Solving Worksheets for Elementary Algebra

Confidence comes with  practice.

In Elementary Algebra we have learned how to solve four types of equations.  The ability to solve a variety of different types of equations takes time and practice to master.  Here are some extra exercises that you can use to hone your skills.

• [ Solving Linear Equations ]
• [ Solve Quadratic Equations by Factoring ]
• [ Solving Rational Equations ]
• [ Solving Radical Equations ]

If you are an instructor, please feel free to use these problems in your Beginning Algebra or Algebra 1 course. More supplementary material can be found on my Elementary Algebra Course page.


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Elementary Algebra TOC - Free to Read Online (No Login!)

Here is the table of contents for the open Elementary Algebra textbook found on the FlatWorld Knowledge website.  You will find that the content is fairly modular, I have written it from the ground up with the expectation that instructors will modify it to suit their particular course.  I am confident that everything needed for a Beginning Algebra (Algebra 1) course is covered, but feel free to MIYO (make it your own) by customizing it as you please. This textbook has the advantage over other open source material in that it has been thoroughly reviewed and vetted through the traditional publishing process.  (Note: Be patient with the website, there are many math elements to be rendered for the web.)


Chapter 1 - Real Numbers and Their Operations

1.1 Real Numbers and The Number Line
1.2 Adding and Subtracting Integers
1.3 Multiplying and Dividing Integers
1.4 Fractions
1.5 Review of Decimals and Percents
1.6 Exponents and Square Roots
1.7 Order of Operations
1.8 Review Exercises and Sample Exam

Chapter 2 - Linear Equations and Inequalities 

2.1 Introduction to Algebra
2.2 Simplifying Algebraic Expressions
2.3 Linear Equations: Part I
2.4 Linear Equations: Part II
2.5 Applications of Linear Equations
2.6 Ratio and Proportion Applications
2.7 Introduction to Inequalities and Interval Notation
2.8 Linear Inequalities (one variable)
2.9 Review Exercises and Sample Exam

Chapter 3 - Graphing Lines

3.1 Rectangular Coordinate System
3.2 Graph by Plotting Points
3.3 Graph using Intercepts
3.4 Graph using the y-intercept and Slope
3.5 Finding Linear Equations
3.6 Parallel and Perpendicular Lines
3.7 Introduction to Functions
3.8 Linear Inequalities (Two Variables)
3.9 Review Exercises and Sample Exam

Chapter 4 - Solving Linear Systems 

4.1 Solving Linear Systems by Graphing
4.2 Solving Linear Systems by Substitution
4.3 Solving Linear Systems by Elimination
4.4 Applications of Linear Systems
4.5 Solving Systems of Linear Inequalities (Two Variables)
4.6 Review Exercises and Sample Exam

Chapter 5 - Polynomials and Their Operations 

5.1 Rules of Exponents (Integer Exponents)
5.2 Introduction to Polynomials and Evaluating
5.3 Adding and Subtracting Polynomials
5.4 Multiplying Polynomials and Special Products
5.5 Dividing Polynomials
5.6 Negative Exponents and Scientific Notation
5.7 Review Exercises and Sample Exam

Chapter 6- Factoring and Solving by Factoring

6.1 Introduction to Factoring
6.2 Factoring Trinomials
6.3 Factoring Trinomials ax^2 + bx + c
6.4 Factoring Binomials
6.5 General Guidelines for Factoring Polynomials
6.6 Solving Equations by Factoring
6.7 Applications involving Quadratic Equations
6.8 Review Exercises and Sample Exam

Chapter 7- Rational Expressions and Equations

7.1 Simplifying Rational Expressions
7.2 Multiplying and Dividing Rational Expressions
7.3 Adding and Subtracting Rational Expressions
7.4 Complex Fractions
7.5 Solving Rational Equations
7.6 Applications of Rational Equations
7.7 Variation
7.8 Review Exercises and Sample Exam

Chapter 8- Radical Expressions and Equations 

8.1 Radicals
8.2 Simplifying Radical Expressions
8.3 Adding and Subtracting Radical Expressions
8.4 Multiplying and Dividing Radical Expressions
8.5 Rational Exponents
8.6 Solving Radical Equations
8.7 Review Exercises and Sample Exam

Chapter 9- Solving Quadratic Equations and Graphing Parabolas 

9.1 Extracting Square Roots
9.2 Completing the Square
9.3 Quadratic Formula
9.4 Guidelines for Solving Quadratic Equations and Applications
9.5 Graphing Parabolas
9.6 Introduction to Complex Numbers and Complex Solutions
9.7 Review Exercises and Sample Exam

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Solving Rational Equations in Beginning Algebra

A thorough explanation for solving rational equations can be read for free in the Elementary Algebra textbook that is available on the Flat World Knowledge website:

[ Solving Rational Equations ] 

Also, you can click on the following rational equation to watch a video solution.

