Solve the inequality 12p 7 139

Elementary Algebra

SENIOR CONTRIBUTING AUTHORS LYNN MARECEK, SANTA ANA COLLEGE MARYANNE ANTHONY-SMITH, FORMERLY OF SANTA ANA COLLEGE

OpenStax Rice University 6100 Main Street MS-375 Houston, Texas 77005 To learn more about OpenStax, visit https://openstax.org. Individual print copies and bulk orders can be purchased through our website. ©2017 Rice University. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0). Under this license, any user of this textbook or the textbook contents herein must provide proper attribution as follows:

- If you redistribute this textbook in a digital format (including but not limited to PDF and HTML), then you must retain on every page the following attribution: “Download for free at https://openstax.org/details/books/elementary-algebra.”

- If you redistribute this textbook in a print format, then you must include on every physical page the following attribution: “Download for free at https://openstax.org/details/books/elementary-algebra.”

- If you redistribute part of this textbook, then you must retain in every digital format page view (including but not limited to PDF and HTML) and on every physical printed page the following attribution: “Download for free at https://openstax.org/details/books/elementary-algebra.”

- If you use this textbook as a bibliographic reference, please include https://openstax.org/details/books/elementary-algebra in your citation.

For questions regarding this licensing, please contact support@openstax.org. Trademarks The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, OpenStax CNX logo, OpenStax Tutor name, Openstax Tutor logo, Connexions name, Connexions logo, Rice University name, and Rice University logo are not subject to the license and may not be reproduced without the prior and express written consent of Rice University. PRINT BOOK ISBN-10 0-9986257-1-X PRINT BOOK ISBN-13 978-0-9986257-1-3 PDF VERSION ISBN-10 1-947172-25-5 PDF VERSION ISBN-13 978-1-947172-25-8 Revision Number EA-2017-001(06/17)-LC Original Publication Year 2017

OPENSTAX OpenStax provides free, peer-reviewed, openly licensed textbooks for introductory college and Advanced Placement® courses and low-cost, personalized courseware that helps students learn. A nonprofit ed tech initiative based at Rice University, we’re committed to helping students access the tools they need to complete their courses and meet their educational goals.

RICE UNIVERSITY OpenStax, OpenStax CNX, and OpenStax Tutor are initiatives of Rice University. As a leading research university with a distinctive commitment to undergraduate education, Rice University aspires to path-breaking research, unsurpassed teaching, and contributions to the betterment of our world. It seeks to fulfill this mission by cultivating a diverse community of learning and discovery that produces leaders across the spectrum of human endeavor.

FOUNDATION SUPPORT OpenStax is grateful for the tremendous support of our sponsors. Without their strong engagement, the goal of free access to high-quality textbooks would remain just a dream.

Laura and John Arnold Foundation (LJAF) actively seeks opportunities to invest in organizations and thought leaders that have a sincere interest in implementing fundamental changes that not only yield immediate gains, but also repair broken systems for future generations. LJAF currently focuses its strategic investments on education, criminal justice, research integrity, and public accountability.

The William and Flora Hewlett Foundation has been making grants since 1967 to help solve social and environmental problems at home and around the world. The Foundation concentrates its resources on activities in education, the environment, global development and population, performing arts, and philanthropy, and makes grants to support disadvantaged communities in the San Francisco Bay Area. Calvin K. Kazanjian was the founder and president of Peter Paul (Almond Joy), Inc. He firmly believed that the more people understood about basic economics the happier and more prosperous they would be. Accordingly, he established the Calvin K. Kazanjian Economics Foundation Inc, in 1949 as a philanthropic, nonpolitical educational organization to support efforts that enhanced economic understanding.

Guided by the belief that every life has equal value, the Bill & Melinda Gates Foundation works to help all people lead healthy, productive lives. In developing countries, it focuses on improving people’s health with vaccines and other life-saving tools and giving them the chance to lift themselves out of hunger and extreme poverty. In the United States, it seeks to significantly improve education so that all young people have the opportunity to reach their full potential. Based in Seattle, Washington, the foundation is led by CEO Jeff Raikes and Co-chair William H. Gates Sr., under the direction of Bill and Melinda Gates and Warren Buffett. The Maxfield Foundation supports projects with potential for high impact in science, education, sustainability, and other areas of social importance.

