 # K12MATH011: Algebra II

Unit 1: Linear Equations and Inequalities   You will begin Algebra II with an introduction to functions and relations. A relation is a matching of values. For instance, the ordered pair (2, 5) matches the value 2 with 5. An ordered triple (3, -1, 4) matches the value 3 with values -1 and 4. A relation is a set of any matchings of this kind. A function is a type of relation that matches each first value with unique second and third values, and so forth. For instance, if (2, 9) is in a function, then any other time we see 2 as a first value in the function, it must be matched with 9 again. It is important to note that this is strictly one way. That is (2, 9) and (2, -4) can’t both be part of the same function, but (2, 9) and (3, 9) can. Linear equations and inequalities are among the easiest to learn and use, and they serve as a basis for the learning of more complex functions. These functions are called linear because they graph as straight lines.

In this section, you will become familiar with linear expressions, equations, and inequalities. There are several different ways to write the same equation, and you will learn the different forms and why each is important, as well as how and when to use each form. Additionally, you will learn how to change one form into another, and you will also learn various ways to graph lines. Once you master, you will learn how to use this knowledge to solve and graph linear inequalities. You will learn how to graph an inequality based on a comparison test, once the inequality is in slope-intercept form, and based on the use of test coordinates. The importance of using test coordinates becomes clearer later, as you use similar techniques throughout the remainder of your mathematics courses. You will complete this unit by learning how to solve absolute value equations and inequalities, which are just variations on linear equations and inequalities.

Completing this unit should take approximately 23 hours.

☐    Subunit 1.1: 7 hours and 45 minutes

☐    Subunit 1.2: 8 hours and 30 minutes

☐    Subunit 1.3: 2 hours and 30 minutes

☐    Subunit 1.4: 4 hours and 15 minutes

Unit1 Learning Outcomes
Upon successful completion of this unit, you will be able to: - Write the equation of a line through two points on a coordinate plane. - Graph a linear function. - Solve linear equations for a given variable. - Write a linear equation in three equivalent forms: standard, point-slope, and slope-intercept. - Determine if two lines are parallel, if they simply intersect, or if they are perpendicular. - Graph linear inequalities. - Find the solution of linear inequalities by comparison. - Find the solution of linear inequalities by using test coordinates.

1.1 Relations and Functions   This subunit concerns mathematical relations and functions. At its core, a relation is just a set of matchings of two or more values. It does not matter if multiple values are matched to one value. Functions are a special case of relations. For every first value (x), only one value (y) may be matched. That is, (2, 5) and (2, -3) may both be part of the same relation, but (2, 5) and (2, -3) cannot both be part of the same function. That is, if (2, 5) is part of a function and 2 is matched again, it must be matched to 5 again for the relation to meet the definition of a function. Subunit 1.1 provides a lot of information on relations and functions.

1.1.1 Introduction to Relations   1.1.1.1 What is a Relation?   - Explanation: Oswego City School District Regents Exam Prep Center: Donna Roberts’s “Definition of a Relation and a Function” Link: Oswego City School District Regents Exam Prep Center: Donna Roberts’s “Definition of a Relation and a Function” (HTML)

Instructions: This is a plain English definition of a relation and function, showing that a function is a special case of relations. The vertical line test is employed to show how to distinguish between them graphically.

``````-   [CCSS.Math.Content.HSF-IF.A.1](http://www.corestandards.org/Math/Content/HSF/IF/A/1)

displayed on the webpage above.
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1.1.1.2 Determining Membership in a Relation   - Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 1, Section 3: Introduction to Functions” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 1, Section 3: Introduction to Functions” (PDF)

Instructions: This precalculus textbook begins with a discussion of relations and functions. It discusses important concepts like the vertical line test for distinguishing between relations and functions on a graph. It also incorporates discussion of the use of graphing calculators with the material. Read pages 43–48, and then try the problems at the end of the chapter. Answers are on the pages that follow.

Completing this activity should take approximately 1 hour.

