Algebrator manual

Contents

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Chapter 1 : Introduction............................................................................................... 1

What is Algebrator ?........................................................................................................... 3

Installing Algebrator on your Computer............................................................. 4

System Requirements.................................................................................................................. 4

Installation................................................................................................................................... 4

Miscellaneous files...................................................................................................................... 6

Typographical Conventions............................................................................................. 7

Chapter 2 : Getting Started.......................................................................................... 9

Basic Workspace Expression Manipulation........................................................ 11

Solving Equations, Inequalities and Graphing.................................................. 23

Solving Systems of Equations, and Graphing..................................................... 33

Chapter 3 : Advanced Features................................................................................ 47

Using Visibility and Domain Options......................................................................... 49

Other Modes of Problem Solution............................................................................. 57

Chapter 4 : Teacher’s Tools...................................................................................... 65

Creating Documents and Workbooks..................................................................... 67

Generating Problems and Documents..................................................................... 77

Changing Supervisor Options......................................................................................... 85

Chapter 5 : Command Reference.............................................................................. 93

Working With Files............................................................................................................... 95

Opening an Expression File................................................................................................... 95

Saving Workspace.................................................................................................................... 96

Saving Workspace under a Different Name......................................................................... 96

Printing in Algebrator............................................................................................................. 96

Setting Printer Options......................................................................................................... 100

Exiting Algebrator................................................................................................................. 100

Basic Workspace manipulations.............................................................................. 102

Creating an Algebraic Expression...................................................................................... 102

Editing an Algebraic Expression........................................................................................ 102

Displaying an Expression..................................................................................................... 103

Deleting an Expression......................................................................................................... 103

Deleting All Workspace Expressions.................................................................................. 103

Generating Algebraic Expressions from a Template....................................................... 104

Simplifying Expressions................................................................................................... 105

Showing Next Simplification Step....................................................................................... 105

Showing Simplification Result............................................................................................. 105

Selecting Simplification Transformation........................................................................... 106

Combining Like Terms........................................................................................................... 107

Combining Like Factors....................................................................................................... 107

Multiplying Powers................................................................................................................ 107

Distributing a Power............................................................................................................. 108

Adding Fractions.................................................................................................................... 108

Multiplying Fractions........................................................................................................... 108

Dividing Fractions................................................................................................................. 109

Reducing a Fraction.............................................................................................................. 109

Rationalizing a Denominator.............................................................................................. 109

Performing One Factoring Step........................................................................................... 110

Complete Factoring............................................................................................................... 110

Expanding a Term.................................................................................................................. 110

Eliminating Parenthesis........................................................................................................ 111

Eliminating Special Cases.................................................................................................... 111

Reducing Order of a Radical................................................................................................ 111

Eliminating Negative Exponents......................................................................................... 112

Making a Base Positive......................................................................................................... 112

Making a Numerator or a Denominator Positive............................................................ 112

Converting a Decimal Number into a Fraction................................................................ 113

Converting a Fraction into a Decimal Number................................................................ 113

Converting Radical Expression into Exponential Expression...................................... 113

Converting Exponential Expression into a Radical Expression................................... 114

Rewriting an Expression in a Common Form................................................................... 114

Evaluating an Expression..................................................................................................... 114

Simplifying a Subexpression................................................................................................ 115

Finding the Greatest Common Factor................................................................................ 115

Finding the Least Common Multiple.................................................................................. 115

Manipulating an Expression................................................................................................ 116

Checking a Manipulation.................................................................................................... 116

Solving Equations and Inequalities....................................................................... 118

Showing Next Solution Step................................................................................................. 118

Showing the Solution............................................................................................................ 119

Checking the Solution........................................................................................................... 119

Selecting Solution Transformation..................................................................................... 120

Adding an Expression to Both Sides of an Equation or Inequality.............................. 120

Subtracting Common Terms from Both Sides of an Equation or Inequality............... 121

Subtracting an Expression from both sides of an Equation or Inequality.................. 121

