alone chapter.

Interface rating: 3

All figures displayed correctly, most links displayed correctly, and all figures were referenced and clearly labeled. Two reasons why I am giving this text a 3 are as follows: 1. Some sections link to learning pods that do not exist, this could easily be cleaned up by removing dead links 2. I feel in some figures are overly linked, if you have a figure directly below text then a hyperlink to that figure is not needed in that text this is especially a problem in section 2.1.

I did not find any grammatical errors and examples seem to be worked out correctly.

I did not find any culturally insensitive or offensive materials in this text. At the beginning of this text it showcases the many different cultures and regions that contributed to the development of our current numbering system. This is the only college algebra book that I have seen that went into this much detail on the history and development of our current numbering system.

After an in-depth review of this text I feel it covers all the topics and does a great job of explaining with many detailed examples and supporting videos. Every textbook has it's flaws, however I feel that this text would be a great open resource alternative to a traditional text.

Reviewed by Lynn Rickabaugh, Math Department Chair, Aiken Technical College on 2/23/21

This book is very comprehensive. All necessary areas and ideas are covered. The index and glossary are effective. The "How To" box is helpful in giving step by step solutions. The highlighted box of equations helps to identify key ideas. read more

Content Accuracy rating: 4

The content is accurate. I did not note any errors. The methodology of problem solving varies sometimes, but that is where the instructor can enhance the material.

Content is up to date. I believe that updates will be easy and straight forward to implement as needed.

Good use of mathematical terminolgy. Good examples to show the meaning of particular concepts. The graphing problems show how to use the graphing calculator, which is very good.

This text is consistent in teminology and framework. I did not notcie any issues.

I think some of the content should be reorganized. Chapter 4 seems to be redundant with chapter 2. I believe the short sections in chapter 4 should be realigned.

I think the concepts of Linear Functions, Equations and Inequalites should be together in the first chapter. Then the Analysis of Graphs including basic graphs, transformations and compostion of functions. Higher Order Polynomials after that. From that point in the book, the material flows nicely.

The interface is appropriate. It is confusing that the Try It problems don't have the solutions that are easy to access. The student has to scroll to the end of the chapter for those solutions, and that is cumbersome. The Solution links with the Example problems are very good. I would suggest Solution links with the Try It problems also for the online version of the textbook.

I did not notice any grammatical errors. There may be some, but in my overview, everything seemed to flow nicely.

There are many real world applications that are very good. I'm not sure how useful they are for the students. There are Lots of word problems. This is generally a stumbling block for students. These word problems can be used by the instructor in Discussion posts to get the students to work on and respond to.

It would be nice to have pop up video links in the Online version of the book. There is a good chapter review at the end of each chapter with key equations, key formulas and key terms. This is very helpful. My main concern is the arrangement of the first couple of chapters. There is some redundancy. Otherwise it is a very thorough book and easy to follow with more infomation than most instuctors will need. Lots of resources is always a good idea.

Reviewed by Sarah Klanderman, Assistant Professor, Marian University on 12/28/20

This textbook covers all of the main topics of a typical College Algebra course, in addition to a few topics (e.g. ellipses, probability) that are less likely to be used. The index lists the main definitions, but usually links to a section in... read more

Because the textbook is available online, the errata are regularly incorporated in the version that many students access.

The text often includes examples that relate the topic to a real-world application that is likely to remain relevant for at least the foreseeable future (e.g. satellite dishes, computer monitor sizes, budgeting for trips). The one improvement that could make it even more appealing for students would be to have alternative media, perhaps associated video content or more infographics. There are existing loosely aligned video options (e.g. Khan Academy) that can be used to supplement the text in its current form however.

The book is certainly written in an approachable format and provides thorough explanations, often explicitly giving the steps involved in a particular solution.

The terminology is consistent throughout the book, and each chapter is organized in a logical manner.

Much of the material could be selected independently and used to supplement an existing curriculum. One of the strongest aspects of this text is the variety of example problems for students to work through: those given as worked out solutions in the content along with supplemental “Try It” opportunities and dozens of problems for each section (including real-world applications), in addition to review, exercises, and a practice test at the end of each chapter.

Although only the first chapter is labeled as “Prerequisites,” the second chapter on Equations and Inequalities should probably fall into that same category. It is a bit repetitive to cover quadratic equations in chapter 2 and within the context of polynomial and rational functions in chapter 5, so perhaps a better curricular approach would begin with chapter 3 material on functions. Depending on your institution however, it may be helpful to have so much prerequisite material available in the same textbook.

The online version defaults to hiding solutions (that are easily revealed at the click of the “[Show Solution]” link), which ideally encourages students to at least ponder or attempt a problem before immediately examining the answer. However, occasionally the PDF has issues with formatting and/or appearance of equations compared to the online version of the textbook.

As would be expected with any textbook, there are a few minor typos. Overall however they are rare and often corrected in the online version.

Although the text does not seem to make specific efforts to be particularly inclusive or diverse, it is neither culturally insensitive nor offensive.

After having looked through a number of different Open Educational Resources for College Algebra, I think Open Stax has one of the best textbooks currently available.

Reviewed by Luanne Gilbert, Part-Time Faculty, Bridgewater State University on 6/27/20

The text thoroughly covers all of the topics I would expect from a College Algebra text and more. It has extensive review material as well. The text has a large index of terms, properties and theorems. A click of your mouse on any index item takes you to the appropriate section. I would like to see an irrational number defined as any number that cannot be expressed as a fraction of two integers and, when written in decimal form, is a non-terminating and non-repeating decimal. It is also nice to show how a repeating decimal can be written as a fraction, thus demonstrating that a repeating decimal is a rational number.

In looking over this text, I did find two errors, one in the solution to Section 3.1, Ex. 14c, Fig. 12, and one in a video in Section 1.3 on Simplifying Radicals. I did report these errors and one typo in 5.2, Ex. 1, so that they could be corrected. There are thousands of exercises in this text and I am sure they are completed with an extremely high degree of accuracy!

Most information for the word problems is general in nature and covers a wide range of topics. However, some word problems were dated. For example, in section 4.2 many of the Real World Applications involve a prediction for a date that is in the past. In section 4.3, Exercise 6 involved gas consumption versus year from 1994-2004. Updated information and dates would be nice and would be fairly easy to implement.

Information is given clearly with terminology explained fully. Each objective has notes with rules and detailed explanation. This is generally followed by a “How To” section with steps to follow. Every example has a detailed solution available with the click of the mouse. There are also timely Q and A’s.

The book is extremely consistent in its framework. Each chapter begins with a list of its sections. Each section begins with the learning objectives for the section followed by a Real World Example to spark interest. Examples are followed by similar “Try It” problems. There are many exercises and every section has extensions and Real World Applications as well. The end of each chapter contains vocabulary, formulas, section concepts, review exercises and a practice test. I did find one case where I felt there was inconsistency in terminology. I would prefer the term “Evaluate,” which is used in Section 1.1, examples 9 and 10, to also be used in 1.1 Section Exercises 28-37 rather than “Solve for the Variable.”

Since each chapter is divided into sections, a section works very favorably as a lesson plan for each class. Each section has its own learning objectives, examples, videos and exercises. Some of the sections could be rearranged or skipped altogether.

Overall, the organization of topics and materials is excellent. The learning objectives are in bold print. The important concepts, “How Tos”, Examples, “Try Its” and Media are in shaded boxes. Every section has many exercises consisting of questions of various types: verbal, numeric, algebraic, graphical and/or technological. There are Extensions and Real World Applications as well. I would like to see a little more detailed explanation of pi, when it is first used in Chapter 1. Also, multiplying binomials and FOIL are introduced in 1.5, but the skill is needed in 1.3 when denominators are rationalized using conjugates.

I found it extremely easy to navigate around from various sections, to problem answers and to media with the click of the mouse. The index is very user friendly as well. I never ran into any distortions or dead ends!

I found nothing culturally offensive in this text. The word problems are primarily general in nature and cover a wide variety of topics.

I believe this text is a great offering for a College Algebra course and you can't do better than free! It is apparent that a great deal of time and effort have been put into this text and its framework. College Algebra, by Jay Abramson, would work extremely well for an online course, and we may see a few this semester!

Reviewed by Abdellah Laamarti, Instructor, NTCC on 4/22/20

The College Algebra textbook contains a wide range of topics that we currently cover in our College Algebra course, such as polynomial, rational, exponential, and logarithmic functions. The course material is presented in a logical order with a seamless transition from one topic to another. The textbook offers many meaningful exercises to practice the concepts covered in each section.

The textbook contains no errors as I read through the examples, or in the section exercises and their solutions.

The textbook includes many word problems and applications; that will continue to support the students in understanding the concepts of algebra for a long time.

The concepts of the textbook are clearly explained. The author includes a lot of examples in each topic and provides many ways while presenting the contents using visual methods such as highlighting the rules, graphs, and illustrations to make it clear for the students to achieve a better understanding of mathematical reasoning.

The text is organized in a logical and consistent format while reading within topics. I noticed that the author used the same mathematical terminology throughout the book.

The chapters are broken down into well-structured sections which are also divided into sub-sections/topics. The students will be pleased with the flow of reading without any disruption.

The topics are presented in a logical order and building in each other concept-wise. I like the “Key Concepts” and “How to …” features as they help students better understand the concepts in each section by questioning their learning and memorizing the rules to solve problems.

The chapters can be accessed easily using the table of contents. I found no interface issues; I navigate through the sections without having any distraction or confusion.

I have not encountered any grammatical errors while reading through the textbook.

The textbook is free of any offensive language or anything that will disturb the reader based on cultural relevance.

As the prices of many commercial textbooks are rising high, this book will be just a good fit for students who are financially challenged. I'm pleased with the material in this textbook and the way it was presented to the reader.

