MATH 021 introductory algebra third edition Third Edition

Introductory Algebra Third Edition September 1, 2019 Preface In 1996, the Brookdale Mathematics Department created an Introductory Algebra course for our non-STEM students. Math faculty consulted other departments with courses that had Introductory Algebra as a prerequisite, and created an algebra course that truly served many non-STEM disciplines. The Department selected a textbook that was as innovative as the course. However, it did not match our preferred order of topics, and it did not include some prerequisite topics requested by other departments, such as Probability and Statistics. For years math faculty created support curriculum in hopes of eventually developing our own textbook. We wanted to provide non-STEM students with a high-quality and academically rigorous book that supported our course learning outcomes, for free or at minimum cost. Finally, with the advent of Open Educational Resources (OER) and the support of Brookdale Community College, we decided to build a book that combined a remix of OER textbook definitions with our own curriculum. This text was piloted in the Spring 2017 semester by math faculty who volunteered to use the book and contribute suggestions. Since then, we received suggestions and edits from faculty who taught the course. This Third Edition reflects those suggestions and changes. We remain appreciative of the support and encouragement we received from our families, as well as Brookdale administrators, staff, and faculty. Barbara Tozzi, Ed.D. Linda C. Wang Brookdale Community College Mathematics Department September 1, 2019Acknowledgements This book is completed because of the support and encouragement we received from our families and our Brookdale family. We remain grateful to: Our families: Robert J. Tozzi, Bob Tozzi, Lisa, Clay, Carolina Marie and Lydia James Rogers Yung-Terng Wang, Jade P. Wang, Jay Muchnij, Raemin, Mariama, Mian and Reo Wang Our Brookdale family: Dr. Matthew Reed, Vice President for Learning Dr. Anoop Ahluwalia, STEM Institute Dean Ann Tickner-Jankowski, Director, Transfer Resources/Articulation Steven Chudnick, Director – Library and College Copyright Officer Valerie Bonilla and Amy Clark, Professors/Librarians and presenters of Intro to OER course Avis McMillon, Executive Director – College Relations Anthony LaGaipa, Graphic Artist/Design Coordinator The Brookdale Mathematics Department, past and present, including the following Math Faculty: Rebecca Rozario, Assistant Mathematics Department Chair Daniel Lopez and Nathalie Darden, MATH 021 Course Coordinators Pilot Instructors/Reviewers: Karina Aliaga, Nathalie Darden, Sally Mulvey, Linda C. Wang, and Arminda WeyTable of Contents Table of Contents Unit 1: Mathematical Expressions................................................................................................... 11 1.1 Problem Solving and Numerical Representations .................................................................. 11 Introduction ................................................................................................................................... 11 A Framework for Problem Solving ............................................................................................ 11 Practice Using Input-Output Tables .......................................................................................... 13 Conditional Rules ......................................................................................................................... 15 1.1 SUMMARY .............................................................................................................................. 17 1.2 Real Numbers and Translating Algebraic Expressions ........................................................... 18 Set Notation and Sets of Numbers ............................................................................................. 18 The Real Number Line ................................................................................................................. 19 Algebraic Expressions .................................................................................................................. 21 Translating Words into Mathematical Symbols ...................................................................... 22 Translating and Building Input-Output Tables ...................................................................... 24 1.2 SUMMARY .............................................................................................................................. 25 1.3 Rectangular Coordinate System and Graphs ........................................................................... 26 The Rectangular Coordinate Plane ............................................................................................ 26 Coordinates of a Point .................................................................................................................. 27 Plotting Points ............................................................................................................................... 27 Four Representations of a Rule................................................................................................... 28 Interpreting Points on a Graph ................................................................................................... 29 1.3 SUMMARY .............................................................................................................................. 31 1.4 Exponents, Fractions, and Square Root Expressions ............................................................... 33 Exponential Notation ................................................................................................................... 33 Simplifying Expressions with Exponents ................................................................................. 34 Simplifying Fractions ................................................................................................................... 36 Square Root Expressions ............................................................................................................. 37 1.4 SUMMARY .............................................................................................................................. 40 1.5 Order of Operations, Properties of Real Numbers, and Simplifying Expressions .............. 41 5Table of Contents Order of Operations...................................................................................................................... 41 Properties of Real Numbers ........................................................................................................ 42 The Difference between Terms and Factors ............................................................................. 45 Like Terms...................................................................................................................................... 46 1.5 SUMMARY .............................................................................................................................. 48 1.6 Polynomials and Evaluating Expressions ................................................................................. 49 Simplifying Polynomial Expressions ........................................................................................ 49 Adding, Subtracting and Multiplying Polynomials ............................................................... 50 Evaluating Expressions ................................................................................................................ 