Topic outline

  • COURSE INTRODUCTION

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    • Time: 35 hours
    • CEUs: 3.5
    • Free Certificate
    Algebra allows us to solve many different types of calculations by using basic formulas and steps that work regardless of the specific numbers in the problem. For example, we can use an algebraic formula to calculate a monthly payment or to pay off a loan of any size using a given interest rate.

    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 problem-solving 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 our Foundations of Real World Math course if you need a refresher!

  • Unit 1: Number Properties

    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 variables 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.

    • 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.

    • 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 for a variable appropriately is an important skill in algebra. You also need to understand how to write variable expressions when multiplication is involved.

    • 1.3: Order of Operations Review

      Mathematicians use the conventional 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.

    • 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.

    • 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.

    • 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.

    • 1.7: Definition and Examples of Like Terms

      Like terms have the same variables and use 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 8x2 are not like terms because, although the variables are the same, the exponents of the variables are different.

    • 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 complicated-looking expressions.

  • Unit 2: Linear Equations

    We use equations every day 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), and estimating how much time it will take to drive to a destination at a certain speed.

    In this unit, you will explore formal procedures for solving equations. After reviewing basic math rules, you will apply the skills you 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 4 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 real-world 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.

    • 3.1: Mathematical Symbols and Expressions for Common Words and Phrases

      Before we can begin solving word problems, we need to understand their language. Common words are used to indicate different types of calculations. For example, when you see the word "difference", it indicates you will be doing subtraction.

    • 3.2: Translating Verbal Expression into Mathematical Equations

      I want to make an extra $3000 for a trip I have planned. If I work an extra job making $15 dollars an hour, how many hours do I need to work? This is a straightforward question, and you may have asked yourself a similar question. In this case, I can quickly rewrite it into an equation, where h is the number of hours I need to work. I want to know when 15h=3000. We can use the tools from Unit 2 and find out that I would need to work 200 hours for my trip. This is just one example of how we use algebra every day; we just don't usually think about it in algebraic terms. Writing equations from word problems helps us to see the connections that we often overlook. It also gives us a tool to solve more difficult problems easily.

    • 3.3: Consecutive Integer Problems

      The next type of problem we see in algebra is consecutive integer problems. Consecutive integers are integers that follow one another. In other words, we can define the variable n as an integer. Then, n + 1 is the next consecutive integer.

    • 3.4: Number Problems

      The first common type of word problem you will encounter is a number problem. These word problems describe numerical operations (addition, subtraction, multiplication, division).

    • 3.5: General Statement Problems

      Often, we can use algebra to solve real-life problems by translating word problems into equations.

    • 3.6: Applying the Uniform Motion Equation

      The uniform motion equation allows us to solve problems involving rate or speed. For example, let's say you need to drive 150 miles on the highway with an average speed of 55 miles per hour. How long will it take you to arrive at your destination? We can use a simple formula to solve for the time it will take to drive that distance.

    • 3.7: Value Mixture Problems

      Other common types of word problems involve percents, price mark-ups or discounts, and mixtures. In this section, we explore these common types of word problems.

      Mixture problems involve mixtures of different variables. We can then relate the variables to each other and reduce the problem to one variable.

  • 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 that your amount of money must be GREATER than the total cost of items. In this unit, we will generalize the procedure for solving inequalities. We will 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.

    • 4.1: Definition and Notation of an Inequality

      Like equations, inequalities have their own set of symbols. This symbol: > is greater than, as in 5 > 2. This symbol < is less than, for example, 8 < 10. There are a few other symbols we need to know before we can solve inequalities and graph them.

    • 4.2: Graphing Inequalities on a Number Line

      One way inequalities differ from equations, is unlike equations which usually only have one solution, inequalities usually have a range of solutions. For example, if we say x > 10, then the solution is all numbers greater than 10. Therefore it is very helpful to see all of the solutions on a number line.

