# Absolute Value Instructor: Dr. Jo Steig

That is, the absolute value of a number, a, is equal to the distance that ‘a’ is from zero. The two statements (above) are needed to define absolute value in order to insure that the output of an absolute value function is NEVER less than zero. Try the following few calculations on your calculator:

• | – 5 |
• | 55.6 |
• | e – 7 | (here, e represents the irrational number that is defined as the base of the natural log)

INSIDE EXPRESSION IS A CONSTANT

If the inside expression is a constant, then writing the expression without absolute values is easy. We just need to insure that out output is nonnegative. That is,

• | – 5 | = 5
• | 55.6 | = 55.6
• | e – 7| = 7 – e, or approximately 4.28

INSIDE EXPRESSION IS A VARIABLE

If the expression inside the absolute value includes a variable, there is not much we can do with it as long as the absolute values are there. Therefore, to get any work done we must first write it without absolute values. This is similar to receiving a computer disk on which the information has been condensed to save space. Before you can access the information you must first run a decompression program. Our decompression program for the absolute value is the definition given above.

The following three examples illustrate

(1.) If x 2 then | x – 2 | = – (x – 2) Justification: | x – 2 | is either x – 2 or – (x – 2) If x 2, then x – 2 is 0 (try a few numbers and see for yourself!)

But | x – 2 | cannot be negative.

So | x – 2 | = – (x – 2) = 2 – x

(2.) If x 8 then | 5 – x | = – (5 – x )

Justification: The reasoning here is similar to the last example. If x 8, then 5 – x 0

So | 5 – x | = – (5 – x) = x – 5

(3.) If x 4 then | x – 4 | = x – 4

Justification: This is true because as long as x is larger than 4, x – 4 will be positive.

BUT, WHAT IF YOU DON’T KNOW ANYTHING ABOUT THE VARIABLES?

When the inside expression includes variables and you do not know anything about the values of the variables, then writing the expression without absolute values takes a bit more care and our reliance on the definition above becomes more obvious. The end result will be a piecewise defined function that is similar to the original definition that was given (above) for absolute value .

The following example shows that process in detail.

Example: Write | 4 – 5x | without using absolute value

Solution: The expression inside the absolute value is (4 – 5x). In the original definition for absolute value, replace each occurance of ‘a’ with (4 – 5x). That is,

When we simplify the inequalities in the last statement, (4 – 5x) 0 becomes x 4/5 and (4 – 5x) 0 becomes x 4/5. The final statement is then

Note: While this last statement looks more complicated than the beginning statement of | 4 – 5x |,it is in a form that can be added, graphed, integrated (calculus) or differentiated (calculus). At this point we are just trying to develop the skill of writing an absolute value expression without the absolute value symbol. You will investigate addition, integration, and differentiation in a later course.

