### Unit 2: Solving Linear Inequalities and Graphing

In this unit, you will learn to apply the concept of solving equations to solve problems involving linear inequalities. You will also learn how to graph a straight line, use different methods to find the slope and intercept of a line, and interpret slope and intercept. You will learn more about types of straight lines.

**Completing this unit should take you approximately 20 hours.**

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

- solve, graph, and give interval notation for basic linear inequalities and tripartite inequalities;
- solve distance, rate, and time problems;
- find the slope of a line from two points or a graph; and
- find the equation of a line from a graph, standard form, two points, parallel lines, and perpendicular lines.

### 2.1: Inequalities

Read this section. Note the difference in meaning among "equal," "less than," and "less than or equal," and the symbols we use for these ideas. Focus on how the steps for solving inequalities are different than for solving equations.

### 2.1.1: Graphing Inequalities

Watch this four-minute video, paying attention to the examples being used to explain graphing inequalities. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary. In this unit, the operations on the terms are very similar to solving equations. Here, we make use of inequality instead of an equal sign. The inequality sign switches/flips when you multiply or divide both sides by a negative number.

Complete page 32 of Wallace's workbook to practice graphing inequalities on a number line. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch these brief optional videos if you feel like you need additional help with this topic.

### 2.1.2: Interval Notation

Complete page 33 of Wallace's workbook to practice with interval notation. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

We use plots, graphs, and pictures to help us understand and communicate. Interval notation is a very visual way to communicate inequalities, and it also gives us visual techniques for solving them.

Watch this three-minute video, paying attention to the examples being used to explain interval notation. Note when to use a square bracket and a round bracket. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

Watch this brief optional video if you feel like you need additional help with this topic.

### 2.1.3: Solving Inequalities

### 2.1.3.1: Solving Linear Inequalities

Complete page 34 of Wallace's workbook to practice solving inequalities. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video, paying attention to the examples which provide a linear approach to solving inequalities. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

### 2.1.3.2: Solve Compound Inequalities (Tripartite)

Watch this five-minute video. Note that compound or tripartite inequalities are very similar to linear inequalities, except for the way they are structured. Bear in mind that in the tripartite inequalities, we will be balancing the left center and right. This means whatever you decide to do to the center in order to get the variable in question by itself, you have to do the same to the other two parts of the inequality. Notice also that if you divide or multiply through by a negative number, then you have to flip the inequality signs.

Complete page 35 of Wallace's workbook to practice solving inequalities. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

### 2.1.4: Homework Assessment

Answer odd-numbered problems 1 to 37.

Answer all problems, from 1 to 19.

### 2.2: Graphing, Slopes, and Rate of Change

### 2.2.1: Point and Lines

Read this section, on pages 89 to 93. One of our most powerful tools is visualization; graphs and XY coordinates comprise the foundation of "seeing" math. Getting comfortable when working with points and lines in a coordinate system is the main point of this section.

Read the following short article for a brief description of the four quadrants of a Cartesian graph.

Complete page 36 of Wallace's workbook for practice with graphing based on points provided or equation of the line provided. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video, which discusses points and lines in graphing and slopes. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

Watch this brief optional video if you feel like you need additional help with this topic.

### 2.2.2: Slope from a Graph

Read this section, which focuses on one fundamental property of the line: its slope. The slope tells us how much the line rises as we move to the right. There are several ways to talk about this one idea.

Complete page 37 of Wallace's workbook to work on determining the slope of a line, given the line on a graph. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this four-minute video, paying attention to the examples being used to explain the slope of a graph. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

### 2.2.3: Slope from Two Points

Complete page 38 of Wallace's workbook to work on finding the slope when given two sets of points. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video. Take note of the examples used in the video to explain how to find the slope using two points on the graph. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

Watch this brief optional video if you feel like you need additional help with this topic.

### 2.2.4: Homework Assessment

Work through problems 1, 2, and every other odd problem from question 3 through 21.

Attempt the odd problems from questions 1 through 29.

### 2.3: Equations of Lines

Read this section. We know what a line is when we see it plotted. There is more than one useful way to write the equation for a line.

### 2.3.1: Slope-Intercept Equations

Complete page 39 of Wallace's workbook to practice providing the slope intercept equation when given information of the slope and y-intercept, or a graph. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video to learn about the slope-intercept equation and its applications. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

Watch this brief optional video if you feel like you need additional help with this topic.