    • 4.3: Solving One-Step Inequalities

      We are now ready to begin solving inequalities. In this section, we will solve one-step inequalities, and in the next section, we will solve more complex inequalities. Solving inequalities is a lot like solving equations with one difference. First, let's consider how they are the same. You know that to solve x + 5 = 20, you would subtract 5 from both sides of the equation and get x = 15. If you had the inequality x + 5 < 20, you would still subtract 5 from both sides to get x < 15. The difference comes when we multiply or divide both sides of an inequality by a negative number. Whenever we multiply or divide both sides of an inequality by a negative number, we need to flip the inequality symbol. So a less than would become a greater than, or a greater than would become a less than. More on that shortly.

    • 4.4: Solving Multi-Step Inequalities

      Now that you have mastered how to solve one-step inequalities, you are ready to move on to more complex inequalities. This will feel a lot like the work you did in unit 2, solving equations. Remember, the steps are the same when solving either an equation or inequalities unless you multiply or divide by a negative number.

  • 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 leaves 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 6 hours.

    • 5.1: Graphing Points in the Rectangular Coordinate Plane

      First, we need to understand the coordinate plane, the space in which we produce graphs. In this section, we will focus on finding and graphing points on the coordinate plane with the goal of becoming comfortable with it.

    • 5.2: Ordered Pairs as Solutions of an Equation in Two Variables

      Whenever we have an equation in two variables, meaning the equation has two variables, we can graph it on the plane. If the equation is of a line, every point on that line will be a solution to that equation. For example, if we have the equation x + y = 4, then we know that the points (0, 4), (1, 3), and (2, 2) are all solutions since each ordered pair adds up to 4. If we graph a line through those points, we will get all of the pairs of numbers that make this equation true. You will see on the line that pairs like (-10, 14) and even (1/2, 3.5) are also solutions. Now, we are ready to begin using graphs to determine if a pair of numbers (an ordered pair) is a solution to an equation.

    • 5.3: Graphing Equations in Two Variables of the Form ax + by = c

      A common way equations of lines can be written is: Ax + By = C, where A, B, and C are real numbers. This is called standard form. One way we can graph an equation written in this format is by creating a table of values.

    • 5.4: Intercepts of a Line

      One of the properties of linear graphs is that they have intercepts on the x and y-axis. The intercept is the point at which the line crosses the axis. So, the x-intercept is the point where the line crosses the x-axis, and the y-intercept is the point where the line crosses the y-axis. Knowing where these two points are tells us a great deal about the graph and can make graphing a line much faster.

    • 5.5: Definition of Slope and Slope Formula

      Another important property of linear graphs is the slope of the graph. The slope tells us how steep the line is. When a linear equation is written in a specific form that we'll discuss later, the slope helps us determine how to graph the line.

    • 5.6: Slopes of Parallel and Perpendicular Lines

      Here, we learn about how the slopes of parallel and perpendicular lines are related. Parallel lines have the same slope, while perpendicular lines have slopes that are opposite reciprocals.

    • 5.7: Graphing Equations in Two Variables of the Form y = mx + b

      One of the most common types of graph is that of a line of the form y = mx + b. In this form, m is the slope of the line, and b is the y-intercept of the line. When an equation is in this form, it is easy to graph the, so it is important to be able to recognize when an equation is in this form.

    • 5.8: Point-Slope Form

      In the last section, we discussed the slope-intercept form of a linear equation. We can also write linear equations in a form known as the point-slope form. This form works when you want to make a line between two known points. This form is: y − y1 = m(x − x1).

    • 5.9: Graphing Linear Inequality of Two Variables on the Coordinate Plane

      The last type of linear graphing we need to study is the graph of an inequality rather than an equation. When we graph inequalities, we must pay attention not only to the numbers and variables but also to the inequality itself. That is, are we graphing a less-than or greater-than inequality?

  • 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.

    • 6.1: Solution of a System of Linear Equations

      The solution to a set of linear systems of equations is the point or points that are on all the lines in the system. Graphically, that would be where the lines intersect. To determine if a point is a solution to a system, we have to verify that it is a solution to all equations in the system. We can do this using techniques you have already learned in this course.

    • 6.2: Solving Systems of Linear Equations by Graphing

      There are several methods for solving systems of linear equations. The first method we explore is using graphing to solve systems of linear equations. When we graph a system of linear equations, three different things could happen. First, the lines could be interesting lines, so the solution would be the one point where the lines intersect. Second, the lines could be parallel, which means the lines will never intersect, and we say this system would have no solution. Finally, the lines could be the same line; the solution to that system would be every point on the line, and we say that the system has an infinite number of solutions.

