### Course Introduction

This introductory mathematics course is for you if you have a solid foundation in arithmetic (that is, you know how to perform operations with real numbers, including negative numbers, fractions, and decimals). Numbers and basic arithmetic are used often in everyday life in both simple situations, like estimating how much change you will get when making a purchase in a store, as well as in more complicated ones, like figuring out how much time it would take to pay off a loan under interest.

The subject of algebra focuses on generalizing these procedures. For example, algebra will enable you to describe how to calculate change without specifying how much money is to be spent on a purchase-it will teach you the basic formulas and steps you need to take no matter what the specific details of the situation are. Likewise, accountants use algebraic formulas to calculate the monthly loan payments for a loan of any size under any interest rate. In this course, you will learn how to work with formulas that are already known from science or business to calculate a given quantity, and you will also learn how to set up your own formulas to describe various situations by translating verbal descriptions to mathematical language. In the later units of this course, you will discover another tool used in mathematics to describe numbers and analyze relationships: graphing. You will learn that any pair of numbers can be represented by a point on a coordinate plane and that a relationship between two quantities can be represented by a line or a curve.

Units 6, 7, and 8 may seem more abstract than the earlier ones, as you will deal with expressions that contain mostly variables and not too many numbers. While the procedures you will master in these units might seem to have little practical application, you have to keep in mind that they result in formulas that describe very real situations in business, accounting, and science. Knowing how to perform various operations with algebraic expressions will eventually enable you to solve quadratic and even more complex equations. You will explore a variety of real-world scenarios that can be described by these kinds of equations. For example, if a ball is thrown up in the air, solving a quadratic equation will help you find out when it will hit the ground. As another example, if you know the area of a rectangular garden, then you can use a quadratic equation to find the length of each side.

### Unit 1: Variables and Variable Expressions

When most people think of math, they think of numbers. However, with this math course, you will be working with letters. In algebra, letters are used to represent numbers. These letters are called variables, because the numbers they represent may vary. For example, if you are paid $10 per hour, your salary can be expressed as 10×

*h*, where*h*is the number of hours you have worked. You can change the number that letter*h*stands for according to your particular case in order to calculate your salary.In this unit, you will learn that the properties of numbers apply to letters as well, and you can use them to work with expressions containing variables. One of the main skills you have to master in this unit is recognizing like terms, because you can add and subtract them as if they were numbers.

**Completing this unit should take you approximately 8 hours.**### Unit 2: Linear Equations

You, like other people, probably use equations in everyday life without even realizing it. Calculating a unit price to figure out which brand is cheaper when making a purchase, converting inches into feet, estimating how much time it would take you to drive to your destination at a certain speed all involve solving equations mentally. In this unit, you will learn formal procedures for solving equations. You can probably recall the basic rules from the math courses you have taken in the past. In this unit, you will review these rules while focusing on the formal logical definition of equation as a statement and its solution as a number that makes this statement true. You will apply the skills from Unit 1 to simplify the sides of an equation before attempting to solve it. You will also work with equations that contain more than one variable, and you will learn that because variables always represent numbers, you can use the same rules to find the specific variable you are looking for.

**Completing this unit should take you approximately 14 hours.**### Unit 3: Word Problems

In this unit, you will apply the skill of solving equations you mastered in Unit 2 to solve various types of word problems. When you encounter a word problem, you have to remember to read it carefully and think critically about what quantity you are asked to find, what quantities are known, and what the relationship is between them. This will help you set up an equation that will give you an answer to the problem. For example, if you know the discounted price of an item and need to find the original price, remember that the percent of the discount is taken from this original price, which is something you do not know!

You probably already know how to solve some of the problems covered in this unit, such as percent problems or uniform motion problems that can be solved by one arithmetic operation. However, you will now approach these types of problems from the algebraic point of view, which will enable you to move on to more complex problems. In these problems, you cannot just add/subtract and multiply/divide the quantities given and arrive at the answer. These problems can only be solved by setting up an equation.