## Sample Problems

1. Write |x + 2| without absolute values if x 3

2. Write |x + 2| without absolute values if x -2

3. Write |x + 2| without absolute values if x -2

4. Write |5 – x| without absolute values if x 5

5. Write |5 – x| without absolute values if x 5

6. Write |x + 2| + |5 + x| without absolute values if x -2

7. Write |x + 2| + |5 – x| without absolute values if x 8

8. Write |x + 2| + |5 – x| without absolute values if x -3

9. Write | x + 2 | without using absolute value

10. Write |5 – x | without using absolute value

11. Write | 2x – 3 | without using absolute value

12. Write | 3x + 6 | without using absolute value

1. If x 3 then | x + 2 | = x + 2

2. If x – 2 then | x + 2 | = x + 2

3. If x – 2 then | x + 2 | = – (x + 2) = -x – 2

4. If x 5 then | 5 – x | = – ( 5 – x ) = x – 5

5. If x 5 then | 5 – x | = ( 5 – x )

6.If x – 2 then | x + 2 | + | 5 + x | = (x + 2) + (5 + x) = 2x + 7

7. If x 8 then | x + 2 | + | 5 – x | = (x + 2) – (5 – x) = x + 2 – 5 + x = 2x – 3

8. If x – 3 then | x + 2 | + | 5 – x | = – (x + 2) + (5 – x) = -x – 2 + 5 – x = 3 – 2x

# American Universities in Europe

The American higher education system has many advantages – a flexible curriculum, a variety of courses, transfer choices, study abroad opportunities and more. But for some American and International students, the dream of studying in Europe is just as strong as their desire to obtain an American degree. Some students may wish to learn a foreign language and travel through Europe without compromising their university education. Others may wish to study in Europe, but have not mastered the language requirements to do so at a European university. And yet others may just want to stay home while obtaining an American degree. For many, the solution is to attend an American university in Europe. Here is a list of universities that follow the American higher education system. Most of these universities offer academic degrees and are accredited by US commissions and local country authorities.

## Austria

Webster University Vienna

## Bulgaria

American University in Bulgaria

## France

American University in Paris

Georgia Institute of Technology Lorraine

Parsons School of Design, Paris

Schiller International University Paris

Schiller International University Strasbourg

Touro College France

## Germany

Jacobs University Bremen

Schiller International University Germany

Touro College Berlin – The American College in Berlin

## Greece

American College of Greece

American College of Thessaloniki

American University of Athens

Hellenic American University

## Hungary

McDaniel College Budapest

## Ireland

American College Dublin

## Italy

American University of Rome

John Cabot University in Rome

Johns Hopkins University SAIS

St John’s University

## Netherlands

Webster University in Leiden

## Spain

Saint Louis University

## Switzerland

Franklin College in Switzerland

International University in Geneva – Business School

Webster University in Geneva

## The United Kingdom

American Intercontinental University

Regent’s American College London

Richmond, The American International University in London

Schiller International University London

### 50 comments to American Universities in Europe

Please tell how to locate some of The canadia universities or collages in europe.

Hi I m currently an Aub (American university in beirut) student and a Red Cross paramedic and mountain rescue volunteer, looking to transfer to an American university in Europe, looking to major in business(risk/insurance major), I would to know if the course is available, if yes I would like to know the requirements.

Ps: I m French, Syrian and Lebanese.

Hi i am currently in college for radiology, but would like to switch out the major i m in now for fashion merchandising (what i always dreamed of).. looking for one of the best fashion schools in Italy or London, any ideas. Please?

I m currently a student at Georgia Regents University located in Augusta, GA. However, I m looking forward to move to Germany in the future. Also, I want to continue my education in Germany as well. Thus, I ve done some research on the internet and I found University of Maryland University College which has several locations in Germany. Based on their website I got the impression that they only accept military members? I m a civilian, and I have never been in the military service.

Do they accept American civilians as well?

Here is their statement on eligibility: Members of the U.S. Armed Forces, American employees of the U.S. Government, and their family members residing within an overseas contracted site may enroll in undergraduate classes through University of Maryland University College Europe. You may want to expand your search to Switzerland, Austria and Italy. Have you contacted the Study Abroad office at your school? Most of the time, they have agreements with universities in Europe where you can spend a semester or year abroad and earn college credits.

I want to study photography in Spain but go to an American university inckuding college, what are my options?

We are moving to Germany from USA, got a nursing job at the US Military Hospital in Landstahl Germany. My son is a sophomore in University in USA and is very excited at the prospect of studying in Germany/Europe. We have been researching schools and found University of Maryland University College in Heidelberg Germany. Is this a reputable college? I don t see it listed here, are there any other schools that offer degrees in Computer Science. It has to be a school where the medium of instruction is English. Would it be better for him to study in England? Otherwise I will have to find out a way of him staying back here and continuing his education.

Thanks and Kind Regards,

UMUC Europe admissions eligibility includes military members, American employees of the U.S. government and their family members. UMUC has a strong reputation in the military and government communities, so it is a very good option for your son. He will always have the option to transfer to a university in England or the US or go for a master s degree in a different country.