Our mission at The Michelson 20MM Foundation is to grow access and success by eliminating unnecessary hurdles to affordability. We support the creation, sharing, and proliferation of more effective, more affordable educational content by leveraging disruptive technologies, open educational resources, and new models for collaboration between for-profit, nonprofit, and public entities. The Bill and Stephanie Sick Fund supports innovative projects in the areas of Education, Art, Science and Engineering.

Access. The future of education.

OpenStax.org

I like free textbooks and I cannot lie.

Give $5 or more to OpenStax and we’ll send you a sticker! OpenStax is a nonprofit initiative, which means that that every dollar you give helps us maintain and grow our library of free textbooks.

If you have a few dollars to spare, visit OpenStax.org/give to donate. We’ll send you an OpenStax sticker to thank you for your support!

1

2

3

4

5

6

Table of Contents

Preface 1

Foundations 5 1.1 Introduction to Whole Numbers 5 1.2 Use the Language of Algebra 21 1.3 Add and Subtract Integers 40 1.4 Multiply and Divide Integers 61 1.5 Visualize Fractions 76 1.6 Add and Subtract Fractions 92 1.7 Decimals 107 1.8 The Real Numbers 126 1.9 Properties of Real Numbers 142 1.10 Systems of Measurement 160

Solving Linear Equations and Inequalities 197 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality 197 2.2 Solve Equations using the Division and Multiplication Properties of Equality 212 2.3 Solve Equations with Variables and Constants on Both Sides 226 2.4 Use a General Strategy to Solve Linear Equations 236 2.5 Solve Equations with Fractions or Decimals 249 2.6 Solve a Formula for a Specific Variable 260 2.7 Solve Linear Inequalities 270

Math Models 295 3.1 Use a Problem-Solving Strategy 295 3.2 Solve Percent Applications 312 3.3 Solve Mixture Applications 330 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem 346 3.5 Solve Uniform Motion Applications 369 3.6 Solve Applications with Linear Inequalities 382

Graphs 403 4.1 Use the Rectangular Coordinate System 403 4.2 Graph Linear Equations in Two Variables 424 4.3 Graph with Intercepts 444 4.4 Understand Slope of a Line 459 4.5 Use the Slope–Intercept Form of an Equation of a Line 486 4.6 Find the Equation of a Line 512 4.7 Graphs of Linear Inequalities 530

Systems of Linear Equations 565 5.1 Solve Systems of Equations by Graphing 565 5.2 Solve Systems of Equations by Substitution 586 5.3 Solve Systems of Equations by Elimination 602 5.4 Solve Applications with Systems of Equations 617 5.5 Solve Mixture Applications with Systems of Equations 635 5.6 Graphing Systems of Linear Inequalities 648

Polynomials 673 6.1 Add and Subtract Polynomials 673 6.2 Use Multiplication Properties of Exponents 687 6.3 Multiply Polynomials 701 6.4 Special Products 717 6.5 Divide Monomials 730 6.6 Divide Polynomials 748 6.7 Integer Exponents and Scientific Notation 760

7

8

9

10

Factoring 789 7.1 Greatest Common Factor and Factor by Grouping 789 7.2 Factor Quadratic Trinomials with Leading Coefficient 1 803 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1 816 7.4 Factor Special Products 834 7.5 General Strategy for Factoring Polynomials 850 7.6 Quadratic Equations 861

Rational Expressions and Equations 883 8.1 Simplify Rational Expressions 883 8.2 Multiply and Divide Rational Expressions 901 8.3 Add and Subtract Rational Expressions with a Common Denominator 914 8.4 Add and Subtract Rational Expressions with Unlike Denominators 923 8.5 Simplify Complex Rational Expressions 937 8.6 Solve Rational Equations 950 8.7 Solve Proportion and Similar Figure Applications 965 8.8 Solve Uniform Motion and Work Applications 981 8.9 Use Direct and Inverse Variation 991

Roots and Radicals 1013 9.1 Simplify and Use Square Roots 1013 9.2 Simplify Square Roots 1023 9.3 Add and Subtract Square Roots 1036 9.4 Multiply Square Roots 1046 9.5 Divide Square Roots 1060 9.6 Solve Equations with Square Roots 1074 9.7 Higher Roots 1091 9.8 Rational Exponents 1107

Quadratic Equations 1137 10.1 Solve Quadratic Equations Using the Square Root Property 1137 10.2 Solve Quadratic Equations by Completing the Square 1149 10.3 Solve Quadratic Equations Using the Quadratic Formula 1165 10.4 Solve Applications Modeled by Quadratic Equations 1179 10.5 Graphing Quadratic Equations 1190

Index 1309

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

PREFACE

Welcome to Elementary Algebra, an OpenStax resource. This textbook was written to increase student access to high- quality learning materials, maintaining highest standards of academic rigor at little to no cost.