Reviewed by Brandie Windham, Mathematics Faculty, Morton College on 1/15/19

The text covers material comparable to the text I currently use. read more

The text covers material comparable to the text I currently use.

Text is accurate.

The content is comparable to the text that I currently use to teach with.

The text is clear and easy to read.

The text is consistent.

The text is broken into logical sections comparable to the text that I currently use to teach with.

The book is organized and logical in order.

The online textbook is interactive and user friendly.

The text is free of grammatical errors.

The book is not culturally insensitive.

The text is comparable to the book a currently use to teach.

Reviewed by Bader Abukhodair, Mathematics Instructor, Fort Hays State University on 11/29/18

This textbook covers all standard topics in College Algebra textbook, including prerequisite materials. There are two versions of the book: a PDF version and an online version. The online version has more capabilities such as hypermedia links to key concepts, “show solution” option right under each question, and a glossary at the end of each section. The textbook has sufficient worked examples and applications that support the learning objectives, many supporting feature such as “How To” and “Try It”, and many online homework software system partners such as “XYZ homework” and “WEBASSIGN” that I did not have the chance to contact nor try yet.

I did not find any math errors.

Overall, the examples and applications are up-to-date and all of the media hyperlinks that I have opened are working properly.

The author uses a consistent narrative text that develops learner curiosity. Each chapter is organized around a set of learning objectives that are listed explicitly at the beginning of each section. Reasonings to why a specific method is used while finding the answer is given and that should eliminate the need of any outside sources.

The Text is internally consistent in terms of terminology and framework with no notable exceptions.

The text is divided into chapters and therefore it would be easy to use only parts of the text. The sections are of a reasonable length to teach in a regular class time setting. The paragraphs are not too long and should be easy for students to read and follow.

The author makes all kind of efforts to keep the textbook clear and to the point in a logical way. For example, each section is supported by one or more worked examples and followed by a set of exercises that are organized by question type.

I believe that the text is free of significant interface issues, including navigation problems, distortion of images/charts, and any other display features that may distract or confuse the reader. I really like the inclusion of many hyperlinks for YouTube explanation videos in the Media part toward the end of each section. I just thought that this could be enhanced by adding more hyperlinks for YouTube videos next to each example.

The text is not culturally insensitive or offensive in any way. The book uses general names and places and gets data from trusted sources such as the United States Census Burea.

I would like the textbook to promote more active learning by requiring student’s engagement while reading the assigned contents. I believe that more questions, animations, and videos could be added to make more connections between the learners and the learning objectives.

Reviewed by Namyong Lee, Professor, Minnesota State University, Mankato on 6/19/18

The textbook contains all standard topics in modern College Algebra textbooks. Compare to our current textbook, which is from a commercial publisher, it covers the "Conic Sections" topics under the " Analytic Geometry". And the textbook does not have a section covering "Mathematical Induction". However, we do not cover both topics in current semester (15 weeks) course. Hence it does not affect our current course.

The textbook seemed to go through proof reading by many people. I didn't check all the details line by line. However, with a scan through review of the textbook, it seems that the textbook has adequate accuracy in both explanation and computation.

I would not worry about the text will be obsolete or out of dated within a short period of time as College Algebra is pretty much settled subject. In addition, the textbook can be mainly distributed by electronic form, such as PDF format or direct web access, the authors or instructors can easily add extra up-to-date material or topic if they need to do so.

The text seems more friendly to students to read than a typical textbooks on the market. It try to deliver the idea in a short sentence rather than a more lengthy but rigorous way. In some sense, the clarity to students and the clarity of instructors might different. However, for the level of mathematics course that students are taking, it should be acceptable level.

From the textbook preface, I found that there are nine additional contributing authors in addition to the main author. As an author of a textbook which was written with several co-authors, I know it is a hard process to keeping the consistency in terminology, notations, and the tone of explanations. However, this text seems maintain the consistency reasonably well. I cannot noticed any particular chapter was written by different author.

Modularity rating: 3

This is a bit too much to asking for the mathematics text. As mathematics textbook authors consider a certain structural development in sequential way, it is hard to write a text that is easily and readily divisible into smaller reading sections that can be assigned at different points within the course. However, I personally do not over concern for this aspect in choosing a text.

As we described the text seems to written for easier access to students. I especially like the blocked features, such as "Q & A", which ask students a right question for better understanding of the concept in the given section. Also I believe students will like "How To…" blocks as they can easily imitate the algorithm to solve the problem.

The textbook has adequate format and spacing. As an electric textbook, it also has several hyperlink (URL) for a short YouTube explanation video. I like the feature and students will appreciate the feature. However, more modern trend is using the QR code instead of URL. Students can use their smart phone to watch the video right away without typing the URL or clicking.

I found no serious grammar issues in my scan through review.

Cultural Relevance rating: 4

Mathematics text seldom has cultural issue. However, i would like to point out some application problems are not so adequate. For example, in chapter 5, a suspension bridge is modeled by quadratic function (parabolic curve) which is not true as the suspension bridge has not only affected by the gravitation but also by "suspension" force at the pole. We need to be a bit more sensitive in application problems.

Overall, the textbook is adequate to be used or be replaced many commercial textbooks. The writing style and the interface are friendly to students and examples and exercise problems are good enough for a typical College Algebra course. Moreover, it is free to students!

Reviewed by Xiaolong Yao, Math Instructor, Portland Community College on 5/21/18

This textbook covers all topics in college algebra, with many chapters to review pre-requisite knowledge, including solving linear equations, factoring, solving quadratic equations and graphing quadratic functions. Instructors who have a need to... read more

The online version’s lesson index can be toggled. This is a great feature. The PDF version's lesson index are all hyperlinks to the corresponding lessons.

I didn't find any grammar or math errors.

Relevance/Longevity rating: 3

Each lesson starts with a real-life application. The textbook uses many real-live scenarios in examples. However, in each lesson’s exercises, there are usually fewer than 5 application problems. There should be more.

Technology content is written for TI-83/84, an outdated model. The textbook should instead use Desmos or Geogebra, free resources for students.

The textbook's language is easy to understand. For some difficult topics, the textbook can be confusing for students who are not used to reading math textbooks. This is a common issue for all math textbooks, though. It's difficult for students to read through textbooks in general.

I didn't find any consistency issues.

The textbook is easily and readily divisible into smaller reading sections that can be assigned at different points within the course. Each lesson's objective is clear.

Lessons are organized into a logical order, with later lessons built on earlier lessons. Functions are introduced early on.

Interface rating: 2

The following are some good features: Important procedures and strategies are summarized in grey boxes. The textbook’s online version has solutions which can be toggled. The PDF version has hyperlinks to figures and tables. Both the online version and the PDF version have hyperlinks to Youtube videos. At the end of each chapter, key concepts are listed with hyperlinks to corresponding examples.

The textbook needs improvement in the following aspects:

The textbook’s online version has errors in figure referencing. Instead of saying “Figure 1.2 shows …”, it says “Figure shows …”. The PDF version doesn’t have this issue. This is a serious issue affecting students’ reading.

In the online version’s exercises, only odd-numbered solutions are shown. This is very confusing to students. They are looking at two problems, but only one solution is available, and it’s not clear which problem’s solution they are looking at. Before this is fixed, it’s hard to use the textbook’s exercises in the online version. The PDF version doesn’t have this problem.

The online version supports search of key words. However, when I search for phrases by using quotation marks, it found some extra results which have only a part of the phrases.

The textbook has a way to receive feedback (reporting errors), but the link to report error is not available when you are reading the textbook. You have to open a new tab, go to the textbook’s homepage to find the link. Once I click on it, I need to sign up to report an error, and the error form gives me an error when I use Safari. The error doesn’t happen when I switch to Chrome. This process should be made easier. Any reader should be able to report an error.

I found no grammar mistakes.

Cultural Relevance rating: 3

This textbook has little mentioning of any cultural issues. I didn't found examples using names.

The textbook covers regression for linear, exponential and logarithm functions, but not for polynomial functions. At least it should cover modeling with quadratic functions, a common application in real life.

Before the following issues are resolved, it's hard for me to use it in my classes:

1. TI-83/84 examples should be replaced by Desmos/GeoGebra examples. 2. The online version should either show solutions to all exercises, or remove even-numbered exercises. 3. The online version should add numbers to hyperlinks to figures and tables. Instead of saying "Please look at Figure", it should say "Please look at Figure 1".

Reviewed by Jill Beals, Assistant Professor, George Fox University on 5/21/18

College Algebra includes all the topics critical to a college algebra course, including review topics. It is complete in its coverage of the building blocks of algebra, many types of functions, their graphs and conic sections. The exercise sets... read more

There have been some inaccurate labels and directions in the PDF version that I have not seen in the online version.

Overall the text includes applications ranging across many subjects, issues, fields, and interests to be meaningful to a wide cross section of students. And the examples and exercises are generic and basic enough that they will not be out of date.

The textbook introduces new concepts and notation in each section with clear explanation. Sections present concepts in a variety of ways from paragraph explanation, to boxed definitions and results, to “How to” boxes and “Q&A” each providing differing facet of the content and level of detail. Pictures, tables, graphs and other visuals effectively illustrate ideas in a clear manner.

The text is consistent in layout and approach to topics. Terminology is used in a consistent way throughout the chapters.

Sections within each chapter break up the material such that it can be covered in a single class session. (One exception to this is 2.2 Linear Equations in One Variable which covers solving linear equations and finding and graphing linear equations. The latter is also included in 4.1 Linear Functions). The content in the sections is divided into self-contained topics which allows for easy reference or for reordering (or skipping) to serve overall course objectives.