52 1.6 SUMMARY .............................................................................................................................. 54 1.7 Functions ....................................................................................................................................... 55 Definition of a Function .............................................................................................................. 55 Numerical Representation of a Function .................................................................................. 56 Graphical Representation of a Function ................................................................................... 57 The Vertical Line Test .................................................................................................................. 58 Function Notation ......................................................................................................................... 58 Interpreting Function Notation .................................................................................................. 60 Unit 2: Equations, Inequalities, and Linear Equations ................................................................. 64 2.1 Solving Linear Equations Three Ways ..................................................................................... 64 Equations ........................................................................................................................................ 64 Solving Equations Symbolically ................................................................................................ 65 Approximate versus Exact Answers ........................................................................................... 67 Translating Words into Equations ............................................................................................. 68 Solving Equations Numerically ................................................................................................. 69 Using Symbolic Techniques to Find f(x) when Given x......................................................... 71 Using Symbolic Techniques to Find x when Given f(x)......................................................... 71 Solving Equations Graphically................................................................................................... 72 2.1 SUMMARY .............................................................................................................................. 75 2.2 Solving Non-Linear Equations with Tables & Graphs; Solving Linear Equations with Parentheses ......................................................................................................................................... 76 Definitions of Linear and Non-Linear Equations.................................................................... 76 6Table of Contents Solving Non-Linear Equations with Tables & Graphs .......................................................... 77 Solving Linear Equations with Parentheses Symbolically .................................................... 79 Translating words into Equations with Parentheses .............................................................. 81 2.2 SUMMARY .............................................................................................................................. 83 2.3 Solving Linear Equations with Variable Terms on Both Sides Symbolically........................ 84 Summary: How to Solve Linear Equations with Variables on Both Sides ......................... 87 Solving Applications by using Equations ................................................................................ 88 2.3 SUMMARY .............................................................................................................................. 89 2.4 Solving Linear Equations with Fractions and Solving Proportions ....................................... 90 Solving Linear Equations with Fractions by using Steps ...................................................... 90 Solving Linear Equations with Fractions by Clearing Denominators ................................. 91 Solving Proportions ...................................................................................................................... 94 Recognizing the Difference between a Proportion and an Equation with Fractions ........ 96 2.4 SUMMARY .............................................................................................................................. 98 2.5 Inequalities, Line Graphs and Solving Linear Inequalities in One Variable ......................... 99 Linear Inequalities ........................................................................................................................ 99 Line Graphs.................................................................................................................................... 99 Solving Linear Inequalities in One Variable ......................................................................... 103 2.5 SUMMARY ............................................................................................................................ 107 2.6 Literal Equations ........................................................................................................................ 108 Variables that are written “In Terms Of” other Variables ................................................... 108 Solving a Formula ....................................................................................................................... 108 Subscripted Variables ................................................................................................................ 112 2.6 SUMMARY ............................................................................................................................ 113 2.7 Linear Functions: Intercepts ..................................................................................................... 114 General Form of a Linear Equation in Two Variables .......................................................... 115 Graphing a Line Using the Horizontal and Vertical Intercepts .......................................... 115 Interpreting Horizontal and Vertical Axis Intercepts in Applications .............................. 117 Using Function Notation to Find Horizontal and Vertical Axis Intercepts ...................... 118 Equations of Horizontal and Vertical Lines ........................................................................... 120 2.7 SUMMARY ............................................................................................................................ 124 7Table of Contents 2.8 Linear Functions: Slope and Rate of Change .......................................................................... 125 Introduction to Slope.................................................................................................................. 125 How to Find the Slope of a Line from the Graph .................................................................. 127 Formula for the Slope of a Line ................................................................................................ 129 Summary of Slope Notation and the Slope Formula ............................................................ 130 The Slope of a Horizontal Line and a Vertical Line .............................................................. 131 Applications with Slope as Rate of Change ........................................................................... 132 2.8 SUMMARY ............................................................................................................................ 