``````-   [CCSS.Math.Content.HSF-IF.A.1](http://www.corestandards.org/Math/Content/HSF/IF/A/1)
-   [CCSS.Math.Content.HSF-IF.B.5](http://www.corestandards.org/Math/Content/HSF/IF/B/5)

It is attributed to Carl Stitz and Jeff Zeager.
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Instructions: This video gives an introduction to domain and range, which are the “input” and “output” of a relation. Basically, these are the sets of values that are matched together by the relation.

Watching the video and writing notes should take approximately 15 minutes.

1.1.2 Functions   1.1.2.1 What is a Function?   - Explanation: Oswego City School District Regents Exam Prep Center: Donna Roberts’s “Functional Notation and Evaluating” Link: Oswego City School District Regents Exam Prep Center: Donna Roberts’s “Functional Notation and Evaluating” (HTML)

Instructions: This is a concise explanation of functional notations and how they relate. It also provides the basics of functional evaluation. Note: At the bottom of the page is a link demonstrating how to use a calculator to evaluate functions, which is a very handy piece of information. Write a short essay (one to three paragraphs) on how to go from hand-worked problems to doing the same problems on calculator. How are the methods similar? How are they different? Does using a calculator require a different way of thinking?

Completing this activity should take approximately 30 minutes.

`````` Standards Addressed (*Common Core*):

-   [CCSS.Math.Content.HSF-IF.A.2](http://www.corestandards.org/Math/Content/HSF/IF/A/2)
-   [CCSS.ELA-Literacy.RST.9-10.3](http://www.corestandards.org/ELA-Literacy/RST/9-10/3)
-   [CCSS.ELA-Literacy.RST.9-10.7](http://www.corestandards.org/ELA-Literacy/RST/9-10/7)

displayed on the webpage above.
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Instructions: This video provides an extended definition of a function. In conjunction with the reading in this unit, this video will help emphasize the concept of functions, in general.

Watching the video and writing notes should take approximately 15 minutes.

1.1.2.2 How Functions Differ from Relations   - Explanation: Monterey Institute: “Unit 3, Lesson 2, Topic 1: Representing Functions and Relations” Link: Monterey Institute: “Unit 3, Lesson 2, Topic 1: Representing Functions and Relations” (HTML)

Instructions: This textbook provides an in-depth introduction to functions and relations. It discusses what relations and functions are and gives a good explanation of the importance of identifying and distinguishing between them. The pages have a series of self-quizzing questions that appear throughout. The answers to the questions are contained on the page and may be seen by clicking on the “Show/Hide Answer” link for each.

``````-   [CCSS.Math.Content.HSF-IF.C.7](http://www.corestandards.org/Math/Content/HSF/IF/C/7)

displayed on the webpage above.
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1.1.2.3 Determining Membership in a Function   - Explanation: Khan Academy’s “Domain and Range of a Function Given a Formula” Link: Khan Academy’s “Domain and Range of a Function Given a Formula” (YouTube)

Instructions: This video provides a discussion of domain and range, which are the sets of “input” and “output” of a function.

Watching the video and writing notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.HSF-IF.A.1](http://www.corestandards.org/Math/Content/HSF/IF/A/1)
-   [CCSS.Math.Content.HSF-IF.A.2](http://www.corestandards.org/Math/Content/HSF/IF/A/2)

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Instructions: This video provides a discussion of domain and range, given an example. This walks through the setup of an equation from the information given and then shows how to find domain and range samples from the real numbers.

Watching the video and writing notes should take approximately 15 minutes.

Instructions: This video provides a discussion of representing domain and range on a coordinate plane, creating ordered pairs, and then plotting them.

Watching the video and writing notes should take approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Domain of a Function” Link: Khan Academy’s “Domain of a Function” (HTML)

Instructions: This is a self-quiz on domains of functions. Note that there are hints available if needed.

Taking the quiz and writing notes should take approximately 45 minutes.

• Did I Get This? Activity: Khan Academy’s “Range of a Function” Link: Khan Academy’s “Range of a Function” (HTML)

Instructions: This is a self-quiz on adding and subtracting polynomials. Note that there are hints available if needed.

Taking the quiz and writing notes should take approximately 45 minutes.