    • 6.3: Solving Systems of Linear Equations Using the Substitution Method

      Sometimes, it is not practical to solve a system of equations by graphing. The substitution method is a non-graphical method for solving systems of equations. It uses algebra techniques you have already learned in this course.

      For the substitution method, we solve one of the equations for one of the variables in terms of the other variable. Then, we substitute this expression into the second equation to solve for the variable. Then, once you have solved for one variable, you can solve for the other.

    • 6.4: Solving Systems of Linear Equations Using the Elimination Method

      The elimination method is another non-graphical method for solving systems of linear equations. For this method, we use the addition property of equality. This states that if we add the same amount to both sides of an equation, the two sides of the equation will still be equal.

    • 6.5: Strategy for Solving Systems of Linear Equations: Choosing a Method

      It can be difficult to decide which of the three methods (graphing, substitution, or elimination) is best for solving a given system of linear equations.

    • 6.6: Solving Word Problems by Using Systems of Equations

      Linear systems of equations appear in many real-world applications. They are often used to conduct comparisons, such as when comparing prices. Translating a word problem into equations and determining the appropriate variables can be difficult.

    • 6.7: Graphing Systems of Linear Inequalities

      We can use graphical methods to solve systems of linear inequalities, much like we graphed single inequalities earlier in this course. Remember when we graphed inequalities? Now, we will look at systems of inequalities. The solution to these systems is the area on the plane that gets shaded in by both inequalities.

    • 6.8: Applications of Systems of Linear Inequalities

      Linear inequalities appear in many real-world applications. These problems are very interesting because this is always a range of solutions. Think about having a business where you want to maximize income while minimizing costs. Your variables could be the number of items you sell and the money involved. You want to make sure your daily productivity is in the part of the graph where profits are maximized.

  • 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 apply rules of exponents, multiply, and divide monomials.

    Completing this unit should take you approximately 2 hours.

    • 7.1: Algebraic Exponential Expressions

      The first mathematical function we explore is the exponential function of monomials. Monomials have one term. Examples would be \(5x\), \(6y^2\), and \(x^2y^3\). Exponents tell us to multiply the number of times in the exponent. For example \(5^2 = 5 * 5 = 25\) and \(2^4 = 2*2*2*2 = 16\). Now consider \(x^2 * x^3\). What do you think that would simplify to? Well,\( x^2\) is \(x*x\) and \(x^3\) is \(x*x*x\) so \(x^2*x^3 = x*x*x*x*x = x^5\).

    • 7.2: Quotient of Exponents and Power of a Quotient

      The next important mathematical operation to study is the quotient of exponents. This means dividing monomials with exponents. When dividing monomials, think of it like when you reduce fractions. You are deleting the common factor that appears on both the numerator and denominator. 25/10 = 5/2 because you can reduce (divide out) a factor of 5 from both. It also holds true for dividing monomials with exponents. This section will show you how it's done.

    • 7.3: Negative Exponents

      Now that we can multiply and divide monomials, it's time to consider negative exponents. Mathematics has defined a negative exponent and given it a close relationship with fractions. In this section, we will learn this definition and how to manipulate monomials with negative exponents.

    • 7.4: Multiplying Monomials

      Now that we are comfortable with the rules of exponents, we are ready to apply them to mathematical functions of monomials. In this section, we will learn to multiply monomials.

    • 7.5: Dividing Monomials

      The last mathematical function we must apply to monomials is division. We have already divided monomials with one variable, so now it's time for more than one variable. Think of this as just reducing fractions.

  • 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 \( 5x^4 + 2x^3 + x^2 + 2x + 10\). In this unit, we discuss recognizing, classifying, adding, subtracting, multiplying, and dividing 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.

    • 8.1: Classifying Polynomials

      When we discuss polynomials, we are referring to an algebraic expression that includes more than one term. We classify polynomials based on the number of terms in the expression.

    • 8.2: Adding and Subtracting Polynomials

      Like all mathematical expressions, we can perform basic mathematical operations on polynomials. In this section we learn how to add and subtract polynomials using different common methods.