135 2.9 Linear Functions: Slope-Intercept Form .................................................................................. 136 Finding the y-Intercept and Slope from a Table and Graph ............................................... 136 Writing an Equation of a Line in Slope-Intercept Form ....................................................... 138 Writing an Equation in Slope-Intercept Form, given the Slope and a Point on the Line 141 Writing an Equation in Slope-Intercept Form if given Two Points ................................... 142 Writing an Equation in Slope-Intercept Form if given the Line in General Form ........... 143 2.9 SUMMARY ............................................................................................................................ 144 Unit 3: Systems of Linear Equations, Statistics and Probability ................................................ 146 3.1 Linear Functions: Graphing ...................................................................................................... 146 Graphing a Linear Equation given the Slope and a Point on the Line .............................. 146 Graphing a Linear Equation given the Slope and the y-Intercept...................................... 147 Graphing a Line given the General Form of a Linear Equation .......................................... 149 Parallel Lines ............................................................................................................................... 150 3.1 SUMMARY ............................................................................................................................ 152 3.2 Linear Functions: Applications ................................................................................................ 153 Identifying the Slope and y-intercept and Interpreting their Meaning ............................ 153 Translating an Application into a Linear Function ............................................................... 154 Solving Applications using Linear Functions ........................................................................ 155 3.2 SUMMARY ............................................................................................................................ 158 3.3 Solving Systems of Equations using Graphs .......................................................................... 159 The Solution to a System of Linear Equations ....................................................................... 159 Solving a System of Linear Equations by Graphing ............................................................. 160 3.4 Solving Systems of Equations by Substitution ....................................................................... 166 8Table of Contents Solving a System of Linear Equations using Substitution .................................................. 166 Substitution that requires using the Distributive Property................................................. 167 Solving One of the Equations for a Variable ......................................................................... 168 Steps for Solving a System of Linear Equations by Substitution ....................................... 170 Solving Systems with Two Expressions for the Same Variable ......................................... 171 Solving Systems with Infinite Solutions or No Solution .................................................... 172 3.4 SUMMARY ............................................................................................................................ 175 3.5 Solving Systems of Equations by Elimination ........................................................................ 176 Solving by Elimination if Opposite Coefficients are not Given ........................................ 177 Solving by Elimination if Both Equations need to be Multiplied by a Constant ............ 179 Steps for Solving a System of Linear Equations by Elimination ........................................ 180 Solving Systems with Infinite Solutions or No Solution .................................................... 181 Deciding Which Method to use when Solving a Linear System of Equations ................. 182 Using a System of Equations to Solve Applications ............................................................. 182 3.5 SUMMARY ............................................................................................................................ 185 3.6 Statistics ....................................................................................................................................... 188 Types of Data ............................................................................................................................... 188 Organizing Data .......................................................................................................................... 188 Mean, Median, and Mode.......................................................................................................... 191 Regression Line ........................................................................................................................... 192 3.6 SUMMARY ............................................................................................................................ 194 3.7 Probability ................................................................................................................................... 195 Sample Space and Tree Diagrams ............................................................................................ 195 The Counting Principle .............................................................................................................. 196 Theoretical Probability .............................................................................................................. 198 Empirical Probability ................................................................................................................. 200 The Law of Large Numbers ....................................................................................................... 201 3.7 SUMMARY ............................................................................................................................ 202 3.8 Applications ................................................................................................................................ 203 Using a System of Equations to Solve Applications ............................................................. 203 Statistics Application ................................................................................................................. 204 9Table of Contents Probability Application ............................................................................................................. 204 Answers ............................................................................................................................................ 205 Bibliography .................................................................................................................................... 252 Index ................................................................................................................................................. 