Multiplying Both Sides of an Equation or Inequality by LCD...................................... 121

Multiplying Both Sides of an Equation or Inequality by an Expression..................... 122

Dividing Both Sides of an Equation or Inequality by Common Factors..................... 122

Dividing Both Sides of an Equation or Inequality by the Variable’s Coefficient...... 122

Dividing Both Sides of an Equation or Inequality by an Expression........................... 123

Exponentiating Both Sides of an Equation or Inequality by a Power......................... 123

Making Bases Equal in an Exponential Equation.......................................................... 123

Using Quadratic Formula to Solve an Equation............................................................. 124

Creating Linear Equations / Inequalities from Higher Order Equation / Inequality 124

Switching Left and Right Side of an Equation or Inequality......................................... 125

Creating Equation or Inequality Solution........................................................................ 125

Deleting Extraneous Root from the Solution.................................................................... 125

Switching Top and Bottom Equation in a System of Two Equations........................... 126

Adding Equations in a System of Two Equations............................................................. 126

Substituting a Selected Variable in a System of Two Equations................................... 126

Solving one of the Equations in a System of Two Equations......................................... 127

Manipulating an Equation or Inequality.......................................................................... 127

Checking a Manipulation.................................................................................................... 128

Changing a Solution Variable............................................................................................ 128

Choosing a Solution Method for Solving a System of Two Equation.......................... 129

Graphing.................................................................................................................................... 130

Showing Next Step in Conversion to Graphable Form................................................... 130

Showing Graphable Form.................................................................................................... 130

Graphing a Curve or a Solutions........................................................................................ 131

Explaining the Solution Process............................................................................... 132

Showing the First Transformation in the Current Step................................................... 132

Showing the Next Transformation in the Current Step................................................... 133

Showing the Previous Transformation in the Current Step........................................... 133

Showing Context Sensitive Explanation for Current Transformation......................... 134

Returning from Step Display................................................................................................ 134

Returning to Step Display..................................................................................................... 135

Displaying Term Definition.................................................................................................. 135

Using Supervisor Options................................................................................................ 136

Changing Transformation Visibility................................................................................... 136

Changing Visibility to None................................................................................................ 136

Changing Visibility to Low.................................................................................................. 136

Changing Visibility to Medium........................................................................................... 137

Changing Visibility to High................................................................................................. 137

Changing Visibility to All..................................................................................................... 137

Changing the Solution Domain........................................................................................... 137

Allowing Calculations with Real Numbers....................................................................... 138

Allowing Calculations with Complex Numbers............................................................... 138

Allowing Use of Quadratic Formula................................................................................... 139

Setting a Number of Steps to Keep...................................................................................... 139

Entering Supervisor Password............................................................................................ 140

Changing Supervisor Password.......................................................................................... 141

Enabling and Disabling Menu Access............................................................................... 141

Changing Default Directories.............................................................................................. 144

Changing Background Color.............................................................................................. 144

Changing Print Colors.......................................................................................................... 145

Black and White Printing..................................................................................................... 145

Color Graph Printing............................................................................................................ 145

Color Graph and Explanation Printing............................................................................ 146

Opening an Option File........................................................................................................ 147

Saving an Option File........................................................................................................... 147

Saving an Option File under a Different Name................................................................ 147

Creating and Generating Documents.................................................................... 148

Creating a Test or Homework Document........................................................................... 148

Creating a Workbook............................................................................................................ 151

Generating Documents......................................................................................................... 154

Editing a Document............................................................................................................... 154

Accessing Help....................................................................................................................... 156

Displaying Help Contents..................................................................................................... 156

Displaying Help Search Dialog........................................................................................... 156

Accessing Algebrator Tutors................................................................................................ 156

Displaying About Box............................................................................................................ 157

Editing Expressions............................................................................................................. 159

Cutting a Subexpression in Algebraic Editor................................................................... 159

Copying a Subexpression in Algebraic Editor Input Line............................................. 159