Linear Algebra is a text for a first US undergraduate Linear Algebra course. You can use it as a main text, as a supplement, or for independent study.

It is Free. The book, the complete answers to all exercises, classroom presentation slides, and a lab manual using Sage, are all available for download, as well as for purchase.

The third edition incorporates many suggestions sent by users, including many more beginning exercises, and a new Topic on coupled oscillators and eigenvalues.

### Highlights

• Standard coverage Linear systems and Gauss’s method, vector spaces, linear maps and matrices, determinants, and eigenvectors and eigenvalues.
• Free The book is Freely available, including its source.
• Developmental approach It covers the requisite material and proves all the results, but it does not start by assuming that students are already able at abstract work. Instead, it proceeds with a great deal of motivation, many computational examples, and exercises that range from routine verifications to a few challenges. The goal is, in the context of developing the material of an undergraduate course, to raise each student’s level of mathematical maturity.
• Extensive exercise sets, with worked answers to all exercises Sometimes material described on the web as a book is really lecture notes. That’s fine but from notes to a book is a long way. That means things like figures and an index, but most importantly means exercises. Each subsection here has many, spanning a range of difficulty. In the Answers book each exercise is covered, completely, including proofs.
• Popular Downloadable for twenty years, this book has been used in hundreds of classes at many schools as well as by thousands of individuals for independent study.
• Applications Each chapter finishes with four or five short supplemental topics. These are good for reading or projects, or for small group work.
• Extras There are beamer slides for classroom presentations, and there is a lab manual using Sage.
• Prerequisite: a semester of calculus.
• Reviews. Here are some: the Mathematical Association of America review, the American Institute of Mathematics, the Open Textbook Library (includes a number of reviews), one from a longstanding site for free texts.

### Get Linear Algebra

Here is the current version of the text and the answers to exercises. Save these files in the same directory so that clicking on an exercise takes you to its answer and clicking on an answer takes you to the exercise. (You need a PDF reader that supports this; one is Adobe Reader.)

You can get a paper copy, an official one, for just about the cost of media. Simply use ISBN-13: 978-1944325039 at your favorite book shopping site, including Amazon; the price there is \$22. (Bookstores can buy from NACSCORP or from Ingram via LightningSource.)

If you are teaching out of this text then have a look at the beamer slides for classroom presentation. These are compiled from the text source, so the theorems and definitions have the exact same wording. But the slides use different examples than the text so that students see twice as many. Along with those is a version of the slides without the pauses, for handing out or posting in your learning management system. (Those slides omit some proofs in favor of examples; you can also get a version that includes all proofs.)

Linear Algebra also has a supplemental, optional, lab manual using Sage.

If you are into LaTeX then you may be interested that Linear Algebra once again has a source repository.

You can also get the problems and their solutions that I assigned for take home work when I taught the class in the Fall of 2014.

### The prior edition of Linear Algebra

The second edition of Linear Algebra is still available, for someone in a legacy situation: book and answers. To use the hyperlinking between questions and answers, rename these files book.pdf and jhanswer.pdf and put them in the same directory,

This text is Free. Use it under either the GNU Free Documentation License or the Creative Commons Attribution-ShareAlike 2.5 License, at your discretion.

For bookstores: thank you for being concerned about my rights. I give instructors permission to make copies of this material, either electronic or paper, and give or sell those to students. But first note that for very little money you can get a paper copy from Amazon, or NACSCORP, or directly from Ingram via LightningSource, using ISBN-13: 978-1944325039.

Please don’t order used or unofficial copies since they can have significantly different content and pagination. With the current edition at pretty much the price of media, any savings from old versions are tiny compared to the cost of confusion, particularly confusion to me since I get contacted about already-fixed bugs. The current version says Third edition on the front cover (it also says 2017-Jan-01 at the end of the Preface).