About OpenStax OpenStax is a nonprofit based at Rice University, and it’s our mission to improve student access to education. Our first openly licensed college textbook was published in 2012, and our library has since scaled to over 25 books for college and AP courses used by hundreds of thousands of students. Our adaptive learning technology, designed to improve learning outcomes through personalized educational paths, is being piloted in college courses throughout the country. Through our partnerships with philanthropic foundations and our alliance with other educational resource organizations, OpenStax is breaking down the most common barriers to learning and empowering students and instructors to succeed.

About OpenStax Resources Customization Elementary Algebra is licensed under a Creative Commons Attribution 4.0 International (CC BY) license, which means that you can distribute, remix, and build upon the content, as long as you provide attribution to OpenStax and its content contributors. Because our books are openly licensed, you are free to use the entire book or pick and choose the sections that are most relevant to the needs of your course. Feel free to remix the content by assigning your students certain chapters and sections in your syllabus, in the order that you prefer. You can even provide a direct link in your syllabus to the sections in the web view of your book. Instructors also have the option of creating a customized version of their OpenStax book. The custom version can be made available to students in low-cost print or digital form through their campus bookstore. Visit your book page on openstax.org for more information.

Errata All OpenStax textbooks undergo a rigorous review process. However, like any professional-grade textbook, errors sometimes occur. Since our books are web based, we can make updates periodically when deemed pedagogically necessary. If you have a correction to suggest, submit it through the link on your book page on openstax.org. Subject matter experts review all errata suggestions. OpenStax is committed to remaining transparent about all updates, so you will also find a list of past errata changes on your book page on openstax.org.

Format You can access this textbook for free in web view or PDF through openstax.org, and for a low cost in print.

About Elementary Algebra Elementary Algebra is designed to meet the scope and sequence requirements of a one-semester elementary algebra course. The book’s organization makes it easy to adapt to a variety of course syllabi. The text expands on the fundamental concepts of algebra while addressing the needs of students with diverse backgrounds and learning styles. Each topic builds upon previously developed material to demonstrate the cohesiveness and structure of mathematics.

Coverage and Scope Elementary Algebra follows a nontraditional approach in its presentation of content. Building on the content in Prealgebra, the material is presented as a sequence of small steps so that students gain confidence in their ability to succeed in the course. The order of topics was carefully planned to emphasize the logical progression through the course and to facilitate a thorough understanding of each concept. As new ideas are presented, they are explicitly related to previous topics.

Chapter 1: Foundations Chapter 1 reviews arithmetic operations with whole numbers, integers, fractions, and decimals, to give the student a solid base that will support their study of algebra. Chapter 2: Solving Linear Equations and Inequalities In Chapter 2, students learn to verify a solution of an equation, solve equations using the Subtraction and Addition Properties of Equality, solve equations using the Multiplication and Division Properties of Equality, solve equations with variables and constants on both sides, use a general strategy to solve linear equations, solve equations with fractions or decimals, solve a formula for a specific variable, and solve linear inequalities. Chapter 3: Math Models Once students have learned the skills needed to solve equations, they apply these skills in Chapter 3 to solve word and number problems. Chapter 4: Graphs Chapter 4 covers the rectangular coordinate system, which is the basis for most consumer graphs. Students learn to plot points on a rectangular coordinate system, graph linear equations in two variables, graph with intercepts,

Preface 1

understand slope of a line, use the slope-intercept form of an equation of a line, find the equation of a line, and create graphs of linear inequalities. Chapter 5: Systems of Linear Equations Chapter 5 covers solving systems of equations by graphing, substitution, and elimination; solving applications with systems of equations, solving mixture applications with systems of equations, and graphing systems of linear inequalities. Chapter 6: Polynomials In Chapter 6, students learn how to add and subtract polynomials, use multiplication properties of exponents, multiply polynomials, use special products, divide monomials and polynomials, and understand integer exponents and scientific notation. Chapter 7: Factoring In Chapter 7, students explore the process of factoring expressions and see how factoring is used to solve certain types of equations. Chapter 8: Rational Expressions and Equations In Chapter 8, students work with rational expressions, solve rational equations, and use them to solve problems in a variety of applications. Chapter 9: Roots and Radical In Chapter 9, students are introduced to and learn to apply the properties of square roots, and extend these concepts to higher order roots and rational exponents. Chapter 10: Quadratic Equations In Chapter 10, students study the properties of quadratic equations, solve and graph them. They also learn how to apply them as models of various situations.