• Did I Get This? Activity: Oswego City School District Regents Exam Prep Center: “Multiple Choice Practice: Patterns, Functions & Relations” Link: Oswego City School District Regents Exam Prep Center: “Multiple Choice Practice: Patterns, Functions & Relations” (HTML)

Instructions: This is a 20-question self-quiz on patterns, functions, and relations. It covers several subtopics in relations and functions and serves as a handy worksheet to test your current level of understanding.

Taking the quiz and writing notes should take approximately 1 hour.

1.2 Linear Equations   Lines are the first exposure to actual algebraic functions that most students see. Linear functions are special cases of polynomial functions (which we will see later in the course). We study lines first because many of the things we can demonstrate with functions, such as translation forms for graphing and solutions of equations, can be shown in an easy-to-understand way with linear functions.

• Explanation: Everything Maths: Grade 10 Mathematics: “Linear Functions” Link: Everything Maths: Grade 10 Mathematics: “Linear Functions” (HTML)

Instructions: This textbook contains a great deal of information on linear equations, including the different forms of a line, slope, the identity function, and constant functions. Although it is packed with information, it is concise, making it an easy read. At the end of the page are a series of self-quizzing questions.

Reading these pages, writing notes, and taking the self-quiz should take approximately 1 hour.

1.2.1 Standard Form of a Line   - Explanation: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 2, Section 1: Linear Functions” Link: Carl Stitz and Jeff Zeager’s Precalculus: “Chapter 2, Section 1: Linear Functions” (PDF)

Instructions: This precalculus textbook offers an in-depth discussion of linear functions. It also incorporates discussion of using graphing calculators with the material. Read pages 151–162, and then try the problems at the end of the chapter. The answers are on the pages that follow.

Completing this activity should take approximately 1 hour.

``````-   [CCSS.Math.Content.HSA-SSE.A.2](http://www.corestandards.org/Math/Content/HSA/SSE/A/2)
-   [CCSS.Math.Content.HSA-REI.D.10](http://www.corestandards.org/Math/Content/HSA/REI/D/10)

It is attributed to Carl Stitz and Jeff Zeager.
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Instructions: This video gives a brief but very useful discussion of the relationship between slope-intercept and standard forms of a line. This video is important for showing how algebraic methods for equations can be used to translate equations from one form to another.

Watching the video and writing notes should take approximately 15 minutes.

• Explanation: YouTube: Eric Buffington’s “Solving Linear Equations in Standard Form (Simplifying Math)” Link: YouTube: Eric Buffington’s “Solving Linear Equations in Standard Form (Simplifying Math)” (YouTube)

Instructions: This video provides a very useful discussion of how to use the standard forms of a line in graphing. Importantly, this video also shows how algebraic methods for equations can be used to find intercepts in general.

Watching the video and writing notes should take approximately 15 minutes.

1.2.2 Slope-Intercept Form of a Line   - Explanation: YouTube: James Sousa’s “Slope-Intercept Form of a Line” Link: YouTube: James Sousa’s “Slope-Intercept Form of a Line” (YouTube)

Instructions: This video gives an excellent discussion of slope-intercept form of a line. Interesting in the instructor’s approach is the graphical representation of a line and how the line changes as the slope changes. He also does a very good job of using several different examples to emphasize his points. As a bonus, he discusses how to take two points on a line and find the slope-intercept form.

Watching the video and writing notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.HSA-SSE.A.2](http://www.corestandards.org/Math/Content/HSA/SSE/A/2)
-   [CCSS.Math.Content.HSA-REI.D.10](http://www.corestandards.org/Math/Content/HSA/REI/D/10)

attributed to James L. Sousa.
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• Web Media: GeoGebra: “Transforming Linear Graphs” Link: GeoGebra: “Transforming Linear Graphs” (HTML)

Instructions: This interactive graph allows you to transform linear graphs by changing the values on the sliders to see how these changes to the slope and y-intercept affect the graph and the equation. Also, use the check box to see a reflection across a vertical axis. Then write a paragraph explaining how transformations work. Writing this will help cement your understanding of this important concept.

Practicing with this application and writing notes should take approximately 30 minutes.

• Web Media: GeoGebra: “Exploring Slope and Y-Intercept” Link: GeoGebra: “Exploring Slope and Y-Intercept” (HTML)

Instructions: This interactive graph has a wealth of information to it. It allows you to change the slope and y-intercept with slide bars to see how other values and the graph change. For an explanation of some of the items seen, “A” is the y-intercept, “B” is the point on the line when x = 2, “a” is the standard form of the line, “f(x)” shows the slope-intercept form, and “m” is the slope. Note: You can only change the slope and y-intercept with this application. Find the blue slope line legend in the middle of the graph. Slide the circle left and right to see how the line changes when the slope is 1 or -1. Change the y-intercept to see how the equation and graph change, as well. Makes notes on how the slope and y-intercept affect the graph of a line.

Practicing with this application and writing notes should take approximately 30 minutes.

1.2.3 Point-Slope Form of a Line   - Web Media: GeoGebra: “Point-Slope Form” Link: GeoGebra: “Point-Slope Form” (HTML)

Instructions: This interactive graph allows you to change the slope using a slide bar or to move a point around to see how the point-slope form of a line changes, as well as the graph. It’s a great way to link the point-slope form visually with the graph. Note: Slider 1 changes the slope in 0.5 unit increments. Slider 2 is simply 0 or 1 (that is, true or false). If you wish to see the slope-intercept form of the equation in point-slope form, slide to 1. If not, slide to 0.

Practicing with this application and writing notes should take approximately 30 minutes.