For instructors who want to modify the text. Please, feel free. But as a favor I ask that you include a statement about your modifications. That way people making reports know who to write. Putting something like this on the cover would be great: \fbox<\parbox<0.75\textwidth>.>>

### Can You Help With Linear Algebra?

I am glad to hear from both teachers and students. I enjoy hearing about the experience that folks have with the book and I find suggestions helpful, especially bug reports. I save these and periodically revise.

If you have something that you are able to share back then I’d be glad to take a look. For instance, I would love to have a topic on Machine Learning, or some sub-topic such as dimension reduction. Of course, I reserve the ability to choose whether to edit or include it. I gratefully acknowledge all the contributions that I use, or I can keep you anonymous.

I would also welcome contrbutions related to the emerging electronic tools. For instance, if you have sets of questions that are suitable for Moodle quizzes and that you could share with other users of this book then write me and we can see about making them available. The same holds for WeBWorK problem sets. And, I would be glad to be able to include in the distribution a deck of Anki cards.

### Related work

You may also like my Introduction to Proofs, an Inquiry-Based approach text, also Freely available. It is for a proofs course taught using Moore method, which is known in other fields as Inquiry-Based or the Discovery Method. You may also like my Cheat sheet for LaTeX math, aimed at undergraduates but useful for anyone.

#### Joshua runs under Linux

Open Source software is a great idea. This project would not have gotten done without it.

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Content Continues Below

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# Online Algebra Tutor

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Algebra is a branch of Mathematics, it is the study of symbols and the rules of Manipulating symbols. Algebra is a technique of writing a word problem in terms of equations and formulae and solve them to get the answer. Elementary algebra has its use in day to day life. It can help you in solving games and puzzles. Algebra finds its use in maths, engineering, statistics and science.

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# College Algebra

We have a series of free College Algebra Video Lessons from UMKC – The University of Missouri-Kansas City.

### UNIT 0 – BASICS: Remembrance of Things Past

Natural Numbers [7 min.]

Rational Numbers [13 min.]

Irrational Numbers [2 min.]

Real Numbers [1 min.]

The Real Line: From Numbers to Points [4 min.]

Real Numbers are Ordered [19 min,]

The Real Line: Distance Between Points [8 min.]

Real Numbers as Decimals [11 min.]

Some Symbols of Algebra [9 min.]

Nouns, Pronouns, The Main Verb of Algebra [4 min.]

Theorems, Corollaries, Lemmas, and All That [7 min.]

Operations with Integer Exponents [31 min.]

Square Roots: A Pair Of Equal Factors [13 min.]

Nth Roots Rational Exponents [12 min.]

Operations with Rational Exponents [21 min.]

Multiplying Polynomials [19 min.]

A Common Error [5 min.]

Handy Polynomial Products [15 min.]

Un-Multiplying (Factoring) Polynomials [28 min.]

Completing A Perfect Square [10 min.]

The Art of Simplification [14 min.]

Solving Some Polynomial Rational Equations [46 min.]

Some Area Formulas [8 min.]

The Pythagorean Theorem A Visual Proof [12 min.]

### UNIT 1 – GRAPHS

Distance Between Points [18 min.]

Midpoint of a Line Segment [7 min.]

Intercepts: Crossing the Axes [12 min.]

Symmetry of Graphs [14 min.]

Lines: Defining “Slope” [34 min.]

Parallel Perpendicular Lines [26 min.]

Circles Their Equations [28 min.]

Some Exercises Explained [16 min.]

### UNIT 2 – FUNCTIONS AND THEIR GRAPHS

The Language Notation of Functions [20 min.]

More On Domains [19 min.]

Function Notation Practice [16 min.]

Increasing Decreasing Functions [23 min.]

Local Maximums Local Minimums [18 min.]

Even Odd Functions [20 min.]

Piecewise Defined Functions [19 min.]

Some Exercises Explained [11 min.]

Graphing Techniques: Compressions Stretches [27 min.]

Graphing Techniques: Reflections Across the Axes [17 min.]