All chapters are broken down into multiple sections, the titles of which can be viewed in the Table of Contents.

Key Features and Boxes Examples Each learning objective is supported by one or more worked examples that demonstrate the problem-solving approaches that students must master. Typically, we include multiple Examples for each learning objective to model different approaches to the same type of problem, or to introduce similar problems of increasing complexity. All Examples follow a simple two- or three-part format. First, we pose a problem or question. Next, we demonstrate the solution, spelling out the steps along the way. Finally (for select Examples), we show students how to check the solution. Most Examples are written in a two-column format, with explanation on the left and math on the right to mimic the way that instructors “talk through” examples as they write on the board in class. Be Prepared! Each section, beginning with Section 2.1, starts with a few “Be Prepared!” exercises so that students can determine if they have mastered the prerequisite skills for the section. Reference is made to specific Examples from previous sections so students who need further review can easily find explanations. Answers to these exercises can be found in the supplemental resources that accompany this title. Try It

The Try It feature includes a pair of exercises that immediately follow an Example, providing the student with an immediate opportunity to solve a similar problem. In the Web View version of the text, students can click an Answer link directly below the question to check their understanding. In the PDF, answers to the Try It exercises are located in the Answer Key. How To

How To feature typically follows the Try It exercises and outlines the series of steps for how to solve the problem in the preceding Example. Media

The Media icon appears at the conclusion of each section, just prior to the Self Check. This icon marks a list of links to online video tutorials that reinforce the concepts and skills introduced in the section. Disclaimer: While we have selected tutorials that closely align to our learning objectives, we did not produce these tutorials, nor were they specifically produced or tailored to accompany Elementary Algebra. Self Check The Self Check includes the learning objectives for the section so that students can self-assess their mastery and make concrete plans to improve.

2 Preface

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

Art Program Elementary Algebra contains many figures and illustrations. Art throughout the text adheres to a clear, understated style, drawing the eye to the most important information in each figure while minimizing visual distractions.

Section Exercises and Chapter Review Section Exercises Each section of every chapter concludes with a well-rounded set of exercises that can be assigned as homework or used selectively for guided practice. Exercise sets are named Practice Makes Perfect to encourage completion of homework assignments.

Exercises correlate to the learning objectives. This facilitates assignment of personalized study plans based on individual student needs. Exercises are carefully sequenced to promote building of skills. Values for constants and coefficients were chosen to practice and reinforce arithmetic facts. Even and odd-numbered exercises are paired. Exercises parallel and extend the text examples and use the same instructions as the examples to help students easily recognize the connection. Applications are drawn from many everyday experiences, as well as those traditionally found in college math texts. Everyday Math highlights practical situations using the concepts from that particular section Writing Exercises are included in every exercise set to encourage conceptual understanding, critical thinking, and literacy.

Chapter Review Each chapter concludes with a review of the most important takeaways, as well as additional practice problems that students can use to prepare for exams.

Key Terms provide a formal definition for each bold-faced term in the chapter. Key Concepts summarize the most important ideas introduced in each section, linking back to the relevant Example(s) in case students need to review. Chapter Review Exercises include practice problems that recall the most important concepts from each section. Practice Test includes additional problems assessing the most important learning objectives from the chapter. Answer Key includes the answers to all Try It exercises and every other exercise from the Section Exercises, Chapter Review Exercises, and Practice Test.

Additional Resources Student and Instructor Resources We’ve compiled additional resources for both students and instructors, including Getting Started Guides, manipulative mathematics worksheets, Links to Literacy assignments, and an answer key to Be Prepared Exercises. Instructor resources require a verified instructor account, which can be requested on your openstax.org log-in. Take advantage of these resources to supplement your OpenStax book.

Partner Resources OpenStax Partners are our allies in the mission to make high-quality learning materials affordable and accessible to students and instructors everywhere. Their tools integrate seamlessly with our OpenStax titles at a low cost. To access the partner resources for your text, visit your book page on openstax.org.

About the Authors Senior Contributing Authors Lynn Marecek and MaryAnne Anthony-Smith have been teaching mathematics at Santa Ana College for many years and have worked together on several projects aimed at improving student learning in developmental math courses. They are the authors of Strategies for Success: Study Skills for the College Math Student.