Overall the text is organized such that concepts build on each other in a logical fashion. However, I have also found that covering 2.1 The Rectangular Coordinate Systems and Graphs right before chapter 3 on Functions works well too, as does covering 3.7 Inverse Functions, before Chapter 6 on Exponential and Logarithmic Functions.

Within chapters, sections move back and forth between explanation, examples, “how to” pull outs, summary boxes, also in a logical sequence, addressing key points as appropriate to the flow of the text.

The interface for both the PDF and the online version work well and navigating around the text with the table of contents is convenient. The PDF version has numbers on the section exercises whereas the online version has links to selected solutions but does not have exercises numbers, which can be confusing.

I have not found any grammatical errors in the text book.

I have not come across anything in the text that raises concerns or questions related to cultural relevance.

I have used several textbooks for college algebra and find this textbook to be just as good or better than ones with high price tags, so being free to students makes it a good choice. I will likely continue to use this textbook.

Reviewed by Justin (Pete) Rusaw, Visiting Professor of Mathematics, George Fox University on 2/1/18

The college algebra text covers all the typical topics for a course of this type. Within each idea there is an appropriate level of both mathematical rigor and application. read more

The college algebra text covers all the typical topics for a course of this type. Within each idea there is an appropriate level of both mathematical rigor and application.

Though there were a few discrepancies between the online book and the hard copy and a couple of questionable solutions in the solutions manual, the accuracy of the text was excellent.

The text has up to date applications and connections. I see no reason why the types of questions and approaches the authors take will not make for a long lasting and relevant text.

The language and organization of the text is excellent. It is very readable and student friendly.

Consistency rating: 4

The text is very consistent in it’s approach to explanations and student engagement. Additionally, the notation used is clear and unambiguous, leading to strong student comprehension.

I did do a little reorganizing of the text. It seemed like some topics, like linear and quadratic equations, appeared and then disappeared, but reappeared later in a slightly new context. I thought it would have been better to keep them together so students could make more connections due to familiarity.

The text is very logically progressive and organized in such a way that both deductive and inductive methods of learning are present. It is consistently varied in that students are asked to think, explore and then do as a part of their learning engagement.

The text has clear visuals. The tables, charts, graphs and diagrams are easy to see and read.

Grammatical Errors rating: 4

The word choice of the text’s authors is excellent. Precise, but not so much jargon that the reader is consistently lost or needing to retrace their previous reading.

I didn’t notice any major flaws or positive aspects of the text, in this regard.

Thank you for putting such a fantastic resource together for students and professors. I will definitely use this again!

Reviewed by Derrick Tucker, Instructor, Ohio University on 2/1/18

For a course titled College Algebra, all the topics covered and the level of rigor for those topics are mostly as one would expect. In addition, the text contains chapters in Analytic Geometry as well as Sequences and Probability. The explanations are thorough yet not overly verbose. Throughout the text, the authors’ take the time to concisely explain the subject matter in natural language. Compared to the textbook currently in use by my institution, I would need to supplement a brief section on circles which serves as a reinforcement of the completing-the-square skill. Also, I would need additional linear application problems. (Problems dealing with motion, value/mixtures, simple interest, etc.) All relevant topics considered, there are only minor differences in content depth and breadth.

A spot check of a few problems per section from chapter 1 through 7 did not reveal any mistakes.

“I am Legend” is the top grossing horror movie of it’s decade? I’m sure I found that more interesting than people will 20 years from now. The content is timeless. The scenarios in which the content is applied are reasonably relevant. As far as I can predict, through the lens of now, the content will age well. In other words, no references to a fad were found. For example, word problems required no knowledge of Tamagotchi.

The clarity of this textbook was excellent. As mentioned before, the authors’ used clear and concise language throughout. As a litmus test, I like have a text-to-speech program narrate some passages. If the text-to-speech program doesn’t struggle, then I am more confident in the overall readability for a general audience. The text-to-speech program performed admirably. This is not a scientific process, but merely a way to inform my opinion. The presentation of concepts was logical and consistent with other textbooks that share the College Algebra title.

This textbook develops a method of presenting the material and sticks to that method throughout.

This textbook does a great job of breaking the content down into easily consumable chunks. It has a relatively high ratio of sections per number of pages. Each of those sections is broken up into: the standard hook example, brief content lecture, a couple example problems, a “try it” section for practice, additional explanation of the topic with a “Q&A” section, then further development of the section topic with more example problems. If I were to adopt this text, I would increase the number of bookmarks on the PDF so students could quickly get to the exercises at the end of each section.

Overall the organization of the text is logical and seems pedagogically justified. The layout is such that the transition from topic-to-topic from within a section and between sections is non-jarring and natural.

Visually they did a nice job of breaking up the content with the use of headings, varying fonts, font styles, and colors. The inclusion of video links at the end of each section is a nice value added feature. Again, personally, I would add more bookmarks to the PDF version. On a few occasions, I would have preferred the graphs right justified and the corresponding explanations left justified such that the size of the graph could have been increased. Similarly, there are a few points where I would have preferred the font to be a larger size. This is especially true in the exponential and logarithm sections. Since I can zoom, this is only a minor inconvenience.

Structure-wise I would like to have more white space between paragraphs. I understand this a trade-off between readability vs. number of pages. If I’m using an electronic version, I’d prefer to error on the side of readability. For example, the four inequalities at the bottom of page 146 in the PDF version would have been more readable if stacked vertically than listed horizontally. These are minor issues. Overall the textbook has an attractive aesthetic and is nearly on par with modern retail offerings.

As far as I can tell the grammar is fine.

I did not discover any culturally insensitive or offensive material.

I appreciate all the time and effort the creators put into this project. It’s incredibly charitable and an impressive achievement in terms of quality. Any issue I mentioned was only to avoid writing the two word review, “It’s great.” I would have no reservations in adopting this textbook.

Reviewed by Zachary McLaughlin-Alcock, Lecturer, Pennsylvania State University Abington on 2/1/18

The text covers the full-set of expected material from a college algebra course with an emphasis on functions. There’s certainly more than a semester’s worth of material here and it would be fairly straightforward to pick and choose sections to tilt the text towards prefers topics. The table of contents contains easy and clear links to chapters and sections (even the PDF has links!). It could be improved with directions to each section’s glossary, key-ideas summary, and practice problems. Every section uses worked and explained examples to showcase ideas and has a full set of practice exercises.

I did not observe any mathematical errors, and the text’s error submission page appears impressively attended and well kept.

I encountered few examples that use time-sensitive data that would need to be updated frequently (teen texting habits from 2012 in 4.1 already feels dated). I do have concerns about the reliance on cell-phone per/minute plans that no longer represent the most common use-pattern and are unlikely to return to prominence. Algebra does not have to worry about obsolescence in quite the same way as other subjects, and there’s nothing to suggest the text will suddenly need major revisions.

Math textbooks are not easy reads for students, and I think most of mine would wish this had more examples with explained steps. Math jargon can be overwhelming and frequently the text and does extra work to clarify or define terms. Q&A’s right after new examples that illustrate potential problem areas preemptively but also in a more conversational tone. The online text has too many things in gray highlighting boxes, and lacks the visual contrast of the pdf. The online text’s graphs and visuals are great, but the tables really need better visual clarity.

Overall I find the text more accessible than a standard math text.

The terminology seemed consistent throughout. Sections also follow similar patterns which make navigation and reading more straightforward.

Several of the sections contain too many topics and would be better served being separated into more manageable parcels. Being over-full of concepts can make it challenging for new students to identify patterns and then build the connections to old/new ideas as they just feel overwhelmed. Section 3.6 deserves special mention for being tightly focused to a specific idea (absolute value functions). There’s no way to make a math text fully modular but College Algebra does work to make topics self contained without repeating all prerequisites every time they’d appear.

The text is clearly organized and follows a logical progression. I found some of the sequencing unusual, but there’s clear reasoning and flow to the concept ordering.

I had no interface problems. However there are a variety of quality of life improvements that would make the text much more readable and approachable for students (not the same grey boxes everywhere please). Many things link in the online reader, but are missing specific labels. For instance, “Which table, Table, Table, or Table, representations a function (if any)? … A: Table and Table define functions…”

I’m unclear why the online reader’s top ‘frame’ is eating as much screen space as it does. The search bar is a very nice feature and appears robust. The pdf’s graphs are too small by default, but can be zoomed in for clarity when needed. Chapter review exercises not being a separate section in the online reader (they’re tacked on to the end of the last section of a chapter) is a strange and disappointing choice. Many sections (it’s more common in the pdf) had links to 3-4 supplementary youtube videos before the practice exercises, which many of my students would appreciate. The end of section key-ideas summary contains links to specific worked examples which is wonderful.

I didn’t notice any egregious typos or grammatical mishaps.

There examples are by no means insensitive, but they are also not doing any particular work to be inclusive in the subject or topic representation. “Real-World Applications” examples vary wildly in quality, some are great and others undermine the “Real-World” label. References to sports and physics in introducing topics has the potential to be exclusionary for students without the needed backgrounds.

Overall College Algebra is a fine textbook, totally in line with common standards. For a potentially free-to-students option, that’s a tremendous achievement. Like many algebra textbooks there are issues with the presentation of mathematics as rote rules & procedures. Potentially attention grabbing introductory questions are left unanswered. Every section has a few extension questions, but there are few questions that would count as "Rich Mathematical Tasks" or demanding critical thinking. I do also wish a fully-digital text would provide a larger variety of fully-worked and explained solutions, and drop the convention of only providing answers to odd practice exercises. Too often ideas are introduced and explained via the fully-abstracted form, which only makes sense to readers already fluent in algebra. There are portions of the text that are vastly more approachable in their discussion of ideas and concepts than is common, and those, like the semi-frequent Q&A asides are where the text is at its best.