    • 8.3: Multiplying a Polynomial by a Monomial

      When we multiply a polynomial, we can multiply it by a monomial or another polynomial. In this section, we learn how to multiply a polynomial by a monomial. We can use the distributive property in these types of calculations.

    • 8.4: Multiplying Binomials (FOIL)

      When multiplying two binomials (polynomials with two terms), we cannot simply use the distributive property. We need to ensure that each term in one binomial is multiplied by each term in the other binomial. The technique we use to do this is called FOIL. FOIL is a mnemonic that stands for first, outer, inner, and last. When we place two binomials next to each other, FOIL reminds us of the order we should use to multiply the terms.

    • 8.5: Complete the Square and Difference of Two Squares

      We sometimes need to take the square of a binomial. In this section, we learn the specific rules for squaring binomials, but we see that these rules are derived from using the FOIL method.

    • 8.6: Multiplying Polynomials with Any Number of Terms

      Polynomials can have more than two terms, and we must develop methods for multiplying these larger polynomials. This can be done easily with the help of the distributive property.

    • 8.7: Dividing a Polynomial by a Monomial

      We also need to be able to divide polynomials. This section teaches techniques for dividing a polynomial by a monomial. This is easy since we already know how to divide monomials.

    • 8.8: Dividing a Polynomial by a Binomial

      The last mathematical operation we need to discuss is dividing polynomials by binomials. This is very different from dividing by monomials, but it is very similar to long division.

  • 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 6 hours.

    • 9.1: Identifying the Greatest Common Factor (GCF) of Two or More Monomials and Other Expressions

      Reading and understanding algebraic expressions, much like translating sentences from a foreign language, is a skill that takes time to develop. Before learning how to factor an algebraic expression (or write the expression as a product), you need to be able to break down the parts.

    • 9.2: Factoring Polynomials by Grouping

      Let's look at the four steps you need to take to factor a four-term polynomial.

      1. Separate the polynomial into two groups of two terms.
      2. Identify a common factor in each group and factor it out.
      3. Check that the resulting expressions contain a common binomial factor. If they do not, then the polynomial cannot be factored, at least not when the terms are grouped this way.
      4. Factor out a common binomial factor and rewrite the polynomial as a product of two binomials.

    • 9.3: Revisiting FOIL: Working Backwards

    • 9.4: Factoring Trinomials of the Form x² + bx + c When c Is Positive

      We will study how to factor trinomials in two different forms: when c is positive and when c is negative. In this section, we will study how to factor trinomials with a positive c value.

    • 9.5: Factoring Trinomials of the Form x² + bx + c When c Is Negative

    • 9.6: Factoring Trinomials of the Form x² + bxy + cy² and ax² + bxy + cy²

      The previous examples of factoring trinomials only include one variable. However, algebraic expressions often include more than one variable, and we need ways to factor polynomials with two variables. The methods used to factor trinomials with two variables are still based on the same principles as factoring trinomials with one variable.

    • 9.7: Identifying and Factoring Complete Square Trinomials

    • 9.8: Identifying and Factoring Difference of Two Squares

      A special case of polynomials is the difference of two squares. For example, the equation \(x^2 - 4\) is a difference of two squares because it can be written as \(x^2 - 2^2\).

    • 9.9: Factoring General Polynomials

      Sometimes, factoring a polynomial requires more than one step. When this occurs, we combine the different techniques we have learned to completely factor the complex polynomial.

    • 9.10: Principle of Zero Product and Identifying Solutions

      The Principle of Zero Product tells us that if the product of two quantities is zero, one of the quantities must also be zero.

    • 9.11: Using Factoring to Solve Quadratic Equations

      We are now ready to use factoring as a method for solving quadratic equations.

    • 9.12: Solving Application Problems

      Now that we have spent so much time learning about factoring, we are ready to see how it applies to everyday life. The whole reason we factor is to solve equations, and we solve equations to answer questions that arise in life. In this section, we will solve geometric problems using factoring.

  • Study Guide

    This study guide will help you get ready for the final exam. It discusses the key topics in each unit, walks through the learning outcomes, and lists important vocabulary. It is not meant to replace the course materials!

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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 7-day waiting period between each attempt. Once you pass this final exam, you will be awarded a free Course Completion Certificate.