Preface 3

Lynn Marecek, Santa Ana College Lynn Marecek has focused her career on meeting the needs of developmental math students. At Santa Ana College, she has been awarded the Distinguished Faculty Award, Innovation Award, and the Curriculum Development Award four times. She is a Coordinator of Freshman Experience Program, the Department Facilitator for Redesign, and a member of the Student Success and Equity Committee, and the Basic Skills Initiative Task Force. Lynn holds a bachelor’s degree from Valparaiso University and master’s degrees from Purdue University and National University. MaryAnne Anthony-Smith, Santa Ana College MaryAnne Anthony-Smith was a mathematics professor at Santa Ana College for 39 years, until her retirement in June, 2015. She has been awarded the Distinguished Faculty Award, as well as the Professional Development, Curriculum Development, and Professional Achievement awards. MaryAnne has served as department chair, acting dean, chair of the professional development committee, institutional researcher, and faculty coordinator on several state and federally- funded grants. She is the community college coordinator of California’s Mathematics Diagnostic Testing Project, a member of AMATYC’s Placement and Assessment Committee. She earned her bachelor’s degree from the University of California San Diego and master’s degrees from San Diego State and Pepperdine Universities.

Reviewers Jay Abramson, Arizona State University Bryan Blount, Kentucky Wesleyan College Gale Burtch, Ivy Tech Community College Tamara Carter, Texas A&M University Danny Clarke, Truckee Meadows Community College Michael Cohen, Hofstra University Christina Cornejo, Erie Community College Denise Cutler, Bay de Noc Community College Lance Hemlow, Raritan Valley Community College John Kalliongis, Saint Louis Iniversity Stephanie Krehl, Mid-South Community College Laurie Lindstrom, Bay de Noc Community College Beverly Mackie, Lone Star College System Allen Miller, Northeast Lakeview College Christian Roldán-Johnson, College of Lake County Community College Martha Sandoval-Martinez, Santa Ana College Gowribalan Vamadeva, University of Cincinnati Blue Ash College Kim Watts, North Lake College Libby Watts, Tidewater Community College Allen Wolmer, Atlantic Jewish Academy John Zarske, Santa Ana College

4 Preface

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

Figure 1.1 In order to be structurally sound, the foundation of a building must be carefully constructed.

Chapter Outline 1.1 Introduction to Whole Numbers 1.2 Use the Language of Algebra 1.3 Add and Subtract Integers 1.4 Multiply and Divide Integers 1.5 Visualize Fractions 1.6 Add and Subtract Fractions 1.7 Decimals 1.8 The Real Numbers 1.9 Properties of Real Numbers 1.10 Systems of Measurement

Introduction Just like a building needs a firm foundation to support it, your study of algebra needs to have a firm foundation. To ensure this, we begin this book with a review of arithmetic operations with whole numbers, integers, fractions, and decimals, so that you have a solid base that will support your study of algebra.

1.1 Introduction to Whole Numbers Learning Objectives By the end of this section, you will be able to:

Use place value with whole numbers Identify multiples and and apply divisibility tests Find prime factorizations and least common multiples

Be Prepared!

A more thorough introduction to the topics covered in this section can be found in Prealgebra in the chapters Whole Numbers and The Language of Algebra.

As we begin our study of elementary algebra, we need to refresh some of our skills and vocabulary. This chapter will focus on whole numbers, integers, fractions, decimals, and real numbers. We will also begin our use of algebraic notation and vocabulary.

FOUNDATIONS1

Chapter 1 Foundations 5

Use Place Value with Whole Numbers The most basic numbers used in algebra are the numbers we use to count objects in our world: 1, 2, 3, 4, and so on. These are called the counting numbers. Counting numbers are also called natural numbers. If we add zero to the counting numbers, we get the set of whole numbers.

Counting Numbers: 1, 2, 3, …

Whole Numbers: 0, 1, 2, 3, …

The notation “…” is called ellipsis and means “and so on,” or that the pattern continues endlessly. We can visualize counting numbers and whole numbers on a number line (see Figure 1.2).

Figure 1.2 The numbers on the number line get larger as they go from left to right, and smaller as they go from right to left. While this number line shows only the whole numbers 0 through 6, the numbers keep going without end.

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Number Line-Part 1” will help you develop a better understanding of the counting numbers and the whole numbers.