Reviewed by Benjamin Gort, Instructor, Chemeketa Community College on 2/8/17

Any instructor teaching a college algebra class will find this text to be the perfect level of comprehensiveness. Some texts try to include too much and some texts leave to much out. With this text you can tell that the authors strive and succeed at finding the perfect amount of information to present and discuss when teaching concepts. The text covers the standard college algebra topics in an order conducive to learning. They start essentially with basics and equations followed by functions and then on to (in standard order) the five main functions: Linear, Polynomial, Rational, Exponential and Logarithmic. Following these standard functions are thorough discussions for every optional subject that shows up in different schools under different instructors. These include: Systems of Equations, Conic Sections, Sequences, Counting and Probability. Each of these sections cover the material thoroughly and in a completely self-contained manner so as to allow the instructor flexibility of including or not including any of these sections without having to supplement.

A review of many of the examples in the sections and many of the homework problems and the solutions found at the end of the text have turned up no errors not already listed on the errata page. The authors have gone to great lengths to eliminate errors from the text. The authors have also decided to separate important definitions, results, concepts, and theorems in blue text boxes that direct the reader's attention as to say “something is really important here.” Extensive review of these important points are error-free concise and clear.

The content of college algebra has for the most part remained unchanged for years. This text is adequately relevant to the subject as it is taught today. There are very useful links to youtube videos that explain concepts. These links will have to be monitored for accuracy as the years go by in the event that the author of the videos moves and deletes videos. Furthermore, the use of real world examples in the chapter and the homework as informational/motivational pieces at the beginning of chapters are relevant and contain recent enough that students won't feel like they're reading a textbook from some other decade. I don't anticipate very many updates to the field of algebra, but if there are any, then the authors will be able to easily incorporate those updates in subsequent editions.

The book is written in a very clear concise manner that aims to teach in a self-contained manner. Student's will be able to pick up this book and learn from it with little outside supplementation. The jargon and technical terminology is perfect for a text at this level and teaches what needs to be taught. The authors walk students through learning in the way an instructor might do so in the classroom. The authors begin every section with clear expectations of learning objectives and then addresses each one of those objectives systematically including a “how to” section followed by examples showing students how to work problems followed by a “try it” section (with answers in the back of the book) allowing students to try their hand at problems. Often there are Q & A sections included that address the most common conceptual questions that instructors are asked by inquisitive students in the classroom. It is this organization/process that makes the book clear and easy to learn from.

This text is very consistent in the language used from section to section. The authors use terminology and symbols consistently from section to section and chapter to chapter. Besides consistency in the language used, the authors have managed to organize the chapters and sections with common fonts, text boxes, and overall layout so that students know what to expect and aren't distracted by the presentation while trying to learn new concepts

The chapters are appropriately designed and ordered. Each chapter is broken up nicely into sections. The text is sometimes self-referential, however, that is the nature of mathematics and scaffolding of learning is very much apart of that. The text could be easily reorganized if needed, but I think that the authors have really managed to organize the material in a way that is conducive to learning and I don't see why reorganization (except in perhaps very minor instances) would be needed. As mentioned before, the advanced chapters following logarithms are self-contained and can be taught in any order skipping chapters without loss of consistency.

The order of the topics presented progress from easier to more difficult in a logical manner starting with a strong foundation and building upon that foundation. Within each chapter, examples and explanations are logical and flow easily from easy to more difficult.

All navigation links to chapters and sections work well. The links to helpful youtube videos all work as well. The text uses appropriate organization of graphics and text highlighting important concepts in a non-distracting manner. Colors for text, fonts, headings are all appropriate and help to focus the reader's attention to what is truly important.

After reviewing this text for several weeks extensively, I have yet to find a grammatical error. The authors have worked very hard to eliminate errors from this text.

The text is not culturally insensitive or offensive in any way, however, there is really not too much effort to include examples or biographies that celebrate other cultures, ethnicities, or lifestyles. For example, using NFL examples are fine for the person that understands football, but this will not benefit students that have now knowledge of the NFL. If there is any room for improvement in this text this would be the area.

Here are some specific things that the book does really well. The examples in the homework do a great job of including problems that cater to the major areas of mathematics instruction namely: Graphical, analytical, numerical, and verbal. Calculator steps look like they cater to Texas Instruments. I'm not sure how this can be expanded to include other calculator manufacturers. Content-wise, the author does a great job of planting seeds early on for future learning. For example the authors introduce some simple transformations during the study of linear graphs before formally learning all of the other transformations in a later chapter. Later on, the authors do an exceptional job of teaching transformations including a very important and thorough discussion of how the order of transformations should be considered. This is something usually left out of most texts. The placement of even/odd functions immediately following reflections looks extremely effective! The authors seem to have found the perfect placement of this topic. There is an example or two of “solving logarithms mentally” which I believe goes a long way towards the goal of getting students to think about the mathematics they do. Discussion of conic sections are very thorough and even include derivations of the standard formulas!

Here are a few things that I think the authors could consider for their next edition. The discussion of graphing non-linear inequalities hearkens back to a method used for graphing linear inequalities. However, the only graphing of linear inequalities I could find in the text is in one variable. There is no real coverage of graphing linear inequalities in two variables (in the xy-plane) which is really what is needed in order to move to the non-linear case analogously. Otherwise, the discussion of graphing non-linear inequalities is adequately thorough. Also, as a personal opinion, I don't think that partial fractions belong in a college algebra text. But if you are going to include such a section, then there should be some sort of discussion as to why you only need constant placeholders when dealing with linear factors in the denominator and why you need linear place holders when you have irreducible quadratics in the denominator. The method presented only gives “this is what you do” and no mention of why. Otherwise the placement right after systems of equations is a perfect application. Finally, in the matrices section the authors present the shortcut method for finding the determinant of a 3x3 matrix which is fine. The authors then point out that the method and not applicable to higher dimension matrices. The follow this with a statement that determinants of higher dimension matrices should be done with technology due to heavy computation. I would like to have seen a discussion of the general method of minor and co-factor expansions or at least a mention that there is a more general method that is “outside the scope of this text.”

Other than these very minor improvements, this is a very well written text that any college algebra instructor could use with success. I plan on adopting this text for my classroom within the next year or so. Thanks to the authors for the hard work and a wonderful product!

Reviewed by Jamie Wirth, Assistant Professor, Valley City State University on 1/7/16

The content of the text is quite comprehensive and certainly includes the relevant content that would be expected of a college algebra text. Some might argue that trigonometric functions should be included in a college algebra text, yet I have... read more

The table of contents (in both the online and PDF versions) is very useful, considering it provides links that allow you to jump straight to a specific section. The section numbering system, however, is not consistent between the online and PDF versions, causing slight confusion (e.g. in the PDF version, "Quadratic Equations" is section 2.5, but in the online version it is section 2.6). This is because the online version assigns section 1 of each chapter to the introduction, whereas the PDF version assigns section 1 to the first section of actual content, thus causing the two versions to be one section "off" from each other.

The index at the end of the book is also very useful, utilizing the links to quickly jump to specific pages where key terms are referenced.

The text appears to be very well done concerning accuracy. Furthermore, the author provides instructors with a quick and easy way to report any errors.

Typically, the main "content" of a college algebra text is not going to change over time, therefore the content in this text is up-to-date, relavent, and will not become obsolete any time soon. The authors do include multiple instances of modern examples within the technology and real-world applications sections of the exercises.

The text is pretty straight-forward with providing relevant content without excess dialogue or commentary.

The structure of the text is quite consistent. It is easy to navigate. From an instructor's point of view it should be easy to work through examples with students and provide ample opportunity to assign exercises.

The text provides a nice flow of commentary, figures, examples, etc.

The layout/order of the various sections within the text is definitely clear and logical.

As previously mentioned, there is a confusing inconsistency with section numbers between the online and PDF versions. Otherwise, the navigation is quite user-friendly. The links within the text that allow the user to jump to specific locations in the text (as opposed to excessive scrolling and manual searching) are quite useful.

The text appears to be grammatically sound.

I found no instances of culturally insensitive or offensive content.

The authors appear to have gone to great lengths to make this a user-friendly, comprehensive text. Furthermore, the inclusion of a wide variety of ancillary content for both instructors and students is much appreciated. I look forward to adopting this text for my college algebra sections.

Introduction to Prerequisites
1.1 Real Numbers: Algebra Essentials
1.2 Exponents and Scientific Notation
1.3 Radicals and Rational Exponents
1.4 Polynomials
1.5 Factoring Polynomials
1.6 Rational
Chapter Review
Introduction to Equations and Inequalities
2.1 The Rectangular Coordinate Systems and Graphs
2.2 Linear Equations in One Variable
2.3 Models and Applications
2.4 Complex Numbers
2.5 Quadratic Equations
2.6 Other Types of Equations
2.7 Linear Inequalities and Absolute Value Inequalities
Introduction to Functions
3.1 Functions and Function Notation
3.2 Domain and Range
3.3 Rates of Change and Behavior of Graphs
3.4 Composition of Functions
3.5 Transformation of Functions
3.6 Absolute Value Functions
3.7 Inverse Functions
Introduction to Linear Functions
4.1 Linear Functions
4.2 Modeling with Linear Functions
4.3 Fitting Linear Models to Data
Introduction to Polynomial and Rational Functions
5.1 Quadratic Functions
5.2 Power Functions and Polynomial Functions
5.3 Graphs of Polynomial Functions
5.4 Dividing Polynomials
5.5 Zeros of Polynomial Functions
5.6 Rational Functions
5.7 Inverses and Radical Functions
5.8 Modeling Using Variation
Introduction to Exponential and Logarithmic Functions
6.1 Exponential Functions
6.2 Graphs of Exponential Functions
6.3 Logarithmic Functions
6.4 Graphs of Logarithmic Functions
6.5 Logarithmic Properties
6.6 Exponential and Logarithmic Equations
6.7 Exponential and Logarithmic Models
6.8 Fitting Exponential Models to Data
Introduction to Systems of Equations and Inequalities
7.1 Systems of Linear Equations: Two Variables
7.2 Systems of Linear Equations: Three Variables
7.3 Systems of Nonlinear Equations and Inequalities: Two Variables
7.4 Partial Fractions
7.5 Matrices and Matrix Operations
7.6 Solving Systems with Gaussian Elimination
7.7 Solving Systems with Inverses
7.8 Solving Systems with Cramer's Rule
Introduction to Analytic Geometry
8.1 The Ellipse
8.2 The Hyperbola
8.3 The Parabola
8.4 Rotation of Axes
8.5 Conic Sections in Polar Coordinates
Introduction to Sequences, Probability, and Counting Theory
9.1 Sequences and Their Notations
9.2 Arithmetic Sequences
9.3 Geometric Sequences
9.4 Series and Their Notations
9.5 Counting Principles
9.6 Binomial Theorem
9.7 Probability

Ancillary Material

About the book.

