### Unit 1: Equations and Inequalities

In this unit, you will learn important terminology and characteristics of linear equations, inequalities, and their graphs. Additionally, we will cover how to solve general linear equations in one variable, literal equations, and compound inequalities. You will also explore how to graph linear equations and inequalities in two variables on a coordinate plane, learn how to solve systems of two or three linear equations, and apply these skills to real-world applications. You may want to refer to this unit as you move through the course. For some, this material may be a repeat, and you will be able to move quickly. For others, it may have been a long time since you practiced the concepts in this unit, and taking the time to become fluent with solving and graphing equations and inequalities will pay off later in the course.

**Completing this unit should take you approximately 4 hours.**

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

- solve linear, rational, quadratic, radical, and absolute value equations in one variable;
- classify solutions to linear, rational, quadratic, radical, and absolute value equations;
- construct linear equations from words;
- represent linear and absolute value inequalities using standard notation and graphs;
- solve linear and absolute value inequalities; and
- perform algebraic operations on complex numbers.

- solve linear, rational, quadratic, radical, and absolute value equations in one variable;

### 1.1: Solving Linear and Rational Equations in One Variable

This passage assumes you have already been exposed to solving a variety of linear equations. This refresher is intended to support you as you explore and manipulate linear functions.

This section provides you with applications of the linear equation and its representation on the Cartesian plane. Examples are given in the context of real-world models and scenarios.

This section will prepare you for studying rational functions and their graphs. Methods for solving rational equations include using the LCD and factoring. Examples also include determining when there are excluded values in the solution.

### 1.2 Quadratic, Radical, and Absolute Value Equations

This summary of algebraic operations on complex numbers will prepare you for solving quadratic equations with no solutions and the related implications for graphing quadratic and polynomial functions.

This is a refresher on solving quadratic equations using factoring methods. This section assumes you have been exposed to methods for factoring previously.

This is a refresher on solving quadratic equations using the square root property. This section assumes you have been exposed to simplifying roots algebraically.

This is a refresher on using the quadratic formula to solve quadratic equations. This section will introduce the discriminant and explain how to use it to classify the number and type of solutions to a quadratic equation. This analysis is an essential step in learning how to analyze the behavior of functions using algebraic and graphical methods.

This section explores solving different types of equations and analyzing solutions.

In this section, you will explore methods for solving absolute value equations and how to analyze solutions to determine their feasibility. This section will help you become familiar with the algebra of absolute value equations in preparation for functions.

This is a refresher on solving various radical equations and determining whether there are extraneous solutions. Radical is another term for root, so you will be solving equations that contain square roots.

### 1.3: Linear Inequalities

This section is an important foundation for learning how to express intervals and sets fluently using words, set-builder notation, and interval notation. You will use the concepts you learn here to describe the solutions to inequalities. Similarly, we can use sets to describe the behavior of all types of functions. You will be required to use the notation and concepts presented here in many future units in the course, so it is vital to make sure that you achieve mastery of the techniques presented here to be successful in the coming sections on functions.

This refresher on solving linear inequalities allows you to practice describing solutions using interval notation, set notation, and graphs. You will also have a chance to practice solving compound linear inequalities.

This refresher on solving absolute value inequalities lets you practice writing solutions using interval notation. You will also practice graphical analysis of absolute value inequalities.