Our number system is called a place value system, because the value of a digit depends on its position in a number. Figure 1.3 shows the place values. The place values are separated into groups of three, which are called periods. The periods are ones, thousands, millions, billions, trillions, and so on. In a written number, commas separate the periods.

Figure 1.3 The number 5,278,194 is shown in the chart. The digit 5 is in the millions place. The digit 2 is in the hundred-thousands place. The digit 7 is in the ten-thousands place. The digit 8 is in the thousands place. The digit 1 is in the hundreds place. The digit 9 is in the tens place. The digit 4 is in the ones place.

EXAMPLE 1.1

In the number 63,407,218, find the place value of each digit:

ⓐ 7 ⓑ 0 ⓒ 1 ⓓ 6 ⓔ 3 Solution

Place the number in the place value chart:

6 Chapter 1 Foundations

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

ⓐ The 7 is in the thousands place. ⓑ The 0 is in the ten thousands place. ⓒ The 1 is in the tens place. ⓓ The 6 is in the ten-millions place. ⓔ The 3 is in the millions place.

TRY IT : : 1.1 For the number 27,493,615, find the place value of each digit:

ⓐ 2 ⓑ 1 ⓒ 4 ⓓ 7 ⓔ 5

TRY IT : : 1.2 For the number 519,711,641,328, find the place value of each digit:

ⓐ 9 ⓑ 4 ⓒ 2 ⓓ 6 ⓔ 7

When you write a check, you write out the number in words as well as in digits. To write a number in words, write the number in each period, followed by the name of the period, without the s at the end. Start at the left, where the periods have the largest value. The ones period is not named. The commas separate the periods, so wherever there is a comma in the number, put a comma between the words (see Figure 1.4). The number 74,218,369 is written as seventy-four million, two hundred eighteen thousand, three hundred sixty-nine.

Figure 1.4

EXAMPLE 1.2

Name the number 8,165,432,098,710 using words.

Solution Name the number in each period, followed by the period name.

HOW TO : : NAME A WHOLE NUMBER IN WORDS.

Start at the left and name the number in each period, followed by the period name. Put commas in the number to separate the periods. Do not name the ones period.

Step 1. Step 2. Step 3.

Chapter 1 Foundations 7

Put the commas in to separate the periods. So, 8, 165, 432, 098, 710 is named as eight trillion, one hundred sixty-five billion, four hundred thirty-two million, ninety-eight thousand, seven hundred ten.

TRY IT : : 1.3 Name the number 9, 258, 137, 904, 061 using words.

TRY IT : : 1.4 Name the number 17, 864, 325, 619, 004 using words.

We are now going to reverse the process by writing the digits from the name of the number. To write the number in digits, we first look for the clue words that indicate the periods. It is helpful to draw three blanks for the needed periods and then fill in the blanks with the numbers, separating the periods with commas.

EXAMPLE 1.3

Write nine billion, two hundred forty-six million, seventy-three thousand, one hundred eighty-nine as a whole number using digits.

Solution Identify the words that indicate periods. Except for the first period, all other periods must have three places. Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas. Then write the digits in each period.

The number is 9,246,073,189.

TRY IT : : 1.5

Write the number two billion, four hundred sixty-six million, seven hundred fourteen thousand, fifty-one as a whole number using digits.

HOW TO : : WRITE A WHOLE NUMBER USING DIGITS.

Identify the words that indicate periods. (Remember, the ones period is never named.) Draw three blanks to indicate the number of places needed in each period. Separate the periods by commas. Name the number in each period and place the digits in the correct place value position.

Step 1. Step 2.

Step 3.

8 Chapter 1 Foundations

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

TRY IT : : 1.6

Write the number eleven billion, nine hundred twenty-one million, eight hundred thirty thousand, one hundred six as a whole number using digits.

In 2013, the U.S. Census Bureau estimated the population of the state of New York as 19,651,127. We could say the population of New York was approximately 20 million. In many cases, you don’t need the exact value; an approximate number is good enough. The process of approximating a number is called rounding. Numbers are rounded to a specific place value, depending on how much accuracy is needed. Saying that the population of New York is approximately 20 million means that we rounded to the millions place.

EXAMPLE 1.4 HOW TO ROUND WHOLE NUMBERS

Round 23,658 to the nearest hundred.

Solution

TRY IT : : 1.7 Round to the nearest hundred: 17,852.

TRY IT : : 1.8 Round to the nearest hundred: 468,751.

HOW TO : : ROUND WHOLE NUMBERS.