College Algebra 2e provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra course. The modular approach and richness of content ensure that the book addresses the needs of a variety of courses. College Algebra 2e offers a wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking students to apply what they’ve learned.

The College Algebra 2e revision focused on improving relevance and representation as well as mathematical clarity and accuracy. Introductory narratives, examples, and problems were reviewed and revised using a diversity, equity, and inclusion framework. Many contexts, scenarios, and images have been changed to become even more relevant to students’ lives and interests. To maintain our commitment to accuracy and precision, examples, exercises, and solutions were reviewed by multiple faculty experts. All improvement suggestions and errata updates from the first edition were considered and unified across the different formats of the text. The first edition of College Algebra by OpenStax is available in web view here .

About the Contributors

Jay Abramson has been teaching Precalculus for over 35 years, the last 20 at Arizona State University, where he is a principal lecturer in the School of Mathematics and Statistics. His accomplishments at ASU include co-developing the university’s first hybrid and online math courses as well as an extensive library of video lectures and tutorials. In addition, he has served as a contributing author for two of Pearson Education’s math programs, NovaNet Precalculus and Trigonometry. Prior to coming to ASU, Jay taught at Texas State Technical College and Amarillo College. He received Teacher of the Year awards at both institutions.

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Chapter 2 Equations and Inequalities

Chapter 2 Practice Test

Chapter practice test.

Graph the following:[latex]\,2y=3x+4.[/latex]

[latex]y=\frac{3}{2}x+2[/latex]

A coordinate plane with the x and y axes ranging from -10 to 10. The line going through the points (0,2); (2,5); and (4,8) is graphed.

Find the x- and y -intercepts for the following: [latex]2x-5y=6[/latex]
Find the x- and y -intercepts of this equation, and sketch the graph of the line using just the intercepts plotted: [latex]3x-4y=12[/latex]

[latex]\left(0,-3\right)[/latex][latex]\left(4,0\right)[/latex]

A coordinate plane with the x and y axes ranging from -10 to 10. The points (4,0) and (0,-3) are plotted with a line running through them.

Find the exact distance between[latex]\,\left(5,-3\right)\,[/latex]and[latex]\,\left(-2,8\right).\,[/latex]Find the coordinates of the midpoint of the line segment joining the two points.
Write the interval notation for the set of numbers represented by[latex]\,\left\{x|x\le 9\right\}.[/latex]

[latex]\left(-\infty ,9\right][/latex]

Solve for x :[latex]\,5x+8=3x-10.[/latex]
Solve for x :[latex]\,3\left(2x-5\right)-3\left(x-7\right)=2x-9.[/latex]

[latex]x=-15[/latex]

Solve for x :[latex]\,\frac{x}{2}+1=\frac{4}{x}[/latex]
Solve for x :[latex]\,\frac{5}{x+4}=4+\frac{3}{x-2}.[/latex]

[latex]x\ne -4,2; \ [/latex][latex]x=\frac{-5}{2},1[/latex]

The perimeter of a triangle is 30 in. The longest side is 2 less than 3 times the shortest side and the other side is 2 more than twice the shortest side. Find the length of each side.
Solve for x . Write the answer in the simplest radical form: [latex]\frac{{x}^{2}}{3}-x=\frac{-1}{2}[/latex]

[latex]x=\frac{3±\sqrt{3}}{2}[/latex]

Solve:[latex]\,3x-8\le 4.[/latex]
Solve:[latex]\,|2x+3|<5.[/latex]

[latex]\left(-4,1\right)[/latex]

Solve:[latex]\,|3x-2|\ge 4.[/latex]
Add these complex numbers:[latex]\,\left(3-2i\right)+\left(4-i\right).[/latex]
Simplify:[latex]\,\sqrt{-4}+3\sqrt{-16}.[/latex]

[latex]14i[/latex]

Multiply:[latex]\,5i\left(5-3i\right).[/latex]
Divide:[latex]\,\frac{4-i}{2+3i}.[/latex]

[latex]\frac{5}{13}-\frac{14}{13}i[/latex]

Solve this quadratic equation and write the two complex roots in[latex]\,a+bi\,[/latex]form:[latex]\,{x}^{2}-4x+7=0.[/latex]
Solve:[latex]\,{\left(3x-1\right)}^{2}-1=24.[/latex]

[latex]x=2,\frac{-4}{3}[/latex]

Solve:[latex]\,{x}^{2}-6x=13.[/latex]
Solve:[latex]\,4{x}^{2}-4x-1=0[/latex]

[latex]x=\frac{1}{2}±\frac{\sqrt{2}}{2}[/latex]

Solve: [latex]\sqrt{x-7}=x-7[/latex]
Solve:[latex]\,2+\sqrt{12-2x}=x[/latex]

[latex]4[/latex]

Solve:[latex]\,{\left(x-1\right)}^{\frac{2}{3}}=9[/latex]

For the following exercises, find the real solutions of each equation by factoring.

[latex]2{x}^{3}-{x}^{2}-8x+4=0[/latex]

[latex]x=\frac{1}{2},2,-2[/latex]

[latex]{\left(x+5\right)}^{2}-3\left(x+5\right)-4=0[/latex]

Media Attributions

Ch 2 Practice Test #1 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license
Ch-2-Practice Test-3 © OpenStax Algebra and Trigonometry, 2e is licensed under a CC BY (Attribution) license

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College Algebra

Unit 1: linear equations and inequalities, unit 2: graphs and forms of linear equations, unit 3: functions, unit 4: quadratics: multiplying and factoring, unit 5: quadratic functions and equations, unit 6: complex numbers, unit 7: exponents and radicals, unit 8: rational expressions and equations, unit 9: relating algebra and geometry, unit 10: polynomial arithmetic, unit 11: advanced function types, unit 12: transformations of functions, unit 13: rational exponents and radicals, unit 14: logarithms.

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2.3: Models and Applications

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Learning Objectives

Set up a linear equation to solve a real-world application.
Use a formula to solve a real-world application.

Josh is hoping to get an $A$ in his college algebra class. He has scores of $75$, $82$, $95$, $91$, and $94$ on his first five tests. Only the final exam remains, and the maximum of points that can be earned is $100$. Is it possible for Josh to end the course with an $A$? A simple linear equation will give Josh his answer.

$Many students studying in a large lecture hall$

Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

Setting up a Linear Equation to Solve a Real-World Application

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $$0.10/mi$, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write $0.10x$. This expression represents a variable cost because it changes according to the number of miles driven.

If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $$0.10/mi$ plus a daily fee of $$50$. We can use these quantities to model an equation that can be used to find the daily car rental cost $C$.

$C=0.10x+50 \tag{2.4.1}$

When dealing with real-world applications, there are certain expressions that we can translate directly into math. Table $\PageIndex{1}$ lists some common verbal expressions and their equivalent mathematical expressions.

How to: Given a real-world problem, model a linear equation to fit it

Identify known quantities.
Assign a variable to represent the unknown quantity.
If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
Write an equation interpreting the words as mathematical operations.
Solve the equation. Be sure the solution can be explained in words, including the units of measure.

Example $\PageIndex{1}$

Find a linear equation to solve for the following unknown quantities: One number exceeds another number by $ 17$ and their sum is $ 31$. Find the two numbers.

Let $ x$ equal the first number. Then, as the second number exceeds the first by $17$, we can write the second number as $ x +17$. The sum of the two numbers is $31$. We usually interpret the word is as an equal sign.

\[\begin{align*} x+(x+17)&= 31\\ 2x+17&= 31\\ 2x&= 14\\ x&= 7 \end{align*}\]

\[\begin{align*} x+17&= 7 + 17\\ &= 24\\ \end{align*}\]

The two numbers are $7$ and $24$.

Exercise $\PageIndex{1}$

Find a linear equation to solve for the following unknown quantities: One number is three more than twice another number. If the sum of the two numbers is $36$, find the numbers.

$11$ and $25$

Example $\PageIndex{2}$: Setting Up a Equation to Solve a Real-World Application

There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $$34$ plus $$.05/min$ talk-time. Company B charges a monthly service fee of $$40$ plus $$.04/min$ talk-time.

Write a linear equation that models the packages offered by both companies.
If the average number of minutes used each month is $1,160$, which company offers the better plan?
If the average number of minutes used each month is $420$, which company offers the better plan?
How many minutes of talk-time would yield equal monthly statements from both companies?

