### Unit 3: Algebra

This unit explores several essential algebra skills: working with algebraic expressions, solving linear and quadratic equations, and representing them on the coordinate plane. You will also learn how to apply algebraic expressions, equations, and graphs to describe and analyze word problems.

**Completing this unit should take you approximately 26 hours.**

Upon successful completion of this unit, you will be able to:

- apply properties of operations to determine equality or inequality of algebra equations, inequalities, and expressions;
- define a real-world problem using algebraic equations and inequalities;
- solve linear and quadratic equations graphically or algebraically;
- graph and interpret linear equations identifying plot points, slope and x & y intercepts; and
- identify and evaluate functions given algebraically and graphically.

### 3.1: Algebraic Expressions

This section reviews basic algebra skills: working with variables, simplifying variable (algebraic) expressions, and translating verbal expressions into mathematics.

While arithmetic deals primarily with operations with numbers, in algebra, you will deal with expressions that involve variables–letters that represent real numbers. Watch this lecture series to review the concept of a variable and the conventional way to write basic expressions involving variables. Complete the interactive exercises.

You can use two main tools to simplify or rewrite algebraic expressions: combining like terms and using the distributive property. Watch this lecture series and complete the interactive exercises to practice these skills.

Complete these exercises, then watch the video to check your solutions.

This section explains how you can write down verbal phrases in terms of variables and mathematical operations. This skill will be used later in this unit to solve word problems.

Complete these exercises and check your answers.

Complete these exercises and check your answers.

### 3.2: Linear Equations with One Variable

In this section, you will learn how to solve general linear equations with one variable and use the equations to describe common situations in word problems.

This lecture series explains the difference between an equation and an algebraic expression. It also defines what it means to solve an equation. Watch the videos and complete the interactive exercises.

Watch this lecture series to review how to solve basic one-step equations involving addition and subtraction.

Watch this lecture series to review how to solve basic one-step equations involving multiplication and division.

Two-step equations can also be solved by "undoing" each operation by applying its inverse to both sides of the equation. Watch this lecture series and complete the interactive exercises.

This lecture series shows how you can apply the principle of doing the same thing to both sides of the equation to equations with variables on both sides. Watch the videos and complete the interactive exercise sets.

Finally, you will look at the most general linear equations with one variable: equations involving parentheses. Here, you have to simplify each side by opening parentheses before attempting to solve by doing the same thing to both sides. Watch this lecture series and complete the interactive exercises.

This section this textbook explains how to translate the situations described in word problems to equations and provides a variety of examples. Read the chapter and work through the problems. Some examples involved the geometric facts you have learned in Unit 2.

This chapter discusses a common type of word problem that can be solved by linear equations: mixture problems. Read the chapter, watch the videos, and work through examples. Complete the review exercise at the end of the chapter.

Complete these exercises and check your answers.

Complete these exercises and check your answers.

### 3.3: Linear Inequalities with One Variable

Solving inequalities is very similar to solving equations, with a few important distinctions. First, the solution to an inequality is a set of numbers rather than just one value. This is why the solutions of inequalities are typically represented on a number line. And, while you can still do the same thing to both sides of inequality to isolate the variable, you have to remember to flip the sign of inequality (less to greater, and vice versa) when multiplying and dividing by a negative number.

Watch this lecture series and complete the interactive exercises to review what an inequality is, what it means to find a solution set, and how to represent it on a number line.

Watch this lecture series and complete the interactive exercises to review how to solve, graph, and represent the solutions to one-step inequalities.

This lecture series provides examples of two-step inequalities and their applications. Watch the videos and complete the interactive exercises.

The approach to solving linear inequalities is similar to equations: first, simplify each side, then isolate a variable by doing the same thing to both sides. Remember to switch the sign when multiplying or dividing by a negative number. This lecture series shows examples of solving inequalities and using them to solve word problems. Watch the videos and complete the interactive exercises.

Complete these exercises and check your answers.

Complete these exercises and check your answers.