``````-   [CCSS.Math.Content.HSA-REI.D.11](http://www.corestandards.org/Math/Content/HSA/REI/D/11)

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Instructions: This video gives an excellent discussion of the slope of a line and how it does not matter which coordinates one uses first in the formula.

Watching the video and writing notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.HSA-SSE.A.2](http://www.corestandards.org/Math/Content/HSA/SSE/A/2)
-   [CCSS.Math.Content.HSA-REI.D.10](http://www.corestandards.org/Math/Content/HSA/REI/D/10)

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1.2.5 Graphing Lines   1.2.5.1 Graphing by Finding Two Points on the Line   - Web Media: GeoGebra: “Line Through Two Given Points” Link: GeoGebra: “Line Through Two Given Points” (HTML)

Instructions: This interactive graph allows you to input the coordinates of two points or to change the coordinates of the points by dragging one or the other. This is to allow you to practice finding the slope and using slope-point form to find the slope-intercept form of the line. To actually see the equation of the line step by step, click the “Text” check box in the upper left corner of the graph. The text begins with finding the slope. Steps two, three, and four demonstrate how to find the y-intercept. Step five demonstrates finding the slope-intercept form of the line. Note: You may have to make sure you have the current version of Java for this applet.

Practicing with this application and writing notes should take approximately 45 minutes.

``````-   [CCSS.Math.Content.HSA-SSE.A.2](http://www.corestandards.org/Math/Content/HSA/SSE/A/2)
-   [CCSS.Math.Content.HSA-REI.D.10](http://www.corestandards.org/Math/Content/HSA/REI/D/10)

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Instructions: This video looks at graphing from several methods, including point-slope form. This is one of the more comprehensive lectures on linear equations, and it reinforces several of the techniques we’ve already seen.

Watching the video and writing notes should take approximately 30 minutes.

``````-   [CCSS.Math.Content.HSA-SSE.A.2](http://www.corestandards.org/Math/Content/HSA/SSE/A/2)
-   [CCSS.Math.Content.HSA-REI.D.10](http://www.corestandards.org/Math/Content/HSA/REI/D/10)

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• Web Media: GeoGebra: “Linear Functions” Link: GeoGebra: “Linear Functions” (HTML)

Instructions: This interactive graph allows you to drag and drop two points to graph a given linear equation. Note: There are several levels, involving different forms of a line, beginning with slope-intercept form, then moving to standard form after you have successfully completed the first five. Also, ignore the “New Point” and “Move Graphics View” buttons and stick with the “Move” button to move the two points on the graph.

Practicing with this application and writing notes should take approximately 45 minutes.

Instructions: This video looks at graphing lines by finding the x and y intercepts for the line. In addition to being an easy, yet important, method for graphing lines, it is also good practice in using substitutions to solve equations, which is a technique used often in algebra and calculus.