The model for Company A can be written as $ A =0.05x+34$. This includes the variable cost of $ 0.05x$ plus the monthly service charge of $$34$. Company B’s package charges a higher monthly fee of $$40$, but a lower variable cost of $ 0.04x$. Company B’s model can be written as $ B =0.04x+$40$.

If the average number of minutes used each month is $1,160$, we have the following:

\[\begin{align*} \text{Company A}&= 0.05(1.160)+34\\ &= 58+34\\ &= 92 \end{align*}\]

\[\begin{align*} \text{Company B}&= 0.04(1,1600)+40\\ &= 46.4+40\\ &= 86.4 \end{align*}\]

So, Company B offers the lower monthly cost of $$86.40$ as compared with the $$92$ monthly cost offered by Company A when the average number of minutes used each month is $1,160$.

If the average number of minutes used each month is $420$, we have the following:

\[\begin{align*} \text{Company A}&= 0.05(420)+34\\ &= 21+34\\ &= 55 \end{align*}\]

\[\begin{align*} \text{Company B}&= 0.04(420)+40\\ &= 16.8+40\\ &= 56.8 \end{align*}\]

If the average number of minutes used each month is $420$, then Company A offers a lower monthly cost of $$55$ compared to Company B’s monthly cost of $$56.80$.

To answer the question of how many talk-time minutes would yield the same bill from both companies, we should think about the problem in terms of $(x,y)$ coordinates: At what point are both the $x$-value and the $y$-value equal? We can find this point by setting the equations equal to each other and solving for $x$.

Check the $x$-value in each equation.

$0.05(600)+34=64$

$0.04(600)+40=64$

Therefore, a monthly average of $600$ talk-time minutes renders the plans equal. See Figure $\PageIndex{2}$.

$Coordinate plane with the x-axis ranging from 0 to 1200 in intervals of 100 and the y-axis ranging from 0 to 90 in intervals of 10. The functions A = 0.05x + 34 and B = 0.04x + 40 are graphed on the same plot$

Exercise $\PageIndex{2}$

Find a linear equation to model this real-world application: It costs ABC electronics company $$2.50$ per unit to produce a part used in a popular brand of desktop computers. The company has monthly operating expenses of $$350$ for utilities and $$3,300$ for salaries. What are the company’s monthly expenses?

$C=2.5x+3,650$

Using a Formula to Solve a Real-World Application

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region,

\[A=LW \tag{2.4.2}\]

the perimeter of a rectangle,

\[P=2L+2W \tag{2.4.3}\]

and the volume of a rectangular solid,

\[V=LWH. \tag{2.4.4}\]

When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

Example $\PageIndex{3}$: Solving an Application Using a Formula

It takes Andrew $30\; min$ to drive to work in the morning. He drives home using the same route, but it takes $10\; min $ longer, and he averages $10\; mi/h$ less than in the morning. How far does Andrew drive to work?

This is a distance problem, so we can use the formula $d =rt$, where distance equals rate multiplied by time. Note that when rate is given in $mi/h$, time must be expressed in hours. Consistent units of measurement are key to obtaining a correct solution.

First, we identify the known and unknown quantities. Andrew’s morning drive to work takes $30\; min$, or $12\; h$ at rate $r$. His drive home takes $40\; min$, or $23\; h$, and his speed averages $10\; mi/h$ less than the morning drive. Both trips cover distance $d$. A table, such as Table $\PageIndex{2}$, is often helpful for keeping track of information in these types of problems.

Write two equations, one for each trip.

\[d=r\left(\dfrac{1}{2}\right) \qquad \text{To work} \nonumber\]

\[d=(r-10)\left(\dfrac{2}{3}\right) \qquad \text{To home} \nonumber\]

As both equations equal the same distance, we set them equal to each other and solve for $r$.

\[\begin{align*} r\left (\dfrac{1}{2} \right )&= (r-10)\left (\dfrac{2}{3} \right )\\ \dfrac{1}{2r}&= \dfrac{2}{3}r-\dfrac{20}{3}\\ \dfrac{1}{2}r-\dfrac{2}{3}r&= -\dfrac{20}{3}\\ -\dfrac{1}{6}r&= -\dfrac{20}{3}\\ r&= -\dfrac{20}{3}(-6)\\ r&= 40 \end{align*}\]

We have solved for the rate of speed to work, $40\; mph$. Substituting $40$ into the rate on the return trip yields $30 mi/h$. Now we can answer the question. Substitute the rate back into either equation and solve for $d$.

\[\begin{align*}d&= 40\left (\dfrac{1}{2} \right )\\ &= 20 \end{align*}\]

The distance between home and work is $20\; mi$.

Note that we could have cleared the fractions in the equation by multiplying both sides of the equation by the LCD to solve for $r$.

\[\begin{align*} r\left (\dfrac{1}{2} \right)&= (r-10)\left (\dfrac{2}{3} \right )\\ 6\times r\left (\dfrac{1}{2} \right)&= 6\times (r-10)\left (\dfrac{2}{3} \right )\\ 3r&= 4(r-10)\\ 3r&= 4r-40\\ r&= 40 \end{align*}\]

Exercise $\PageIndex{3}$

On Saturday morning, it took Jennifer $3.6\; h$ to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer $4\; h$ to return home. Her speed was $5\; mi/h$ slower on Sunday than on Saturday. What was her speed on Sunday?

$45\; mi/h$

Example $\PageIndex{4}$: Solving a Perimeter Problem

The perimeter of a rectangular outdoor patio is $54\; ft$. The length is $3\; ft$ greater than the width. What are the dimensions of the patio?

The perimeter formula is standard: $P=2L+2W$. We have two unknown quantities, length and width. However, we can write the length in terms of the width as $L =W+3$. Substitute the perimeter value and the expression for length into the formula. It is often helpful to make a sketch and label the sides as in Figure $\PageIndex{3}$.

$A rectangle with the length labeled as: L = W + 3 and the width labeled as: W.$

Now we can solve for the width and then calculate the length.

\[\begin{align*} P&= 2L + 2W\\ 54&= 2(W+3)+2W\\ 54&= 2W+6+2W\\ 54&= 4W+6\\ 48&= 4W\\ W&= 12 \end{align*}\]

\[\begin{align*} L&= 12+3\\ L&= 15 \end{align*}\]

The dimensions are $L = 15\; ft$ and $W = 12\; ft$.

Exercise $\PageIndex{4}$

Find the dimensions of a rectangle given that the perimeter is $110\; cm$ and the length is $1\; cm$ more than twice the width.

$L=37\; cm$, $W=18\; cm$

Example $\PageIndex{5}$: Solving an Area Problem

The perimeter of a tablet of graph paper is $48\space{in.}^2$. The length is $6\; in$. more than the width. Find the area of the graph paper.

The standard formula for area is $A =LW$; however, we will solve the problem using the perimeter formula. The reason we use the perimeter formula is because we know enough information about the perimeter that the formula will allow us to solve for one of the unknowns. As both perimeter and area use length and width as dimensions, they are often used together to solve a problem such as this one.

We know that the length is $6\; in$. more than the width, so we can write length as $L =W+6$. Substitute the value of the perimeter and the expression for length into the perimeter formula and find the length.

\[\begin{align*} P&= 2L + 2W\\ 48&= 2(W+6)+2W\\ 48&= 2W+12+2W\\ 48&= 4W+12\\ 36&= 4W\\ W&= 9 \end{align*}\]

\[\begin{align*}L&= 9+6\\ L&= 15 \end{align*}\]

Now, we find the area given the dimensions of $L = 15\; in$. and $W = 9\; in$.

\[\begin{align*} A&= LW\\ A&=15(9)\\ A&= 135\space{in.}^2 \end{align*}\]

The area is $135\space{in.}^2$.

Exercise $\PageIndex{5}$

A game room has a perimeter of $70\; ft$. The length is five more than twice the width. How many $ft^2$ of new carpeting should be ordered?

$250\space{ft}^2$

Example $\PageIndex{6}$: Solving a Volume Problem

Find the dimensions of a shipping box given that the length is twice the width, the height is $8\; $ in, and the volume is $1,600\space{in.}^3$.

The formula for the volume of a box is given as $V =LWH$, the product of length, width, and height. We are given that $L =2W$, and $H =8$. The volume is $1,600\; \text{cubic inches}$.

The dimensions are $L = 20\; in$, $W= 10\; in$, and $H = 8\; in$.

Note that the square root of $W^2$ would result in a positive and a negative value. However, because we are describing width, we can use only the positive result.

Access these online resources for additional instruction and practice with models and applications of linear equations.

Problem solving using linear equations
Problem solving using equations
Finding the dimensions of area given the perimeter
Find the distance between the cities using the distance = rate * time formula
Linear equation application (Write a cost equation)

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Houston Community College
Eagle Online

Charles Gabi
Contemporary Mathematics (Math 1332)

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2.1 The Rectangular Coordinate Systems and Graphs

x -intercept is ( 4 , 0 ) ; ( 4 , 0 ) ; y- intercept is ( 0 , 3 ) . ( 0 , 3 ) .