Pasting a Subexpression in Algebraic Editor................................................................... 160

Clearing Input Line in Algebraic Editor........................................................................... 160

Deleting a Subexpression in Algebraic Editor................................................................. 160

Undoing Changes in Algebraic Editor.............................................................................. 161

Selecting Expressions....................................................................................................... 162

Copying a Subexpression in Algebraic Two-dimensional Display.............................. 162

Replacing a Subexpression in Algebraic Two-dimensional Display........................... 162

Substituting a Subexpression in Algebraic Two-Dimensional Display....................... 163

Importing Expressions...................................................................................................... 164

Inserting an Existing Expression in Algebraic Editor.................................................... 164

Inserting a Solution Form in Algebraic Editor................................................................ 164

Creating Templates........................................................................................................... 166

Defining a Template in Algebraic Editor........................................................................... 166

Clearing a Template in Algebraic Editor.......................................................................... 167

Exiting Algebraic Editor................................................................................................ 168

Exiting Algebraic Editor....................................................................................................... 168

Canceling Algebraic Editor................................................................................................. 168

Renaming Expression in Algebraic Editor........................................................................ 169

Saving and Closing Documents.................................................................................. 171

Saving a Document................................................................................................................ 171

Saving a Document under Different Name......................................................................... 171

Closing a Document............................................................................................................... 172

Editing Problems.................................................................................................................. 173

Cutting a Problem.................................................................................................................. 173

Pasting a Problem.................................................................................................................. 173

Copying a Problem................................................................................................................ 174

Clearing a Problem................................................................................................................ 174

Deleting a Range of Problem............................................................................................... 174

Inserting Problems............................................................................................................. 176

Inserting a Problem................................................................................................................ 176

Inserting a File........................................................................................................................ 176

Moving between Problems............................................................................................. 177

Moving to the Next Problem................................................................................................. 177

Moving to the Previous Problem......................................................................................... 177

Moving to the Last Problem................................................................................................. 178

Moving to the First Problem................................................................................................ 178

Moving to a Specific Problem.............................................................................................. 178

Creating Answers............................................................................................................... 179

Generating Problem Answers............................................................................................... 179

Duplicating Problem Answers............................................................................................. 180

Miscellaneous non-Menu Operations.................................................................... 181

Focusing an Expression........................................................................................................ 181

Scrolling through Workspace Expressions....................................................................... 181

Selecting a Subexpression on Algebraic Editor Input Line........................................... 182

Selecting a Subexpression in Algebraic Editor Two-dimensional Display................ 182

Entering an Expression in Select Transformation Editor............................................... 183

Chapter 6 : Appendices................................................................................................. 185

Algebrator’s Limitations............................................................................................. 187

Valid Characters in Algebraic Editor.................................................................. 189

Workspace and Expressions......................................................................................... 191

Expression Workspace Defined............................................................................................ 191

Expression Defined................................................................................................................. 192

Types of Workspace Expressions......................................................................................... 192

Step Expressions..................................................................................................................... 193

Graphable Expression........................................................................................................... 193

Group of Expressions............................................................................................................. 194

Solution Check Expression.................................................................................................. 194

Solution Expression............................................................................................................... 194

Template Expression.............................................................................................................. 195

Instance Expression............................................................................................................... 195

Ambiguous Expressions in Algebraic Editor.................................................................... 195

Common Form of an Algebraic Expression....................................................................... 196

Expression Evaluation.......................................................................................................... 196

Files and Documents......................................................................................................... 197

Expression File....................................................................................................................... 197

Option File............................................................................................................................... 197

Document File......................................................................................................................... 198

Workbook Definition File..................................................................................................... 198

Test or Homework Document............................................................................................... 198

Workbook Document............................................................................................................. 199

Template Document................................................................................................................ 199

Instance Document................................................................................................................. 199

Solution Process.................................................................................................................. 201

Solution Process Defined...................................................................................................... 201