**Completing this unit should take you approximately 13 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 the Amount of Money has to be GREATER than the Total Cost of Items. In this unit, very similar to Unit 2: Equations, you will generalize the procedure for solving inequalities. You will learn that an inequality is a statement and its solution is a set of numbers that makes this statement true. You 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 5 hours.**### Unit 5: Graphs of Linear Equations and Inequalities

This unit is an introduction to graphing relationships between the two quantities on the coordinate plane. A graph helps visualize how one quantity depends on another. In this course, you will only graph relationships that can be described by a linear equation, and their graphs are always straight lines. Graphing is an important tool that aids in analyzing relationships, both in abstract and in applied math. The material in this unit will help you become comfortable with graphing pairs of numbers on the coordinate plane and understand how the equations and relationships can be represented by lines. As an example, you might want to graph how the location of a train depends on the time since the train departed from the station. If the train is moving with constant speed, this graph would be a straight line. The slant of this line (which, as you will learn in this unit, is called the slope) will depend on the speed: the greater the speed, the steeper the line. If the line is going up (looking from left to right), it tells you that the distance is growing with time, which means that the train is moving away from the station. Otherwise, if the line is going down, it tells you that the distance is decreasing, which means that the train is approaching the station. You will see that a lot of information about the train's journey can be gathered from just one graph.

**Completing this unit should take you approximately 21 hours.**### Unit 6: Systems of Linear Equations and Inequalities

You have seen in Units 2 and 5 that linear equations in one variable usually have 1 solution and linear equations in two variables have infinitely many solutions. What would happen if two linear equations in two variables had to be solved together? That would mean that a pair of numbers would have to satisfy both equations at the same time. This pair of numbers would be the solution of a system of linear equations. Some of the mixture problems from Unit 3 could be solved by setting up a system of linear equations, because they involve two given relationships between two variables. In this unit, you will learn three different methods of solving systems of linear equations and use them to solve a variety of world problems. You will also find solutions of systems of linear inequalities in two variables and to apply them to real-life situations. For example, if you want to determine the price for and amount of two types of candy for a party, you have the constraints of the total amount of candy needed (GREATER than a given amount) and the amount of money you can spend (LESS than the amount you have). The quantities of two types of candy you buy have to satisfy both constraints. In this unit, you will learn how to identify all the possibilities for similar problems.

**Completing this unit should take you approximately 18 hours.**### Unit 7: Operations with Monomials

Starting with this unit, you will work with expressions that consist mostly of letters (variables) and do not seem to have very much to do with numbers. You will see how the rules governing operations with these expressions arise from the properties of operations with numbers, particularly distributive property and order of operations. This unit focuses on expressions called monomials. These are expressions that contain only one term (recall from Unit 1 what "term" means). The fact that monomial contains the Greek word mono, which means one, can help you remember this definition. For example, expression

*ab*is a monomial, but*a + b*is not, because it contains two terms.**Completing this unit should take you approximately 7 hours.**### Unit 8: Operations with Polynomials

In this unit, you will become familiar with a special type of algebraic expressions, called a polynomial. A polynomial, as opposed to a monomial, is an expression that contains two or more terms. (The word poly means many in Greek.) Usually the polynomials you will work with will look like

*x*^{5}+ 2*x*^{3}+*x*+ 2, or similar. Polynomials have various special properties that you will be analyzing in the future math courses. In this course, you will learn how to recognize, classify, add, subtract, multiply, and divide polynomials. You will apply the skills of combining like terms and using distributive property in order to perform these operations. These skills are helpful when you are dealing with the motion of two or more objects: for example, when you need to calculate when and where one runner will overtake another runner. For another example, these skills are useful if you need to know how much interest you are earning from two or more savings accounts.**Completing this unit should take you approximately 13 hours.**### Unit 9: Factoring Polynomials

Factoring, the procedure you will learn to perform in this unit, is multiplication in reverse: instead of multiplying two polynomials, you will need to write a given polynomial as a product of two or more different expressions. Factoring is an important tool that you will use in solving quadratic, and, later, higher degree polynomial equations. Quadratic equations occur a lot in problems that involve rectangular objects and their areas: planning gardens, framing photographs, carpeting the floors, and so on.

**Completing this unit should take you approximately 23 hours.**### Final Exam