125 = 5 5 125 = 5 5

( − 5 , 5 2 ) ( − 5 , 5 2 )

2.2 Linear Equations in One Variable

x = −5 x = −5

x = −3 x = −3

x = 10 3 x = 10 3

x = 1 x = 1

x = − 7 17 . x = − 7 17 . Excluded values are x = − 1 2 x = − 1 2 and x = − 1 3 . x = − 1 3 .

x = 1 3 x = 1 3

m = − 2 3 m = − 2 3

y = 4 x −3 y = 4 x −3

x + 3 y = 2 x + 3 y = 2

Horizontal line: y = 2 y = 2

Parallel lines: equations are written in slope-intercept form.

y = 5 x + 3 y = 5 x + 3

2.3 Models and Applications

C = 2.5 x + 3 , 650 C = 2.5 x + 3 , 650

L = 37 L = 37 cm, W = 18 W = 18 cm

2.4 Complex Numbers

−24 = 0 + 2 i 6 −24 = 0 + 2 i 6

( 3 −4 i ) − ( 2 + 5 i ) = 1 −9 i ( 3 −4 i ) − ( 2 + 5 i ) = 1 −9 i

5 2 − i 5 2 − i

18 + i 18 + i

−3 −4 i −3 −4 i

2.5 Quadratic Equations

( x − 6 ) ( x + 1 ) = 0 ; x = 6 , x = − 1 ( x − 6 ) ( x + 1 ) = 0 ; x = 6 , x = − 1

( x −7 ) ( x + 3 ) = 0 , ( x −7 ) ( x + 3 ) = 0 , x = 7 , x = 7 , x = −3. x = −3.

( x + 5 ) ( x −5 ) = 0 , ( x + 5 ) ( x −5 ) = 0 , x = −5 , x = −5 , x = 5. x = 5.

( 3 x + 2 ) ( 4 x + 1 ) = 0 , ( 3 x + 2 ) ( 4 x + 1 ) = 0 , x = − 2 3 , x = − 2 3 , x = − 1 4 x = − 1 4

x = 0 , x = −10 , x = −1 x = 0 , x = −10 , x = −1

x = 4 ± 5 x = 4 ± 5

x = 3 ± 22 x = 3 ± 22

x = − 2 3 , x = − 2 3 , x = 1 3 x = 1 3

2.6 Other Types of Equations

{ −1 } { −1 }

0 , 0 , 1 2 , 1 2 , − 1 2 − 1 2

1 ; 1 ; extraneous solution − 2 9 − 2 9

−2 ; −2 ; extraneous solution −1 −1

−1 , −1 , 3 2 3 2

−3 , 3 , − i , i −3 , 3 , − i , i

2 , 12 2 , 12

−1 , −1 , 0 0 is not a solution.

2.7 Linear Inequalities and Absolute Value Inequalities

[ −3 , 5 ] [ −3 , 5 ]

( − ∞ , −2 ) ∪ [ 3 , ∞ ) ( − ∞ , −2 ) ∪ [ 3 , ∞ )

x < 1 x < 1

x ≥ −5 x ≥ −5

( 2 , ∞ ) ( 2 , ∞ )

[ − 3 14 , ∞ ) [ − 3 14 , ∞ )

6 < x ≤ 9 or ( 6 , 9 ] 6 < x ≤ 9 or ( 6 , 9 ]

( − 1 8 , 1 2 ) ( − 1 8 , 1 2 )

| x −2 | ≤ 3 | x −2 | ≤ 3

k ≤ 1 k ≤ 1 or k ≥ 7 ; k ≥ 7 ; in interval notation, this would be ( − ∞ , 1 ] ∪ [ 7 , ∞ ) . ( − ∞ , 1 ] ∪ [ 7 , ∞ ) .

2.1 Section Exercises

Answers may vary. Yes. It is possible for a point to be on the x -axis or on the y -axis and therefore is considered to NOT be in one of the quadrants.

The y -intercept is the point where the graph crosses the y -axis.

The x- intercept is ( 2 , 0 ) ( 2 , 0 ) and the y -intercept is ( 0 , 6 ) . ( 0 , 6 ) .

The x- intercept is ( 2 , 0 ) ( 2 , 0 ) and the y -intercept is ( 0 , −3 ) . ( 0 , −3 ) .

The x- intercept is ( 3 , 0 ) ( 3 , 0 ) and the y -intercept is ( 0 , 9 8 ) . ( 0 , 9 8 ) .

y = 4 − 2 x y = 4 − 2 x

y = 5 − 2 x 3 y = 5 − 2 x 3

y = 2 x − 4 5 y = 2 x − 4 5

d = 74 d = 74

d = 36 = 6 d = 36 = 6

d ≈ 62.97 d ≈ 62.97

( 3 , − 3 2 ) ( 3 , − 3 2 )

( 2 , −1 ) ( 2 , −1 )

( 0 , 0 ) ( 0 , 0 )

y = 0 y = 0

not collinear

A: ( −3 , 2 ) , B: ( 1 , 3 ) , C: ( 4 , 0 ) A: ( −3 , 2 ) , B: ( 1 , 3 ) , C: ( 4 , 0 )

d = 8.246 d = 8.246

d = 5 d = 5

( −3 , 4 ) ( −3 , 4 )

x = 0 y = −2 x = 0 y = −2

x = 0.75 y = 0 x = 0.75 y = 0

x = − 1.667 y = 0 x = − 1.667 y = 0

15 − 11.2 = 3.8 mi 15 − 11.2 = 3.8 mi shorter

6 .0 42 6 .0 42

Midpoint of each diagonal is the same point ( 2 , –2 ) ( 2 , –2 ) . Note this is a characteristic of rectangles, but not other quadrilaterals.

2.2 Section Exercises

It means they have the same slope.

The exponent of the x x variable is 1. It is called a first-degree equation.

If we insert either value into the equation, they make an expression in the equation undefined (zero in the denominator).

x = 2 x = 2

x = 2 7 x = 2 7

x = 6 x = 6

x = 3 x = 3

x = −14 x = −14

x ≠ −4 ; x ≠ −4 ; x = −3 x = −3

x ≠ 1 ; x ≠ 1 ; when we solve this we get x = 1 , x = 1 , which is excluded, therefore NO solution

x ≠ 0 ; x ≠ 0 ; x = − 5 2 x = − 5 2

y = − 4 5 x + 14 5 y = − 4 5 x + 14 5

y = − 3 4 x + 2 y = − 3 4 x + 2

y = 1 2 x + 5 2 y = 1 2 x + 5 2

y = −3 x − 5 y = −3 x − 5

y = 7 y = 7

y = −4 y = −4

8 x + 5 y = 7 8 x + 5 y = 7

Perpendicular

m = − 9 7 m = − 9 7

m = 3 2 m = 3 2

m 1 = − 1 3 , m 2 = 3 ; Perpendicular . m 1 = − 1 3 , m 2 = 3 ; Perpendicular .

y = 0.245 x − 45.662. y = 0.245 x − 45.662. Answers may vary. y min = −50 , y max = −40 y min = −50 , y max = −40

y = − 2.333 x + 6.667. y = − 2.333 x + 6.667. Answers may vary. y min = −10 , y max = 10 y min = −10 , y max = 10

y = − A B x + C B y = − A B x + C B

The slope for ( −1 , 1 ) to ( 0 , 4 ) is 3. The slope for ( −1 , 1 ) to ( 2 , 0 ) is − 1 3 . The slope for ( 2 , 0 ) to ( 3 , 3 ) is 3. The slope for ( 0 , 4 ) to ( 3 , 3 ) is − 1 3 . The slope for ( −1 , 1 ) to ( 0 , 4 ) is 3. The slope for ( −1 , 1 ) to ( 2 , 0 ) is − 1 3 . The slope for ( 2 , 0 ) to ( 3 , 3 ) is 3. The slope for ( 0 , 4 ) to ( 3 , 3 ) is − 1 3 .

Yes they are perpendicular.

2.3 Section Exercises

Answers may vary. Possible answers: We should define in words what our variable is representing. We should declare the variable. A heading.

2 , 000 − x 2 , 000 − x

v + 10 v + 10

Ann: 23 ; 23 ; Beth: 46 46

20 + 0.05 m 20 + 0.05 m

90 + 40 P 90 + 40 P

50 , 000 − x 50 , 000 − x

She traveled for 2 h at 20 mi/h, or 40 miles.

$5,000 at 8% and $15,000 at 12%

B = 100 + .05 x B = 100 + .05 x

R = 9 R = 9

r = 4 5 r = 4 5 or 0.8

W = P − 2 L 2 = 58 − 2 ( 15 ) 2 = 14 W = P − 2 L 2 = 58 − 2 ( 15 ) 2 = 14

f = p q p + q = 8 ( 13 ) 8 + 13 = 104 21 f = p q p + q = 8 ( 13 ) 8 + 13 = 104 21

m = − 5 4 m = − 5 4

h = 2 A b 1 + b 2 h = 2 A b 1 + b 2

length = 360 ft; width = 160 ft

A = 88 in . 2 A = 88 in . 2

h = V π r 2 h = V π r 2

r = V π h r = V π h

C = 12 π C = 12 π

2.4 Section Exercises

Add the real parts together and the imaginary parts together.

Possible answer: i i times i i equals -1, which is not imaginary.

−8 + 2 i −8 + 2 i

14 + 7 i 14 + 7 i

− 23 29 + 15 29 i − 23 29 + 15 29 i

8 − i 8 − i

−11 + 4 i −11 + 4 i

2 −5 i 2 −5 i

6 + 15 i 6 + 15 i

−16 + 32 i −16 + 32 i

−4 −7 i −4 −7 i

2 − 2 3 i 2 − 2 3 i

4 − 6 i 4 − 6 i

2 5 + 11 5 i 2 5 + 11 5 i

1 + i 3 1 + i 3

( 3 2 + 1 2 i ) 6 = −1 ( 3 2 + 1 2 i ) 6 = −1

5 −5 i 5 −5 i

9 2 − 9 2 i 9 2 − 9 2 i

2.5 Section Exercises

It is a second-degree equation (the highest variable exponent is 2).

We want to take advantage of the zero property of multiplication in the fact that if a ⋅ b = 0 a ⋅ b = 0 then it must follow that each factor separately offers a solution to the product being zero: a = 0 o r b = 0. a = 0 o r b = 0.