Transformation........................................................................................................................ 201

Special Cases Transformations............................................................................................ 202

Visibility................................................................................................................................... 202

Next Step.................................................................................................................................. 202

Next Step in Select Transformation..................................................................................... 203

Expression Manipulation Modes........................................................................................ 203

Context Sensitive Explanations........................................................................................... 204

Error, Warning and Information Messages...................................................... 207

License Agreement.............................................................................................................. 231

Index............................................................................................................................................ 235

What is Algebrator ? EnterExpT_ctx

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Algebrator is a computer algebra system specifically designed to teach pre-college algebra. It is meant to be used students as well as teachers. Algebrator is not dependent on predefined, ‘hardwired’ problems. A user can enter and manipulate a wide variety of symbolic expressions found in a typical algebra textbook. Specifically, Algebrator covers the following areas of algebra :

· Simplification of rational, radical and complex expressions

· Solving linear and higher order equations and systems of two linear equations

· Graphing conic sections and equation/inequality solutions

If you are a student...

You can enter a problem from your algebra practice workbook, and Algebrator will solve it for you, using natural solving strategy; the problem solution will be displayed in the same fashion a teacher would write it on a blackboard - step by step. If there is a step you don’t understand, explanation is just a mouse click away. Algebrator doesn’t only give you a general rule; it also relates it to a particular problem at hand. If you would like to see how Algebrator does this, go to Getting Started chapter now.

If you are a teacher...

If you install Algebrator in your school PC lab, you can control how your students use it. For example, you could disable Result Only option to force step by step solution mode. Algebrator also lets you design your own problem templates that can be used to generate a large number of similar problems. Finally, Algebrator is a powerful tool for creating printable test, homework and workbook documents (it will even automatically generate correct and incorrect multiple choice answers!). To familiarize yourself with basic operations, go to Getting Started section now. More advanced control options are described in Advanced Features section. Supervisor controls and document creation topics can be found in Teacher’s Tools section.

Installing Algebrator on your Computer EnterExpT_ctx

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System Requirements

Algebrator will work on any IBM-PC compatible microcomputer running Windows 3.1 or Windows 95. Currently, Algebrator does not work under Windows NT.

Installation

To install Algebrator software:

From a diskette :

· place the enclosed diskette in drive a:

· choose File | Run (Win 3.1) or Start | Run (Win95)

· type a:\setup

· press <Enter>

· follow on-screen instructions

From a CD (assuming that d: drive is CD ROM drive) :

· place the enclosed CD in CD ROM drive d:

· If Autorun (Win95) is enabled, startup process will start automatically.

· If Autorun is not enabled, choose :

· Start | Run (Win95) or

· File | Run (Win 3.1)

· type d:\setup

· press <Enter>

· follow on-screen instructions

Notice that Algebrator window contains three icons. If you want to :

· start Algebrator, double-click on

· start Algebrator help only, double-click on

· remove Algebrator from your system, double-click on

Troubleshooting :

If you are running Windows 3.1 and installing Algebrator from a CD-ROM, on an older CD-ROM drive, you might get a message “Error writing directory” at the very end of the installation process. This message can usually be ignored. Click on <OK> and start Algebrator. If the problem persists (i.e. you get some other kind of error message while trying to run the program), reinstall the software from the diskette.

Miscellaneous files

Beside the Algebrator program, an extensive collection of problem, template and document files has also been installed. Assuming that Algebrator is placed in c:\alg, these files can be found in \problem and \doc subdirectories.

Algebrator is shipped with no supervisor password. We strongly recommend that you create your own password as soon as possible. In case you ever forget it, delete the option file that Algebrator is currently using. This will delete the password. For security reasons it is recommended that only supervisors have write access to a disk containing option files.

Typographical Conventions

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In this manual, the following fonts and styles have specific meanings :

Example Meaning

Open | File Arial (‘|’ delimited) . Specifies a sequence of menu items. Usually these menu items have to be ‘clicked’ for a particular operation to be performed.