Watching the video and writing notes should take approximately 30 minutes.

``````-   [CCSS.Math.Content.HSA-REI.D.11](http://www.corestandards.org/Math/Content/HSA/REI/D/11)

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• Web Media: GeoGebra: “Linear Equation Mixed Forms” Link: GeoGebra: “Linear Equation Mixed Forms” (HTML)

Instructions: This interactive graph allows you to plot points for equations of different forms. Once you have the x and y intercepts of the line plotted correctly, by dragging and dropping the two points they provide (it does not matter which you use for x and y intercepts), the line appears. By clicking on the “Refresh” icon in the top right corner of the graph, you can get a new equation to practice. This is great practice for finding the x and y intercepts.

Practicing with this application and writing notes should take approximately 45 minutes.

• Web Media: GeoGebra: “Exploring Linear Equations” Link: GeoGebra: “Exploring Linear Equations” (HTML)

Instructions: This interactive graph allows you look at the graphs of linear equations of different forms, simply by clicking on a check box by each type. By changing the graph and equations by clicking on the “Refresh” icon in the top right corner of the graph, you can receive new problems. This is a good way to learn how to match the equation with the visual graph.

Practicing with this application and writing notes should take approximately 45 minutes.

1.3 Linear Inequalities   The first practical relationships to which we are introduced are linear inequalities. Practical in this case means they have real-world applications. We study linear equations first, since many of the techniques used to solve inequalities are extensions of solving equations. Ironically, graphing is sometimes not very useful in solving equations, since any real number can be a solution and it is hard to plot some decimals. On the other hand, graphs of inequalities are very handy for solving equations, as the solution sets are parts of planes rather than a point on a line. We can’t list all the points in an inequality, but we can graph them.

1.3.1 Graphing Linear Inequalities by Using a Comparison   - Web Media: GeoGebra: “Linear Inequalities N1v2 y ≤,≥ mx + b” Link: GeoGebra: “Linear Inequalities N1v2 y ≤,≥ mx + b” (HTML)

Instructions: This interactive graph allows you to change the y-intercept and slope to adjust the line and change from ≤ to ≥in order to affect the shading. This is good practice for using the comparison method of solving inequalities. Note: the boundary equation follows the left side of the line around as the slope and intercepts change. The possible slopes change from ­10 to 10 and the possible y-intercepts from -15 to 15.

Practicing with this application and writing notes should take approximately 45 minutes.

``````-   [CCSS.Math.Content.HSA-REI.D.12](http://www.corestandards.org/Math/Content/HSA/REI/D/12)

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1.3.2 Graphing Linear Inequalities by Using Test Coordinates   - Explanation: Monterey Institute: “Unit 5, Lesson 2, Topic 1: Solving and Graphing Inequalities in Two Variables” Link: Monterey Institute: “Unit 5, Lesson 2, Topic 1: Solving and Graphing Inequalities in Two Variables” (HTML)