### 3.4: Quadratic Equations

Quadratic equations are equations that contain a variable in the second power (squared). This section introduces the primary method for solving quadratic equations: quadratic formula. Quadratic equations can have two real solutions (rational or irrational), one real rational solution, or no real solutions.

Watch these videos and complete the interactive exercises.

Read this section to review the process of solving quadratic equations by using the quadratic formula.

This section describes using quadratic equations to solve word problems involving numbers, geometrical figures, and motion. Read this section and work through the examples.

Complete these exercises and check your answers.

Complete these exercises and check your answers.

### 3.5: Graphs of Linear Equations

This section explores how linear equations can be represented by graphs on the coordinate plane and the properties of these graphs.

This lecture series reviews the basic concepts related to graphing points on the Cartesian coordinate plane and associated terminology.

While the solution of a linear equation in one variable is one value of x, the solution of an equation in two variables is an ordered pair of values, x and y. When these solutions are plotted on the coordinate plane, they form a line (hence the term "linear" equation). Watch this lecture series, which explains how to find and graph the solutions of a linear equation in two variables. Complete the interactive exercises.

To describe a line, it is important to indicate how steep it is. This property of the line is called slope. Slope can be any number, including zero (when the line is horizontal). Vertical lines have an infinitely large slope. This lecture series explains how to find the slope of a line given two points and how to graph a line given its slope. Watch the videos and complete the interactive exercises.

Another important property of a line (or any curve on a coordinate plane) are its x- and y-intercepts: the points where the line intersects coordinate axes. Watch this lecture series and complete the interactive exercises.

This lecture series explores the meaning of slope and intercepts in the context of real-life situations. Watch the videos and complete the interactive exercises.

As you have seen from examples, you can write linear equations in different ways. There are three main forms of linear equations: slope-intercept, point-slope, and standard. This lecture series introduces the point-slope form. Watch the videos and complete the interactive exercises. This lecture series focuses on graphing linear equations when they are given in slope-intercept form.

Watch this lecture series and complete the interactive exercises to learn how to write an equation of a line in slope-intercept form.

Point-slope form might be less familiar and more formal-looking. It is a general form of a linear equation with a known slope and one of the points. Watch this lecture series and complete the interactive exercises.

When a linear equation is written in standard form, both variables x and y are on the same side of the equation. Watch this lecture series and practice converting equations to standard form.

Finally, review how to get information about the line using any form of a linear equation representing this line.

Complete these exercises and check your answers.

### 3.6: Graphs of Quadratic Equations

This section explores how quadratic equations can be represented by graphs on the coordinate plane and the properties of these graphs.

While the graph of a linear equation is a straight line, the graph of a quadratic equation is a curve called a parabola. Parabolas are more complicated to graph than lines, but they have distinct features and properties that you can use to help with graphing. This lecture series explores what all parabolas have in common and how to use them to model real-life situations. Watch the videos and complete the interactive exercises.

As you have seen, all parabolas have a vertex and an axis of symmetry. You can write a quadratic equation in vertex form, making it easy to find the vertex and graph. Watch this lecture series and complete the interactive exercises.

If a quadratic equation is written in standard form, it can be converted to vertex form by using the vertex formula or completing the square. Watch this lecture series and complete the interactive exercises.

Complete these exercises and check your answers.

### 3.7: Functions

This section will introduce the concepts of relations and functions.

A relation is a rule that describes a relationship between two variables. It can be represented in various ways: verbally, as a set of ordered pairs, as an equation, or as a graph on a coordinate plane. A function is a particular kind of relation. This lecture series discusses how to recognize functions when they are given by different representations. Watch the videos and complete the interactive exercises.

This lecture series focuses on working with functions that are represented by equations and graphs. Watch the videos and complete the interactive exercises.

Complete these exercises and check your answers.

### Unit 3 Assessment

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Take this assessment to see how well you understood this unit.

- This assessment
**does not count towards your grade**. It is just for practice! - You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
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