<Enter> Arial (‘<>‘ enclosed). Specifies a control key to be pressed.

test.doc Arial. Specifies expression and file names

x+y^2 Arial bold. Specifies a string of characters that has to be typed (usually within algebraic editor).

double-click on ... Book Antiqua, bold. This typeface is used in tutor chapters. It describes a specific action that user needs to take.

edit mode Book Antiqua, italic. This typeface is used for terms that have specific meaning in Algebrator environment

Tutor 1

Basic Workspace Expression ManipulationEnterExpT_ctx

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When you first start Algebrator by double-clicking on your screen should look like this :

We will refer to this main Algebrator window as workspace. You can think of it as of an electronic blackboard. So, how do we write problems on our blackboard? In order to create an algebraic expression, we need to invoke a special algebraic editor. Let’s do that by either choosing Workspace | New or by clicking on buttonbuttonerror_ctx. The following editor window should now appear on your screen :

Notice that algebraic editor consists of two parts :

· input line (one line input box right underneath the buttons)

· two-dimensional display (larger window bellow the input line)

Input line is used to enter an expression in ‘calculator’ notation. For example, a over b would be entered as a/b and a squared would be entered as a^2. As an expression is being entered, it will also appear in its standard ‘book’ form on the two dimensional expression display. By glancing at this display, you can immediately see whether your entry is correct.

Go ahead and enter the following expression in algebraic editor input line :

(x^2-y^4)/(x+yy)

Your screen should now look like this :

Input line cursor is green informing you that the editor is in check mode. In this mode characters are checked for correct algebraic syntax as soon as they are entered. If an error is detected the cursor turns red, algebraic editor goes into error mode and displays appropriate error message. Try entering ++ at the end of the current expression to see how this works. Delete both pluses before proceeding. If you move the cursor to the left (‘inside’ the expression), algebraic editor goes into edit mode and stops performing the syntax check, so that you can freely edit the expression. You can always get back to check mode by pressing <Enter> or <End> key. In edit mode the cursor is yellow.

Here are some of the more important features of algebraic editor :

· If two variables are entered in sequence and editor is in check modesyntax_ctx, multiplication sign will be inserted between the variables.

· All open parenthesis can be closed by ] (parenthesis fill character) at any time.

· If you want an existing workspace expression to become a part of the expression that you are editing, enter its full name in caps.

· Blank space is not a valid character. For a complete list of valid characters see Appendix B.

Two-dimensional display always displays a completed algebraic expression. This means :

· When algebraic editor expects another operand (i.e. in expression ’a+’ ), a ‘dummy’ operand (?) will be inserted in two-dimensional display, wherever the character is expected.

· TTTTWhen algebraic editor expects the closing parenthesis (i.e. in expression ‘(a+b’ ) a closing parenthesis will be displayed in two-dimensional display, even if user still hasn’t entered it on the input line.

Let’s go on with defining our first expression. If you are in edit mode (yellow cursor) press <Enter> key once to go to check mode (green cursor). Once you are in check mode press <Enter> key one more time to let Algebraic Editor know you are done. The following dialog box should now be displayed on your screen.

Enter the following name : myfirst. Expression that you entered has now been placed at the bottom of workspace window. Its full name is myfirst.exp .

Using the same procedure enter expression :

(x+y)^2/(x+y)

and name it mysecond. Your screen should now look like this :

Notice that mysecond.exp is now displayed in dark red color, while myfirst.exp is black. This means that mysecond.exp is focused. Transformations such as simplification or graphing are always applied to the focused expression. Practice changing expression focus by clicking the right mouse button on the expression that you wish to focus. Notice that when you position the mouse over a particular expression, its name will be displayed in the status line at the bottom of the screen.

It is possible that some large workspace expressions will not entirely fit on the screen. In order to improve overall clarity, scroll bars for each individual expression in workspace are not provided. When you need to see the whole expression, click on it, to bring up the algebraic editoralged_ctx and then use the scroll bars.

