Course Introduction

Time: 38 hours

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In this course, we explore how to use standard mathematical and business formulas, and how to translate verbal descriptions into mathematical equations to establish relationships and create predictions. In the later units, we explore how to use graphs to make these same predictions. You can apply the problemsolving strategies we discuss in this course to business, science, and many other fields.
To succeed in this introductory course you should know how to perform operations with real numbers, including negative numbers, fractions, and decimals. Be sure to review RWM101: Foundations of Real World Math if you need a refresher!
First, read the course syllabus. Then, enroll in the course by clicking "Enroll me in this course". Click Unit 1 to read its introduction and learning outcomes. You will then see the learning materials and instructions on how to use them.

Unit 1: Variables and Variable Expressions
In algebra, we use letters to represent numbers in equations. We call these letters variables, because the numbers they represent vary. For example, let’s say your salary is $10 per hour. If you worked two hours, you would be paid $10 × 2, or $20. If you worked five hours you would be paid $10 × 5, or $50. To generalize this we can say that if you work h hours, you will be paid $10 × h.
In this unit, we learn that the properties of numbers also apply to the letters we use in these formulas: you treat variables the same way you treat numbers in calculations. We discuss how to recognize like terms, which you can add and subtract as if they were numbers.
Completing this unit should take you approximately 4 hours.
Upon successful completion of this unit, you will be able to:
 identify parts of algebraic expressions, including terms, factors, and coefficients;
 evaluate algebraic expressions for the given values of the variables; and
 apply commutative, associative, and distributive properties of real numbers to simplify algebraic expressions.
1.1: Variables, Constants, and Coefficients
The first step in understanding algebra is knowing the definitions of variables, constants, and coefficients. These are all important parts of an algebraic equation.
Read these notes to get a handle on the definitions we'll be using and to see a few examples. After you read, complete a few practice problems and check your answers.
1.2: Replacing Variables with Their Values
We often know the value of a variable for a given problem. For the example given at the start of the chapter, we said that if you make $10 per hour and you work h hours, your pay would be $10 × h. Let's say you worked 40 hours in a given week. For that week, we say h = 40. So, your pay would be $10 × 40 = $400.
Being able to substitute a value in for a variable appropriately is an important skill in algebra. You also need to understand how to write variable expressions when multiplication is involved.
Watch this video, which highlights some common ways to write multiplication in algebra. It also shows examples of how to substitute values in algebraic expressions.
After you watch, complete these assessments that involve substituting values into algebraic expressions with variables. If you need help, use the review tools at the bottom of the page.
Continue with this second set of assessments. If you need help, use the review tools at the bottom of the page.
1.3: Order of Operations Review
Mathematicians use the convention order of operations to ensure we always evaluate expressions with many operations in the same way. This ensures we complete complicated calculations the same way every time.
Watch this video to review the rules for order of operations. Many use the acronym PEMDAS to remember the order of operations: parentheses, exponents, multiplication and division, addition and subtraction.
Try each of the examples on this worksheet without looking at the solution. Then, check your answers. If you need more practice, you can do the review problems that follow the examples.
1.4: Commutative Property of Addition and Multiplication
In addition to knowing the order of operations, we need to understand some properties of arithmetic before we can begin evaluating more complicated algebraic expressions.
The commutative property of addition and multiplication states that the order of the input numbers does not matter. For example, 3 + 2 = 5 and 2 + 3 = 5. Likewise, 3 × 2 = 6 and 2 × 3 = 6.
Watch these videos for examples of how to apply these laws in arithmetic.
1.5: Associative Property of Addition and Multiplication
The associative property states that we can add or multiply several numbers in any order. We use this property frequently, such as when we switch terms around to simplify algebraic expressions. For example, (1 + 1) + 3 = 5 and 1+ (1 + 3) = 5.
Watch these videos for examples of how to use this property in arithmetic.
1.6: Distributive Property of Multiplication over Addition/Subtraction
The distributive property is another property we use extensively to simplify algebraic expressions. It allows us to "distribute" a multiplicative factor over an addition or subtraction. For example, for the expression 4 × (3 + 2), we can distribute the multiple of four across the addition in the parentheses to get (4 x 3) + (4 × 2) = 12 + 8 = 20.
Watch this video for examples of how to use this property in arithmetic.
1.7: Definition and Examples of Like Terms
Like terms have the same variables using the same exponents, but may have different numerical coefficients. You need to recognize like terms to add and subtract them. For example, 3x and 8x are like terms because they have the same variables and exponents. The terms 3x and 8x^{2} are not like terms because, although the variables are the same, the exponents on the variables are different.
Review this brief definition and examples in this encyclopedia entry.
1.8: Simplifying Expressions by Combining Like Terms
In general algebraic expressions, we open the parentheses to combine like terms. Due to the commutative and associative properties, we can move like terms around to combine them. This makes it easier to solve complicatedlooking expressions.