The basic steps for solving these equations are as follows:

Step 1: Factor all denominators and determine the LCD.

Step 2: Identify the restrictions. 

Step 3: Multiply both sides of the equation by the LCD. Distribute carefully and then simplify.

Step 4: Solve the resulting equation. 

Step 5: Check for extraneous solutions.

The complete solution is presented below:

Basically, we multiply both sides of the equation by the LCD as a means to clear the fractions.  This introduces the possibility for extraneous solutions and so we must check our answers.  Look for more Chapter 7 videos here.

[ Visually Searchable Algebra Problems ] 

Elementary Algebra (Ch. 6) Factoring and Solving by Factoring

Elementary Algebra is an open textbook that is available to read for free on the flatworld knowledge website. Chapter 6 is an introduction to factoring and solving by factoring. In section 6.5 you will find general guidelines for factoring and steps for using factoring to solve certain quadratic equations.

6.1 Introduction to Factoring
6.2 Factoring Trinomials
6.3 Factoring Trinomials ax^2 + bx + c
6.4 Factoring Binomials
6.5 General Guidelines for Factoring Polynomials
6.6 Solving Equations by Factoring
6.7 Applications involving Quadratic Equations  

You will find a complete review exercise section and a practice test.

6.8 Review Exercises and Sample Exam

Feel free to copy and paste the links found here and post them in your LMS. Links to all of the chapters can be found by clicking the Elementary Algebra course button at the top of this page.

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Video Examples of Polynomials and Their Operations

All of the "Try This!" problems in the open textbook Elementary Algebra have been solved and are available for free on YouTube.  Click on the problems below to view the videos found in chapter 5.

Polynomials and Their Operations

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For further updates please feel free to circle the Elementary Algebra Google+ page for this book.
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Beginning Algebra in the Early 20th Century

Image via WikipediaIt is interesting to see how content changes and stays the same over time. Check out these two beginning Algebra books that I found on the Project Gutenberg webiste:

A First Book in Algebra by Wallace C. Boyden 1895

The First Steps in Algebra by G. A. Wentworth 1904

"162. A root which can be found exactly is called an exact or
rational root. Such roots are either whole numbers or fractions.
A root which is indicated but can be found only approxi-
mately is called a surd. Such roots involve the roots of imperfect
powers. 

Rational and surd roots are together called real roots. 

A root which is indicated but cannot be found, either exactly
or approximately, is called an imaginary root. Such roots in-
volve the even roots of negative numbers."

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Elementary Algebra (Ch.1) Real Numbers and Their Operations

Elementary Algebra is an open textbook that is available to read for free on the flatworld knowledge website. Chapter 1 is considered a review chapter covering many of the topics found in a Pre-Algebra course. You will find written examples, video examples, and exercises with answers to the odd problems.


1.1 Real Numbers and the Number Line
1.2 Adding and Subtracting Integers
1.3 Multiplying and Dividing Integers
1.4 Fractions
1.5 Review of Decimals and Percents
1.6 Exponents and Square Roots
1.7 Order of Operations

In addition, you will find a complete review exercise section and a practice test.

1.8 Review Exercises and Sample Exam

Feel free to copy and paste the links found here and post them in your LMS.  Links to all of the chapters can be found by clicking the Elementary Algebra course button at the top of this page.

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Solve by Factoring in Elementary Algebra

Learning how to solve equations is one of our main goals in Algebra. At this point, we know how to solve linear equations,

And now we begin to learn how to solve quadratic equations,

One way to solve quadratic equations is to use factoring and the zero-product property. The zero-product property sounds fancy but it simply states that,

So if we can factor a quadratic equation in standard form, equal to zero, we can then set each factor equal to zero and solve the resulting linear equations. For example, solve the following quadratic equation:

Step 1: Express the quadratic equation in standard form, equal to zero.

Step 2: Factor the quadratic expression.

Step 3: Apply the zero-product property and set each variable factor equal to zero.

Step 4: Solve the resulting linear equations.

The solutions are -5, and -3/2. You can check that these are solutions by substituting them into the original equation and verify that they lead to a true statement.  Here is another example on video:

Hopefully you can see that the factoring step is very important in this process. Take care to factor your quadratic equations carefully because this is the step where most students make an error.

• Elementary Algebra Textbook - Chapter 6 Factoring and Solving by Factoring
• Elementary Algebra Textbook - Chapter 6 YouTube Playlist

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