You will now learn how to combine the existing expressions into a single one. First, click left mouse button anywhere on myfirst.exp window to bring it into algebraic editor. Then, type : +MYSECOND.EXP at the end of the expression. Make sure that you use uppercase letters. If you don’t, Algebrator will try to parse string mysecond.exp as an algebraic expression instead of interpreting it as an expression name. Press <Enter> key twice to exit the editor. If you did everything correctly, the expression at the bottom of the screen (myfirst.exp) should now look like this :

In order to save the workspace (all three expressions), click on buttonerror_ctx on main button bar and enter myfile as a file name.

Now, let’s see if this file really got saved. First, clear the workspace by choosing Workspace | Clear All menu item. Then, click on to retrieve the file that you have just saved. Enter myfile as a file name. The screen should now look exactly the same as it did before the workspace got cleared.

OK, let’s do some real work now. Why did we create this expression in the first place ? To have it simplified, of course. Algebrator can simplify expressions in a variety of different ways. Right now, let’s assume that we are only interested in the result of simplification. Click on to create the simplified form of myfirst.exp.

Your screen should now look like this :

Lots of computer algebra systems (including some calculators) can do what Algebrator just did. The real power of Algebrator is in the ability to solve a problem in a step by step fashion and provide context sensitive explanations. That’s what we are going to do next.

It is usually a good idea to get rid of old solution, before resolving a problem. So, let’s do that first : press <Del> key, select myfirst@1.exp and then press <Enter> key. The solution is now gone.

Make sure that myfirst.exp is focused. Click on on main button bar to perform the first step toward simplifying expression myfirst.exp. Notice that buttonerror_ctx button on main button bar is no longer disabled. This means that context sensitive explanation is available for this step.

Click on buttonerror_ctx on main button bar to highlight the first transformation in step myfirst@1.exp. Click on buttonerror_ctx on main button bar to display explanation window for the first transformation. Your screen should now look like this :

Notice that in addition to displaying the difference of two squares formula, Algebrator also tells you how that formula relates to the problem at hand (A = x, B = y, M = 2).

If you click on any green term in explanation window, its definition will be provided. You can remove the definition window by clicking anywhere on the screen. To remove the explanation window from the screen click on .

Notice that buttonerror_ctx and buttonerror_ctx buttons are enabled. This means that there are more explanations available for this step. You can click on buttonerror_ctx if you want to see the next transformation. Practice displaying different transformations and explanations.

Now, let’s exit step display. First, make sure that no explanation windows are displayed. Then, click on buttonerror_ctx to return to standard workspace expression display.

Use buttonerror_ctx and buttons to see all the step of the simplification process for myfirst.exp. You will know that you are done when a message ‘Further simplification is not possible’ appears on the screen. At this point expression myfirst.exp is completely simplified.

Your screen should now look like this :

Notice that only the last three steps are kept in workspace. This is done so that you don’t have to manage an excessive number of expression. Number of steps kept in workspace can be changed by choosing Options | Keep Steps menu item.

Your workspace now contains a number of expressions. Practice moving through workspace by using the following browse buttons :

buttonerror_ctxScrolls to the ‘top’ of expression space. The first workspace expression will be focused and displayed at the bottom of the screen.

buttonerror_ctxScrolls to the ‘bottom’ of expression space. The last workspace expression will be focused and displayed at the bottom of the screen.

buttonerror_ctxScrolls ‘up’ by one expression. Expression focus does not change.

buttonerror_ctxScrolls ‘down’ by one expression. Expression focus does not change.

Now, let’s save the workspace expressions into a new file. Choose File | Save As and name the file : myfile2.

Notice that, if you had clicked on buttonerror_ctx instead, the previous contents of myfile would have been overwritten by the current workspace contents.

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Congratulations ! You have now learned how to perform basic Algebrator operations.

