Introductory Algebra

Chris Nord

Description

Introductory Algebra provides precollege algebra students with the essentials for understanding what algebra is, how it works, and why it useful. It is written in plain language and includes annotated examples and practice exercises so that even students with an aversion to math will understand these ideas and learn how to apply them. This precollege algebra textbook introduces students to the building blocks of algebra that they need to progress with mathematics at the college level, including concepts such as whole numbers, integers, rational numbers, expressions, graphs and tables, and proportional reasoning. Written by faculty at Chemeketa Community College for the students in the classroom, Introductory Algebra is a classroom-tested textbook that sets students up for success.

Sujets

• Sciences et techniques
• Algebra
• Mathematics

Informations

• Publié par: Chemeketa Press
• Date de parution: 01 août 2021
• Nombre de lectures: 0
• EAN13: 9781943536887
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

Introductory Algebra
by Chris Nord
with additional contributions by Matthew Schmidgall and Martin Prather
Introductory Algebra
ISBN: 978-1-943536-56-6
Edition 1.0 Fall 2019
2019, Chemeketa Community College. All rights reserved.
Chemeketa Press
Chemeketa Press is a nonprofit publishing endeavor at Chemeketa Community College. Working together with faculty, staff, and students, we develop and publish affordable and effective alternatives to commercial textbooks. All proceeds from the sale of this book will be used to develop new textbooks. For more information, please visit www.chemeketapress.org .
Publisher: David Hallett
Director: Steve Richardson
Managing Editor: Brian Mosher
Manuscript Editors: Steve Richardson, Matt Schmidgall
Instructional Editor: Stephanie Lenox
Design Editors: Ronald Cox IV
Cover Design: Ronald Cox IV
Interior Design: Ronald Cox IV, Kristi Etzel
Layout: Ronald Cox IV, Matthew Sanchez, Brice Spreadbury
Chemeketa Math Faculty
The development of this text and its accompanying MyOpenMath classroom has benefited from the contributions of many Chemeketa math faculty in addition to the author, including:
Dan Barley, Lisa Healey, Kelsey Heater, Tim Merzenich, Nolan Mitchell, Chris Nord, Martin Prather, and Matthew Schmidgall.
Printed in the United States of America.
Contents
Chapter 1: Whole Numbers
1.1 Addition and Subtraction
1.2 Solving Equations
1.3 Multiplication and Division
1.4 Order of Operations
1.5 Solving Equations
Chapter 2: Integers
2.1 Introduction to Integers
2.2 Addition and Subtraction
2.3 Multiplication and Division
2.4 Solving Equations
Chapter 3: Rational Numbers
3.1 Introduction to Rational Numbers
3.2 Divisibility
3.3 Prime Factorization
3.4 Equivalent Fractions
3.5 Multiplication and Division
3.6 Comparing Rational Numbers
3.7 Addition and Subtraction
3.8 Solving Equations
Chapter 4: Expressions
4.1 Introduction to Expressions
4.2 Combining Like Terms
4.3 Simplifying Expressions with Algebraic Fractions
4.4 Modeling with Expressions and Equations
4.5 Solving Equations
Chapter 5: Graphs and Tables
5.1 Interpreting Graphical Representations of Data
5.2 Mean, Median, and Mode
5.3 Equations with Two Variables
5.4 The Cartesian Plane
5.5 Graphs of Linear Equations
Chapter 6: Proportional Reasoning
6.1 Ratios and Rates
6.2 Solving Proportions
6.3 Dimensional Analysis
Solutions to Odd-Numbered Exercises
Glossary/Index
CHAPTER 1
Whole Numbers
What Are Numbers?
You know the answer to this question, of course, because you ve been using numbers for most of your life. You use them every day. However, as you will see in the chapters to come, numbers can be a little more complicated than those things you use to count the money in your wallet or the people in front of you in the express lane at the grocery store. We will return to this question about what numbers are at the beginning of each chapter, and with each return, we ll add a new layer of sophistication to the answer.
To start, we ll just say that numbers are symbols that we use to count objects in a collection. This set of numbers is called the whole numbers . The whole numbers are {0, 1, 2, 3, 4, }. With the exception of zero, which has only been considered a number for about 1400 years, the set of whole numbers is as old as human civilization. In fact, there is some evidence of a surprisingly sophisticated innate understanding of numbers among several non-human species in the animal kingdom, too, so it may be that some version of the set of whole numbers predates humanity.
While there is a smallest whole number, zero, there is no greatest whole number. With any given whole number, there is always a next whole number that is greater than the given number.
In this chapter, we will explore the arithmetic of whole numbers and introduce the concept of an equation in the context of whole numbers. This chapter lays the foundation for the rest of the book, but it s important to remember that this chapter also rests on its own foundation - the ancient and basic concept of counting.