Instructions: This is a great description of a common mathematical method, which involves using test coordinates to determine the solution spaces for inequalities. This is not only important for linear inequalities and systems of inequalities but can be extended to use for any type of inequality.

``````-   [CCSS.Math.Content.HSA-REI.D.12](http://www.corestandards.org/Math/Content/HSA/REI/D/12)

displayed on the webpage above.
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• Web Media: GeoGebra: “Linear Inequality Solution” Link: GeoGebra: “Linear Inequality Solution” (HTML)

Instructions: This interactive graph allows you to move a point around in order to check regions of an inequality. You only need one test point in a region, and if any point in the region works, that is the area that gets shaded with the test-point method. You can test multiple problems. Note: This is a test that can apply to most inequalities, not just linear inequalities.

Practicing with this application and writing notes should take approximately 45 minutes.

1.4 Absolute Value Equations and Inequalities   Absolute value equations and inequalities are the second type of functions and relations, respectively, to which most students are exposed. For now we will concentrate on the linear forms, and we will see that, once again, solving them tends to involve extensions of solving linear equations. Also, studying the graphs of linear absolute value equations is priceless for studying reflections, stretching, compressions, and shifts of basic graphs. Absolute value equations are very easy to write in translation form (the form that shows how to take a basic function, such as f(x) = |x|, and shift it left or right, up or down; reflect across the x-axis, etc).

Instructions: This video introduces you to the concept of absolute value equations. It uses the definition of absolute value with methods for solving linear equations to find the solutions for these equations. This video provides a simple example.

Watching the video and writing notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.HSF-IF.C.7](http://www.corestandards.org/Math/Content/HSF/IF/C/7)
-   [CCSS.Math.Content.HSF-IF.C.7b](http://www.corestandards.org/Math/Content/HSF/IF/C/7/b)

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Instructions: This video contains a more complex example than the first, but it continues to use the same techniques as before. These videos are important in that they not only demonstrate how to solve absolute value equations, but they also demonstrate how mathematical definitions can be used in conjunction with equation properties to solve problems.

Watching the video and writing notes should take approximately 15 minutes.

• Web Media: GeoGebra: “Absolute Value Function” Link: GeoGebra: “Absolute Value Function” (HTML)

Instructions: This interactive graph allows you to change the function shifts (horizontal and vertical) and the graph scaling. This is great practice toward understanding absolute value functions and shifts and scaling for any basic functions. Note: The interactive graph uses three sliders to change a, h, and k. Take note of each one in the equation. Almost all function transformations look similar to this.

Practicing with this application and writing notes should take approximately 45 minutes.

1.4.2 Solving Absolute Value Inequalities   - Explanation: Khan Academy’s “Absolute Value Inequalities Example 1” Link: Khan Academy’s “Absolute Value Inequalities Example 1” (YouTube)

Instructions: This video introduces you to the concept of absolute value inequalities, providing a simple example. It uses the definition of absolute value with methods for solving linear inequalities to find the solutions for these problems.

Watching the video and writing notes should take approximately 15 minutes.

``````-   [CCSS.Math.Content.HSF-IF.C.7](http://www.corestandards.org/Math/Content/HSF/IF/C/7)

a [Creative Commons Attribution-NonCommercial-ShareAlike 3.0 United
States
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Instructions: This video continues the concept of absolute value inequalities, providing a more complex example. It uses the definition of absolute value with methods for solving linear inequalities to find the solutions for these problems.

Watching the video and writing notes should take approximately 15 minutes.

Instructions: This video continues the concept of absolute value inequalities. This example is important, for it demonstrates that not all equations have a solution. This is the first time we are introduced to the idea of checking the solution. That is, we see that only one solution is possible, but that potential solution creates an untrue statement when substituted back for the variable in the original equation. As we move forward, it is advisable to begin checking all answers.

Watching the video and writing notes should take approximately 15 minutes.

• Web Media: GeoGebra: “Basic Absolute Value Inequalities” Link: GeoGebra: “Basic Absolute Value Inequalities” (HTML)

Instructions: This interactive grid allows you to drag an inequality to an area next to corresponding interval graphs. Once you are ready, you can check your answers to see how many are correct. This is good practice toward finding the solutions of inequalities in one variable.

Practicing with this application and writing notes should take approximately 45 minutes.

• Checkpoint: MyOpenMath: “Beginning and Intermediate Algebra: Chapter 1 Review” Link: MyOpenMath: “Beginning and Intermediate Algebra: Chapter 1 Review” (HTML)

Instructions: This is an assessment on understanding linear equations in one variable. Work all 16 problems. Answers are available for viewing by a button on each problem. There is also a link for a printed version at the bottom left of the page. Note: New versions of the 16 problems can be created by accessing the link in the upper right corner of the page. You may take a nearly unlimited number of attempts with different problems.

Completing this assessment should take approximately 30 minutes.

• Checkpoint: MyOpenMath: “Beginning and Intermediate Algebra: Chapter 2 Review” Link: MyOpenMath: “Beginning and Intermediate Algebra: Chapter 2 Review” (HTML)

Instructions: This is an assessment on understanding linear equations in two variables. Work all 16 problems. Answers are available for viewing by a button on each problem. There is also a link for a printed version at the bottom left of the page. Note: New versions of the 16 problems can be created by accessing the link in the upper right corner of the page. You may take a nearly unlimited number of attempts with different problems.

Completing this assessment should take approximately 30 minutes.