Locate the given place value and mark it with an arrow. All digits to the left of the arrow do not change. Underline the digit to the right of the given place value. Is this digit greater than or equal to 5?

◦ Yes–add 1 to the digit in the given place value. ◦ No–do not change the digit in the given place value.

Replace all digits to the right of the given place value with zeros.

Step 1.

Step 2. Step 3.

Step 4.

Chapter 1 Foundations 9

EXAMPLE 1.5

Round 103,978 to the nearest:

ⓐ hundred ⓑ thousand ⓒ ten thousand Solution

ⓐ

Locate the hundreds place in 103,978.

Underline the digit to the right of the hundreds place.

Since 7 is greater than or equal to 5, add 1 to the 9. Replace all digits to the right of the hundreds place with zeros.

So, 104,000 is 103,978 rounded to the nearest hundred.

ⓑ

Locate the thousands place and underline the digit to the right of the thousands place.

Since 9 is greater than or equal to 5, add 1 to the 3. Replace all digits to the right of the hundreds place with zeros.

So, 104,000 is 103,978 rounded to the nearest thousand.

ⓒ

Locate the ten thousands place and underline the digit to the right of the ten thousands place.

Since 3 is less than 5, we leave the 0 as is, and then replace the digits to the right with zeros.

So, 100,000 is 103,978 rounded to the nearest ten thousand.

10 Chapter 1 Foundations

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

TRY IT : : 1.9 Round 206,981 to the nearest: ⓐ hundred ⓑ thousand ⓒ ten thousand.

TRY IT : : 1.10 Round 784,951 to the nearest: ⓐ hundred ⓑ thousand ⓒ ten thousand.

Identify Multiples and Apply Divisibility Tests The numbers 2, 4, 6, 8, 10, and 12 are called multiples of 2. A multiple of 2 can be written as the product of a counting number and 2.

Similarly, a multiple of 3 would be the product of a counting number and 3.

We could find the multiples of any number by continuing this process.

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Multiples” will help you develop a better understanding of multiples.

Table 1.4 shows the multiples of 2 through 9 for the first 12 counting numbers.

Counting Number 1 2 3 4 5 6 7 8 9 10 11 12

Multiples of 2 2 4 6 8 10 12 14 16 18 20 22 24

Multiples of 3 3 6 9 12 15 18 21 24 27 30 33 36

Multiples of 4 4 8 12 16 20 24 28 32 36 40 44 48

Multiples of 5 5 10 15 20 25 30 35 40 45 50 55 60

Multiples of 6 6 12 18 24 30 36 42 48 54 60 66 72

Multiples of 7 7 14 21 28 35 42 49 56 63 70 77 84

Multiples of 8 8 16 24 32 40 48 56 64 72 80 88 96

Multiples of 9 9 18 27 36 45 54 63 72 81 90 99 108

Multiples of 10 10 20 30 40 50 60 0 80 90 100 110 120

Table 1.4

Multiple of a Number

A number is a multiple of n if it is the product of a counting number and n.

Another way to say that 15 is a multiple of 3 is to say that 15 is divisible by 3. That means that when we divide 3 into 15, we get a counting number. In fact, 15 ÷ 3 is 5, so 15 is 5 · 3.

Divisible by a Number

If a number m is a multiple of n, then m is divisible by n.

Look at the multiples of 5 in Table 1.4. They all end in 5 or 0. Numbers with last digit of 5 or 0 are divisible by 5. Looking for other patterns in Table 1.4 that shows multiples of the numbers 2 through 9, we can discover the following divisibility tests:

Chapter 1 Foundations 11

Divisibility Tests

A number is divisible by: • 2 if the last digit is 0, 2, 4, 6, or 8. • 3 if the sum of the digits is divisible by 3. • 5 if the last digit is 5 or 0. • 6 if it is divisible by both 2 and 3. • 10 if it ends with 0.

EXAMPLE 1.6

Is 5,625 divisible by 2? By 3? By 5? By 6? By 10?

Solution Is 5,625 divisible by 2? Does it end in 0, 2, 4, 6, or 8? No.

5,625 is not divisible by 2.

Is 5,625 divisible by 3? What is the sum of the digits? 5 + 6 + 2 + 5 = 18 Is the sum divisible by 3? Yes. 5,625 is divisible by 3.

Is 5,625 divisible by 5 or 10? What is the last digit? It is 5. 5,625 is divisible by 5 but not by 10.

Is 5,625 divisible by 6? Is it divisible by both 2 and 3? No, 5,625 is not divisible by 2, so 5,625 is

not divisible by 6.