1.1 Addition and Subtraction
When we re presented with two collections of objects, we can count them separately, or we can combine them into a single collection and count them together. In this section, we will introduce the mathematical operation called addition , which encodes the relationship between these two ways of counting, and explore its properties.
If we wish to remove any objects from a collection of objects, we can either count the number of objects before any are removed and then count the number of objects that are removed, or we can count the number of objects that remain after we are finished removing objects. In this section, we will also introduce the mathematical operation called subtraction , which encodes the relationship between these two ways of counting, and explore its properties.
A. Addition
Whole numbers are the symbols we use to count the objects in a collection. So suppose that you have two collections of - anything . First you count the things in one collection. We ll use the letter a to represent that whole number. Then you count all the things in the second collection. We ll use the letter b to represent that whole number. If you combine these two collections into a single collection, you can figure out the total number of things you have by adding a and b .
Addition is the process of bringing two numbers together to make a new total. The math symbol for addition is the plus sign (+). The numbers being added, a and b , are called the terms , and the result of the addition is called the sum .

Addition

Addition is the operation where 2 numbers, let s call them a and b , are added together to create a sum, c . Symbolically, a + b = c .
For example, suppose that Ben and Joaquin go door to door in their neighborhoods to raise money for their baseball team. Ben secures 7 donations, and Joaquin brings his baby sister along to make himself look like a good brother and secures 11 donations. The combined number of donations that Ben and Joaquin secure is represented by the sum of the terms 7 and 11.
By the way, we use the word sum to mean the addition itself and also the result of the addition. So both of the following statements use sum correctly:
The sum of 7 and 11 is 7 + 11.
The sum of 7 and 11 is 18.
To evaluate the sum of two numbers is to calculate the result of their addition. If your teacher asks you to evaluate the sum of 7 and 11, then the sum you give your teacher is the result. Eighteen, you say, or, The sum of 7 and 11 is 18.
When we count Ben s and Joaquin s combined donations, it doesn t matter whether we count Ben s donations first and Joaquin s second. With 7 + 11, the sum is 18. With 11 + 7, the sum is also a combined total of 18 donations. This is true whenever we add two numbers together. Being able to add numbers in any order and get the same sum is a property of addition that we call the commutative property of addition .

Commutative Property of Addition

When a and b represent any two numbers, then a + b = b + a .
Addition as an operation only happens between two numbers at one time. In real life, however, we often use addition to combine three or more numbers. When we do this, we re actually calculating a series of two-number sums.
For example, suppose Luke is a teammate of Ben and Joaquin. Luke is kind of lazy, to be honest, and only secures 3 donations for the fundraiser. If we use addition to calculate the total number of donations secured by Ben, Joaquin, and Luke, we represent the total with the three-number sum of 7 + 11 + 3. To evaluate this sum, we can either add 7 and 11 first and then add 3 to the result of that sum, or we can get the same result by adding 7 to the sum of 11 and 3:
( 7 + 11 ) + 3 = 18 + 3 = 21
7 + ( 11 + 3 ) = 7 + 14 = 21
This is true whenever we add three or more numbers together. It s another property of addition, and it s called the associative property of addition .

Associative Property of Addition

When a, b , and c represent any three numbers, then ( a + b ) + c = a + ( b + c ).
The associative property of addition ensures that there is only one possible meaning for an expression such as a + b + c . Taken together, the commutative and associative properties of addition mean that we can add two or more numbers together in any order we please. The sum will always be the same no matter what the order is.
If we needed to add 7, 3, 5, and 4 together, we can write 7 + 3 + 5 + 4, or even change the order and write 3 + 4 + 5 + 7 and still get the same sum:
7 + 3 + 5 + 4 = 19
3 + 4 + 5 + 7 = 19
Since we know that we ll get the same sum regardless of the order in which numbers are added together, we can make evaluating addition problems easier. For example, to more easily evaluate 9 + 3 + 8 + 7 + 1, some people would mentally change the order of the terms to put numbers that add to 10 together:
( 9 + 1 ) + ( 3 + 7 ) + 8 = 10 + 10 + 8 = 28

Example 1
Evaluate the following multi-term sums.
1. 8 + 4 + 2 + 7 + 6
2. 13 + 9 + 2 + 21 + 7
Solutions
1. 8 + 4 + 2 + 7 + 6

Re-order so that the terms adding to 10 are together.
2. 13 + 9 + 2 + 21 + 7

Re-order so that the terms adding to 20 or 30 are together.
Now let s go back to the story of the baseball team that is trying to raise money. Suppose that Carlos, the catcher, catches the chicken pox. Get it? He s the catcher , and he catches the chicken pox. Hilarious. Anyway, because he has chicken pox, Carlos can t go anywhere and doesn t secure any donations from his neighborhood. For Carlos, we use the whole number 0 to represent the number of donations that he secures. To calculate the combined number of donations that Carlos and Ben secure, we use the sum 0 + 7. But the result of this addition is the same as the number of donations the Ben secured by himself, 7. We repres