Tutor 2

Solving Equations, Inequalities and GraphingEnterExpT_ctx

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In this tutoring session you will learn how to solve some simple equations and inequalities. You will also discover how to represent solutions graphically.

After you start Algebrator, enter the following equation in Algebraic Editor :

2x+6=0

and name it firsteq. Click on button couple of times to see how Algebrator solves this equation. If you need explanations for any step, click on buttonerror_ctx button. Message ‘Further solution is not possible’ will appear on the screen, once the problem is completely solved.

Your screen should now look like this:

Notice that the equation name has extension sol. When graphing, this extension tells Algebrator to use a number line instead of (x,y) coordinate system. So, let’s see the graphical representation of this solution.

Click on to obtain the following graph :

To remove the solution graph from the screen double-click on graph window close button.

Now we are going to see what happens when we try to solve an equation containing a variable other than x. Click on button to create another equation. Instead of typing the entire equation, we will import firsteq into the editor. You probably remember from the first tutoring session that this can be done by entering equation name in upper case letters. There is an even easier method; simply choose Quick Pick | Expression from algebraic editor menu and select firsteq.equ from the list of existing expressions. Then, click on buttonerror_ctx button to import firsteq.equ into algebraic editor.

Now, replace x in 2*x+6=0 by y . Exit the editor andbuttonerror_ctx name the new equation secondeq .

Click on buttonerror_ctx button on main button bar to see the first step in the solution process.

A message ‘Can not solve equations of this type’ should now be displayed on your screen. Press <Enter> key to acknowledge the message. This message has appeared because Algebrator tried to solve the equation for variable x , which doesn’t exist any more. As you can see, it is important to know which variable the equation is being solved for, before the solution process is started.

OK, so how do we change the solution variable? It is really very simple; choose Solve | Solve for menu item from the main menu bar and then select the appropriate variable (y). Try solving the equation again. This time, there should be no problems.

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Now it’s time for more complicated equations. First, clear the workspace. Then type in the following equation :

x^2-(x-1)/(x-1)=0

and name it nonlin.

Make sure that x is the solution variable. Let Algebrator solve this equation step by step. Look at explanations if any steps are unclear.

By the time ‘Further solution is not possible’ message is displayed, your screen should look like this :

Are we done now ? The answer is no ! Since the original equation contains the variable x in the denominator, the solutions {1,1} need to be checked.

Click on buttonerror_ctx button on main button bar to start the solution check process. The first solution has now been substituted for variable x in the original equation. Notice that the focused check equation is named nonlin@1.ck1 (first step in checking the first solution).

Click on buttonerror_ctx button on main button bar again to see the next step in solution check process . The messages ‘Division by zero’ and ‘Original solution not valid’ will by displayed on your screen.

Good thing we decided to check our solution! ‘1’ turns out not to be a valid solution. Acknowledge the message(s) by pressing the <Enter> key.

Click on buttonerror_ctx button couple of times to check the second solution. After Algebrator determines the validity of this solution, the solution set will be updated. Your screen should now look like this :

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So far we have solved equations only. Now let’s try some inequalities. Clear the workspace and type in the following inequality :

x^2+5x+6>0.

Name it nonlin. Make sure that x is the solution variable.

Click on buttonerror_ctx button on main button bar several times to see how Algebrator solves this inequality. If you need explanations for any step, click on buttonerror_ctx button.

After a few steps, messages ‘Only some of the following intervals are valid solutions’ and ‘To get valid solution, perform check solution operation’ will appear on the screen. Acknowledge them by pressing the <Enter> key. Your screen should now look like this :

Notice that focused expression name is nonlin?.sol .The ‘?’ indicates that solution intervals have not been checked.

Click on buttonerror_ctx button on main button bar to start the process of eliminating invalid solution intervals. A value from the leftmost solution interval (-4) has now been substituted for variable x in the original inequality.

Keep clicking on buttonerror_ctx button on main button bar until a series of messages appears on the screen informing you about the validity of the solution intervals. The correct solution should now be displayed at the bottom of your screen :