### 2.3.2: Putting Equations in Intercept Form

Complete page 40 of Wallace's workbook to practice putting an equation in intercept form. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this four-minute video to learn about putting equations in intercept form. Understand that sometimes there will be no intercept or slope, in which case the slope or intercept will be zero.

### 2.3.3: Equation from Graph

Complete page 41 of Wallace's workbook to work on graphing equations. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video, paying attention to the examples being used to instruct you on how to graph equations. Be sure that you understand the different parts of the straight line equation and the names that go with them. You may practice by graphing your own lines and seeing if you can label them completely.

### 2.3.4: Vertical and Horizontal Lines

Complete page 42 of Wallace's workbook to practice graphing or finding the equations for vertical and horizontal lines. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this two-minute video, which discusses the different characteristics of vertical and horizontal lines. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

### 2.3.5: Point-Slope Equations

Complete page 43 of Wallace's workbook to practice finding the point slope equation, given a point the line passes through and the slope. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video, paying attention to the examples being used to demonstrate the point-slope equation.

Watch this brief optional video if you feel like you need additional help with this topic.

### 2.3.6: Using Two Points to Find the Equation of a Line

Complete page 44 of Wallace's workbook to practice with finding an equation when given two points. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video, focusing on the examples used to find the equation of the line with two points given.

### 2.3.7: Homework Assessment

Answer every other odd-numbered problem in questions 1 through 41.

Answer every other odd-numbered problem in questions 1 through 51.

### 2.4: Parallel and Perpendicular

Read this section. Parallel lines never intersect, which is good to know if you are looking for an intersection! The shortest distance from a line to something else is perpendicular to the line. The slopes for parallel and perpendicular lines have simple relationships to each other (which you should memorize!).

Watch this brief optional video if you feel like you need additional help with this topic.

### 2.4.1: Slope

Complete page 45 of Wallace's workbook to practice determining whether lines will be parallel, perpendicular, or neither. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video, focusing on the examples used to explain how to find the slope of parallel and perpendicular lines. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

### 2.4.2: Equations

Complete page 46 of Wallace's workbook to find the equations of parallel and perpendicular lines. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video to learn about equations used for parallel and perpendicular lines.

### 2.4.3: Homework Assessment

Answer the odd-numbered problems for questions 1 through 47.

### 2.5: Distance and Rate Problems

Read this section. The relationship among distance, rate, and time is one of the simplest and most fundamental of physics and engineering. Notice how d = rt is a relationship that can be solved for any of the three variables.

### 2.5.1: Opposite Directions

Complete page 47 of Wallace's workbook to practice solving distance problems. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video, which demonstrates how to solve distance problems involving opposite directions. In working with distance and rate problems, it will be helpful if you begin every question by first drawing a picture and translating the picture into a table as demonstrated in the video. Finally, apply the formula to get the required answer. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

### 2.5.2: Catch-Up

Complete page 48 of Wallace's workbook to practice solving distance problems. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Two objects may be moving in the same direction or in different directions. This is the simplest relationship between two moving objects. "Catch-Up" problems are about moving in the same direction but at different speeds.

Watch this five-minute video, which demonstrates distance problems in which one person is trying to catch up with another person. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

### 2.5.3: Total Time

Complete page 49 of Wallace's workbook to practice solving distance problems. Try to complete this exercise before watching the video in this subunit, and then review the worksheet as you follow along with the video for solutions.

Watch this five-minute video, paying attention to the distance and rate problems shown to demonstrate the concept of total time. You may watch the video as often as you please. You may refer to the video when doing your homework, if necessary.

### 2.5.4: Homework Assessment

Answer the odd-numbered problems for questions 1 through 37.

### Unit 2 Practice Test

Review Unit 2 before taking this practice test. Be sure that you are ready before taking the practice test, as it will give you a clear picture of what you know and the areas you need to review, if necessary. You may review the problems in the work pages in addition to watching the videos to prep for the practice test. When you have finished this practice test, you may check your answers against the "Unit 2 Practice Test - Answer Key".

### Unit 2 Assessment

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!
- You can take this assessment as many times as you want, whenever you want.

- This assessment