TRY IT : : 1.11 Determine whether 4,962 is divisible by 2, by 3, by 5, by 6, and by 10.

TRY IT : : 1.12 Determine whether 3,765 is divisible by 2, by 3, by 5, by 6, and by 10.

Find Prime Factorizations and Least Common Multiples In mathematics, there are often several ways to talk about the same ideas. So far, we’ve seen that if m is a multiple of n, we can say that m is divisible by n. For example, since 72 is a multiple of 8, we say 72 is divisible by 8. Since 72 is a multiple of 9, we say 72 is divisible by 9. We can express this still another way. Since 8 · 9 = 72, we say that 8 and 9 are factors of 72. When we write 72 = 8 · 9, we say we have factored 72.

Other ways to factor 72 are 1 · 72, 2 · 36, 3 · 24, 4 · 18, and 6 · 12. Seventy-two has many factors: 1, 2, 3, 4, 6, 8, 9, 12, 18, 36, and 72.

Factors

If a · b = m, then a and b are factors of m.

Some numbers, like 72, have many factors. Other numbers have only two factors.

12 Chapter 1 Foundations

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Model Multiplication and Factoring” will help you develop a better understanding of multiplication and factoring.

Prime Number and Composite Number

A prime number is a counting number greater than 1, whose only factors are 1 and itself. A composite number is a counting number that is not prime. A composite number has factors other than 1 and itself.

MANIPULATIVE MATHEMATICS

Doing the Manipulative Mathematics activity “Prime Numbers” will help you develop a better understanding of prime numbers.

The counting numbers from 2 to 19 are listed in Figure 1.5, with their factors. Make sure to agree with the “prime” or “composite” label for each!

Figure 1.5

The prime numbers less than 20 are 2, 3, 5, 7, 11, 13, 17, and 19. Notice that the only even prime number is 2. A composite number can be written as a unique product of primes. This is called the prime factorization of the number. Finding the prime factorization of a composite number will be useful later in this course.

Prime Factorization

The prime factorization of a number is the product of prime numbers that equals the number.

To find the prime factorization of a composite number, find any two factors of the number and use them to create two branches. If a factor is prime, that branch is complete. Circle that prime! If the factor is not prime, find two factors of the number and continue the process. Once all the branches have circled primes at the end, the factorization is complete. The composite number can now be written as a product of prime numbers.

EXAMPLE 1.7 HOW TO FIND THE PRIME FACTORIZATION OF A COMPOSITE NUMBER

Factor 48.

Solution

Chapter 1 Foundations 13

We say 2 · 2 · 2 · 2 · 3 is the prime factorization of 48. We generally write the primes in ascending order. Be sure to multiply the factors to verify your answer! If we first factored 48 in a different way, for example as 6 · 8, the result would still be the same. Finish the prime factorization and verify this for yourself.

TRY IT : : 1.13 Find the prime factorization of 80.

TRY IT : : 1.14 Find the prime factorization of 60.

EXAMPLE 1.8

Find the prime factorization of 252.

HOW TO : : FIND THE PRIME FACTORIZATION OF A COMPOSITE NUMBER.

Find two factors whose product is the given number, and use these numbers to create two branches. If a factor is prime, that branch is complete. Circle the prime, like a bud on the tree. If a factor is not prime, write it as the product of two factors and continue the process. Write the composite number as the product of all the circled primes.

Step 1.

Step 2. Step 3. Step 4.

14 Chapter 1 Foundations

This OpenStax book is available for free at http://cnx.org/content/col12116/1.2

Solution

Step 1. Find two factors whose product is 252. 12 and 21 are not prime.

Break 12 and 21 into two more factors. Continue until all primes are factored.

Step 2. Write 252 as the product of all the circled primes. 252 = 2 · 2 · 3 · 3 · 7

TRY IT : : 1.15 Find the prime factorization of 126.

TRY IT : : 1.16 Find the prime factorization of 294.

One of the reasons we look at multiples and primes is to use these techniques to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators. Two methods are used most often to find the least common multiple and we will look at both of them. The first method is the Listing Multiples Method. To find the least common multiple of 12 and 18, we list the first few multiples of 12 and 18:

Notice that some numbers appear in both lists. They are the common multiples of 12 and 18. We see that the first few common multiples of 12 and 18 are 36, 72, and 108. Since 36 is the smallest of the common multiples, we call it the least common multiple. We often use the abbreviation LCM.