One, when no linear term is present (no x term), such as x 2 = 16. x 2 = 16. Two, when the equation is already in the form ( a x + b ) 2 = d . ( a x + b ) 2 = d .

x = 6 , x = 6 , x = 3 x = 3

x = − 5 2 , x = − 5 2 , x = − 1 3 x = − 1 3

x = 5 , x = 5 , x = −5 x = −5

x = − 3 2 , x = − 3 2 , x = 3 2 x = 3 2

x = −2 , 3 x = −2 , 3

x = 0 , x = 0 , x = − 3 7 x = − 3 7

x = −6 , x = −6 , x = 6 x = 6

x = 6 , x = 6 , x = −4 x = −4

x = 1 , x = 1 , x = −2 x = −2

x = −2 , x = −2 , x = 11 x = 11

z = 2 3 , z = 2 3 , z = − 1 2 z = − 1 2

x = 3 ± 17 4 x = 3 ± 17 4

One rational

Two real; rational

x = − 1 ± 17 2 x = − 1 ± 17 2

x = 5 ± 13 6 x = 5 ± 13 6

x = − 1 ± 17 8 x = − 1 ± 17 8

x ≈ 0.131 x ≈ 0.131 and x ≈ 2.535 x ≈ 2.535

x ≈ − 6.7 x ≈ − 6.7 and x ≈ 1.7 x ≈ 1.7

a x 2 + b x + c = 0 x 2 + b a x = − c a x 2 + b a x + b 2 4 a 2 = − c a + b 4 a 2 ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 x + b 2 a = ± b 2 − 4 a c 4 a 2 x = − b ± b 2 − 4 a c 2 a a x 2 + b x + c = 0 x 2 + b a x = − c a x 2 + b a x + b 2 4 a 2 = − c a + b 4 a 2 ( x + b 2 a ) 2 = b 2 − 4 a c 4 a 2 x + b 2 a = ± b 2 − 4 a c 4 a 2 x = − b ± b 2 − 4 a c 2 a

x ( x + 10 ) = 119 ; x ( x + 10 ) = 119 ; 7 ft. and 17 ft.

maximum at x = 70 x = 70

The quadratic equation would be ( 100 x −0.5 x 2 ) − ( 60 x + 300 ) = 300. ( 100 x −0.5 x 2 ) − ( 60 x + 300 ) = 300. The two values of x x are 20 and 60.

2.6 Section Exercises

This is not a solution to the radical equation, it is a value obtained from squaring both sides and thus changing the signs of an equation which has caused it not to be a solution in the original equation.

He or she is probably trying to enter negative 9, but taking the square root of −9 −9 is not a real number. The negative sign is in front of this, so your friend should be taking the square root of 9, cubing it, and then putting the negative sign in front, resulting in −27. −27.

A rational exponent is a fraction: the denominator of the fraction is the root or index number and the numerator is the power to which it is raised.

x = 81 x = 81

x = 17 x = 17

x = 8 , x = 27 x = 8 , x = 27

x = −2 , 1 , −1 x = −2 , 1 , −1

y = 0 , 3 2 , − 3 2 y = 0 , 3 2 , − 3 2

m = 1 , −1 m = 1 , −1

x = 2 5 , ±3 i x = 2 5 , ±3 i

x = 32 x = 32

t = 44 3 t = 44 3

x = −2 x = −2

x = 4 , −4 3 x = 4 , −4 3

x = − 5 4 , 7 4 x = − 5 4 , 7 4

x = 3 , −2 x = 3 , −2

x = 1 , −1 , 3 , -3 x = 1 , −1 , 3 , -3

x = 2 , −2 x = 2 , −2

x = 1 , 5 x = 1 , 5

x ≥ 0 x ≥ 0

x = 4 , 6 , −6 , −8 x = 4 , 6 , −6 , −8

2.7 Section Exercises

When we divide both sides by a negative it changes the sign of both sides so the sense of the inequality sign changes.

( − ∞ , ∞ ) ( − ∞ , ∞ )

We start by finding the x -intercept, or where the function = 0. Once we have that point, which is ( 3 , 0 ) , ( 3 , 0 ) , we graph to the right the straight line graph y = x −3 , y = x −3 , and then when we draw it to the left we plot positive y values, taking the absolute value of them.

( − ∞ , 3 4 ] ( − ∞ , 3 4 ]

[ − 13 2 , ∞ ) [ − 13 2 , ∞ )

( − ∞ , 3 ) ( − ∞ , 3 )

( − ∞ , − 37 3 ] ( − ∞ , − 37 3 ]

All real numbers ( − ∞ , ∞ ) ( − ∞ , ∞ )

( − ∞ , − 10 3 ) ∪ ( 4 , ∞ ) ( − ∞ , − 10 3 ) ∪ ( 4 , ∞ )

( − ∞ , −4 ] ∪ [ 8 , + ∞ ) ( − ∞ , −4 ] ∪ [ 8 , + ∞ )

No solution

( −5 , 11 ) ( −5 , 11 )

[ 6 , 12 ] [ 6 , 12 ]

[ −10 , 12 ] [ −10 , 12 ]

x > − 6 and x > − 2 Take the intersection of two sets . x > − 2 , ( − 2 , + ∞ ) x > − 6 and x > − 2 Take the intersection of two sets . x > − 2 , ( − 2 , + ∞ )

x < − 3 or x ≥ 1 Take the union of the two sets . ( − ∞ , − 3 ) ∪ [ 1 , ∞ ) x < − 3 or x ≥ 1 Take the union of the two sets . ( − ∞ , − 3 ) ∪ [ 1 , ∞ )

( − ∞ , −1 ) ∪ ( 3 , ∞ ) ( − ∞ , −1 ) ∪ ( 3 , ∞ )

[ −11 , −3 ] [ −11 , −3 ]

It is never less than zero. No solution.

Where the blue line is above the orange line; point of intersection is x = − 3. x = − 3.

( − ∞ , −3 ) ( − ∞ , −3 )

Where the blue line is above the orange line; always. All real numbers.

( − ∞ , − ∞ ) ( − ∞ , − ∞ )

( −1 , 3 ) ( −1 , 3 )

( − ∞ , 4 ) ( − ∞ , 4 )

{ x | x < 6 } { x | x < 6 }

{ x | −3 ≤ x < 5 } { x | −3 ≤ x < 5 }

( −2 , 1 ] ( −2 , 1 ]

( − ∞ , 4 ] ( − ∞ , 4 ]

Where the blue is below the orange; always. All real numbers. ( − ∞ , + ∞ ) . ( − ∞ , + ∞ ) .

Where the blue is below the orange; ( 1 , 7 ) . ( 1 , 7 ) .

x = 2 , − 4 5 x = 2 , − 4 5

( −7 , 5 ] ( −7 , 5 ]

80 ≤ T ≤ 120 1 , 600 ≤ 20 T ≤ 2 , 400 80 ≤ T ≤ 120 1 , 600 ≤ 20 T ≤ 2 , 400

[ 1 , 600 , 2 , 400 ] [ 1 , 600 , 2 , 400 ]

Review Exercises

x -intercept: ( 3 , 0 ) ; ( 3 , 0 ) ; y -intercept: ( 0 , −4 ) ( 0 , −4 )

y = 5 3 x + 4 y = 5 3 x + 4

72 = 6 2 72 = 6 2

620.097 620.097

midpoint is ( 2 , 23 2 ) ( 2 , 23 2 )

x = 4 x = 4

x = 12 7 x = 12 7

y = 1 6 x + 4 3 y = 1 6 x + 4 3

y = 2 3 x + 6 y = 2 3 x + 6

females 17, males 56

x = − 3 4 ± i 47 4 x = − 3 4 ± i 47 4

horizontal component −2 ; −2 ; vertical component −1 −1

7 + 11 i 7 + 11 i

−16 − 30 i −16 − 30 i

−4 − i 10 −4 − i 10

x = 7 − 3 i x = 7 − 3 i

x = −1 , −5 x = −1 , −5

x = 0 , 9 7 x = 0 , 9 7

x = 10 , −2 x = 10 , −2

x = − 1 ± 5 4 x = − 1 ± 5 4

x = 2 5 , − 1 3 x = 2 5 , − 1 3

x = 5 ± 2 7 x = 5 ± 2 7

x = 0 , 256 x = 0 , 256

x = 0 , ± 2 x = 0 , ± 2

x = 11 2 , −17 2 x = 11 2 , −17 2

[ − 10 3 , 2 ] [ − 10 3 , 2 ]

( − 4 3 , 1 5 ) ( − 4 3 , 1 5 )

Where the blue is below the orange line; point of intersection is x = 3.5. x = 3.5.

( 3.5 , ∞ ) ( 3.5 , ∞ )

Practice Test

y = 3 2 x + 2 y = 3 2 x + 2

( 0 , −3 ) ( 0 , −3 ) ( 4 , 0 ) ( 4 , 0 )

( − ∞ , 9 ] ( − ∞ , 9 ]

x = −15 x = −15

x ≠ −4 , 2 ; x ≠ −4 , 2 ; x = − 5 2 , 1 x = − 5 2 , 1

x = 3 ± 3 2 x = 3 ± 3 2

( −4 , 1 ) ( −4 , 1 )

y = −5 9 x − 2 9 y = −5 9 x − 2 9

y = 5 2 x − 4 y = 5 2 x − 4

5 13 − 14 13 i 5 13 − 14 13 i

x = 2 , − 4 3 x = 2 , − 4 3

x = 1 2 ± 2 2 x = 1 2 ± 2 2

x = 1 2 , 2 , −2 x = 1 2 , 2 , −2

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Book title: Algebra and Trigonometry
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Answer Key Chapter 2
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2.2 Models and Applications
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