 Watch these videos to see examples of how to combine like terms to simplify algebraic expressions.
After you watch, complete this first set of assessments. If you need help, use the review tools at the bottom of the page.
Continue with this second set of assessments. If you need help, use the review tools at the bottom of the page.
Unit 2: Linear Equations
We use equations everyday without realizing it. Examples include calculating the unit price to compare the price of brands in the grocery store, converting inches into feet (or centimeters into meters), estimating how much time it will take to drive to a destination at a certain speed.
In this unit, we explore formal procedures for solving equations. After reviewing basic math rules, we apply the skills we learned in Unit 1 to simplify the sides of an equation before attempting to solve it and work with equations that contain more than one variable. Because variables represent numbers, we use the same rules to find the specific variables we are looking for.
Completing this unit should take you approximately 5 hours.
Unit 3: Word Problems
Now, let's apply what we learned about solving equations to various types of word problems. To set up the equation, read the word problem carefully to identify the quantity you are being asked to find, the known quantities, and the relationship between them. This is an important skill in algebra since we use algebra to solve many realworld problems.
In this unit, we will identify common types of word problems and discuss how to translate these problems into algebraic equations that can be solved.
Completing this unit should take you approximately 5 hours.
Unit 4: Inequalities
You probably use inequalities, just like equations, in everyday life without thinking about it. Every time you go to the store, you need to decide whether you have enough money to pay for the items you need to purchase. The inequality you need to solve is: your amount of money must be GREATER than the total cost of items.
In this unit, we generalize the procedure we use to solve inequalities. We explore which properties of inequalities are the same and which are different from the properties of equations.
Completing this unit should take you approximately 2 hours.
Unit 5: Graphs of Linear Equations and Inequalities
We use graphs to help us visualize how one quantity relates to another. This unit will help you become comfortable with graphing pairs of numbers on the coordinate plane and understand how we can use lines to represent equations and relationships.
For example, we can graph how the location of a train depends on when it left the station. If the train is moving at constant speed, the line in the graph is straight. The slope or slant of the line depends on the speed: the greater the speed, the steeper the line. If the line is going up (from left to right), it tells you the distance is growing with time: the train is moving away from the station. If the line is going down, it tells you the distance is decreasing: the train is approaching the station. You can gather a lot of information about the train's journey from just one graph.
Completing this unit should take you approximately 5 hours.
Unit 6: Systems of Linear Equations and Inequalities
In previous units, we learned that linear equations with one variable generally have one solution. However, linear equations with two variables have an infinite number of solutions. If we pair two linear equations together, we can solve for the pair of numbers that would solve both equations. This is called a system of linear equations. In this unit, we will learn how to solve systems of linear equations.
Completing this unit should take you approximately 3 hours.
Unit 7: Operations with Monomials
As we have seen, algebra involves the use of variables to represent unknown quantities in equations. Here we will begin the study of expressions that primarily consist of variables.
The rules that govern operations with these expressions arise from the properties of operations with numbers, such as the distributive property and the order of operations. In this unit we focus on monomials, which are expressions that contain only one term. We will learn how to multiply, divide, and apply rules of exponents to monomials.
Completing this unit should take you approximately 3 hours.
Unit 8: Operations with Polynomials
Polynomials are a special type of algebraic expression that contain two or more terms. For example, a polynomial might look like 5x + 2x^{3} + x^{2} + 2. In this unit, we discuss how to recognize, classify, add, subtract, multiply, and divide polynomials by combining like terms and using the distributive property. These skills help us make calculations that pertain to the motion of two or more objects. For example, we can calculate when and where a runner will overtake a competitor, or how much interest you will earn from two or more savings accounts.
Completing this unit should take you approximately 4 hours.
Unit 9: Factoring Polynomials
Factoring is multiplication in reverse: rather than multiplying two polynomials, you write a given polynomial as a product of two or more different expressions. Factoring is an important tool for solving advanced equations, such as quadratic equations. Quadratic equations occur in problems that involve rectangular objects and their areas, such as planning gardens, framing photographs, or carpeting a floor.
Completing this unit should take you approximately 7 hours.
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Certificate Final Exam
Take this exam if you want to earn a free Course Completion Certificate.
To receive a free Course Completion Certificate, you will need to earn a grade of 70% or higher on this final exam. Your grade for the exam will be calculated as soon as you complete it. If you do not pass the exam on your first try, you can take it again as many times as you want, with a 7day waiting period between each attempt.
Once you pass this final exam, you will be awarded a free Course Completion Certificate.