253 101.1 Problem Solving and Numerical Representations Unit 1: Mathematical Expressions 1.1 Problem Solving and Numerical Representations Introduction Everyone must have felt at least once in his or her life how wonderful it would be if we could easily solve a problem using one method that works every time. Problem solving can be challenging, and there are no universal approaches one can take to solving problems. Basically one must explore possible methods until recognizing the path to a solution. Guessing can help, and algebraic thinking can be involved. In general, as one gains experience in solving problems, one develops one's own techniques and strategies. Thus the guessing is not arbitrary. Rather, it is educated. In this section we are going to learn a framework for problem solving and get a glimpse of strategies that are often used by experts. The strategies are based on the work of George Polya. For further study, read: G. Polya, How to Solve It, A New Aspect of Mathematical Method, Second Ed., Princeton University Press, Princeton, NJ, 1985. A Framework for Problem Solving The following four phases can be identified in the process of solving problems: (1) Understanding the problem (2) Making a plan of solution (3) Carrying out the plan (4) Looking back i.e. verifying Before we apply this process, it is important to introduce some algebraic vocabulary. Definitions and Key Concepts Variable: A variable, usually represented by a letter or symbol, can be defined as:  A quantity that may change within the context of a mathematical problem.  A placeholder for a specific value.  Download for free at 111.1 Problem Solving and Numerical Representations Definitions and Key Concepts Input-Output table: An input-output table is the numerical representation of a rule. This table has at least two columns. The first column (input) is for values that you know; the second column (output) lists the answer that you get for each input value, after applying an appropriate rule. The rule for an input-output table is the symbolic representation. Rules are equations that begin with: y = . A rule can also be called a formula. Example 1: When the Mathematics Department moved to its current location, faculty considered various ways to place cubicles in the large, open space. Each cubicle contains a desk for one professor, and uses three partitions for sides. One suggestion for how to place the cubicles, was to make one long row. Figure 1 (below) shows this arrangement for five faculty. How many partitions will be needed to arrange the cubicles in one long row for 30 faculty? 1 2 3 4 5 Figure 1. Solution: (1) Understanding the problem – This is a “find it” type of problem. If we use the cubicle layout in Figure 1 above, we need to find how many partitions will be needed for 30 cubicles – one cubicle for each professor. (2) Making a plan of solution – Since a pattern seems to be involved in this problem (each new cubicle requires an additional two partitions), it may be helpful to create an input-output table that lists the number of cubicles, and the number of partitions that would be needed. Creating a table is known as a numerical representation. (3) Carrying out the plan – The first cubicle needs three partitions, but the second cubicle only needs two more partitions since one partition is shared. Each new cubicle needs two additional partitions.  Download for free at: 121.1 Problem Solving and Numerical Representations To read the table, read across each row individually. For example, if you only wanted to make three cubicles, then you would need 7 partitions. Input, x Number of Cubicles 1 2 3 4 5 Output, y Number of Partitions 3 5 7 9 11 Following this pattern, it would be cumbersome to make a table that has 30 rows to represent 30 cubicles, so this is where algebra can be helpful. Using the variables x (to represent the number of cubicles) and y (to represent the number of partitions), the rule (or formula) to find the number of partitions is y = 2x + 1. For now, you will be given the rule when you work with tables. Eventually, you will learn how to come up with the rule yourself. Some of you may even be able to guess the rule now! (4) Looking back i.e. verifying – Let’s see if the rule works: for 4 cubicles, we replace x with 4: y = 2(4) + 1 y=8+1 y=9 Look back at Figure 1. For four cubicles, were there 9 partitions? YES! The rule works, so now we can answer the original question and apply it for 30 faculty, or x = 30: y = 2(30) + 1, y = 61. Answer: 61 partitions will be needed to make cubicles in one long row for 30 faculty. Practice Using Input-Output Tables Example 2: American Fitness charges a one-time registration fee of $50, plus $25 per month of membership at its gym. Answer the following questions using the rule y = 25x + 50, where x represents the number of months of membership and y represents the total cost of membership for that number of months. a) How much will it cost a student to become a member for two years? b) How much will it cost a student to become a member for thirty months? 131.1 Problem Solving and Numerical Representations Solution part a): (1) Understanding the problem - If we let x represent the number of months of membership, and y represent the total cost of membership for that number of months, the rule y = 25x + 50 represents this situation. (2) Making a plan of solution – Again a table will be helpful since there is a pattern – the same amount is being added to the cost, depending on the number of months. (3) Carrying out the plan – We use the rule: y = 25x + 50 to complete the table below: Input, x Number of months of membershipOutput, y = 25x + 50 Total cost of membership 050 6200 12350 18500 24650 (4) Looking back i.e. verifying – This is where you need to make sure that your answer “makes sense.” Since you are asked for the price of a two-year membership, be careful to realize that two years is the amount paid for 24 months. If you only let x = 2, the answer will be the cost of membership for two months! Answer: A two-year (24 month) membership will cost the student $650. Solution part b): Since we know the rule: y = 25x + 50 works for this situation, where x represents the number of months of membership, we can let x = 30. We have: y = 25(30) + 50, y = 800. Answer: A 30 month membership will cost the student $800. 141.1 Problem Solving and Numerical Representations Example 3: Some phones can be used on a “pay as you go” system. For example, Company T charges $0.20 per minute for airtime on its prepaid cell phones. Customers pay $20.00 to “fill” their phone with airtime. The rule is y = 20 – 0.20x, where x represents the number of minutes and y represents the amount of money remaining on the Company T phone. Find how many minutes of airtime a customer will receive for $20. Solution: The rule y = 20 – 0.20x gives the amount of money remaining on the phone (y) after x minutes of airtime have been used. Complete the table below: Answer: Input, x Minutes of airtime that have been usedOutput, y = 20 – 0.20x Amount of money remaining on the phone 1018 2016 3014 5010 1000 For $20, the customer will receive 100 minutes of airtime. Conditional Rules Some input-output tables require more than one rule, or conditional rules, depending on the input value. Example 4 illustrates that shipping and handling fees can depend on the amount of money you are spending. There are five different fees (output), depending on the total of purchases (input). 151.1 Problem Solving and Numerical Representations Example 4: Lydia is shopping at an online website. She selects $99.85 in merchandise, and then uses the table below to determine the additional shipping and handling fees. Input, Total of purchasesOutput, Shipping and handling fee Up to $25$4.95 $25.01 - $50$5.95 $50.01 - $75$6.95 $75.01 - $100$7.95 More than $100Free a) How much will she spend on shipping and handling for her $99.85 purchase? b) If she decides to purchase one more item for $5, how much will she pay for sh