Putting It All Together: Moving The Graphs of Functions Around The Plane [36 min.]

A New Operation Unique To Functions: Composition [24 min.]

### UNIT 3 – EQUATIONS INEQUALITIES

Solving Linear Equations: The “Linear Formula” or, Graphing [23 min.]

Solving Non-Linear Equations That Lead To Linear Equations [13 min.]

A Complex Reminder The Principal Square Root of a Negative Number [12 min.]

Solving Quadratic Equations: The “Quadratic Formula” The Discriminant or, Graphing [50 min.]

Solving Equations “Quadratic in Form” [15 min.]

Solving Factorable Equations [7 min.]

Solving Inequalities In General [17 min.]

Solving Linear Inequalities [19 min.]

Solving Higher-Degree Polynomial Inequalities [14 min.]

Solving Rational Inequalities [27 min.]

When Absolute Value Appears: Inequalities [17 min.]

More Exercises Explained [15 min.]

### UNIT 4 – POLYNOMIAL RATIONAL FUNCTIONS

Quadratic Functions As Mathematical Models [25 min.]

Special Case: Power Functions Their Graphs [18 min.]

Graphing General Polynomial Functions: Zeros, Multiplicity, Turning Points End Behavior [40 min.]

How Many Zeros Are Real? [13 min.]

How Many Real Zeros Are Positive? Negative? [16 min.]

Where (On What Interval) Are All Real Zeros? [22 min.]

How Can You Guess The Location of Real Zeros? [4 min.]

How Can You Reduce The Number of Real Zeros? [21 min.]

Some Polynomial Exercises Explained [22 min.]

What Is An Asymptote? [17 min.]

Finding Asymptotes of Rational Functions [35 min.]

Graphing Rational Functions [19 min.]

### UNIT 5 – EXPONENTIAL LOGARITHMIC FUNCTIONS

Exponential Functions Their Graphs [26 min.]

The Natural Exponential Function [15 min.]

Logarithmic Functions Their Graphs [24 min.]

The Natural Logarithmic Function [4 min.]

All Logarithms are Natural (or Common)! [15 min.]

Solving Logarithmic Equations [20 min.]

Logarithmic Models: Sound (Loudness) Fury (Earthquakes) [16 min.]

Exponential Models: Compounded Interest, and Growth Decay [32 min.]

### UNIT 6 – SYSTEMS OF EQUATIONS

Solving A System of 2 (or 3) Linear Equations in 2 (or 3) Variables: Substitution [20 min.]

Solving A System of 2 (or 3) Linear Equations in 2 (or 3) Variables: Elimination [36 min.]

An Application: Writing Proper Rational Functions As Sums of Simpler Proper Rational Functions (Partial Fractions) [43 min.]

### UNIT 7 – SOME DISCRETE TOPICS

The Factorial Symbol: ! [8 min.]

Adding The First n Terms of a Sequence: nth Partial Sums Summation Notation [17 min.]

Geometric Sequences: Multiplying Your Way to Infinity [30 min.]

The Principle of Mathematical Induction [30 min.]

Pascal’s Triangle [19 min.]

The Binomial Theorem: How to Expand (x+a) n [32 min.]

### THE X-Y FILES: The Proof is in Here

• How Can You Find Rational Zeros (if any) of a Polynomial Function? [21 min.]

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# How to Do Well in College Algebra

College algebra is usually a pre-requisite for higher level math courses and science degrees. Although it can be a little bit tricky, mastering these concepts is necessary to moving forward in math. There is no fast and simple way to pass college algebra. Whether you know you aren’t strong in math, have taken this class at least once before, or are an adult learner coming back to math as part of a career change, know that you are not alone in wanting to get through this difficult course. You must put in the work to understand the concepts and that takes time. Doing well in college algebra is all about attending class, doing the homework, asking for help when needed, and reviewing the material throughout the course (no cramming!).

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# Algebra Word Problems

## Get Interactive Learning CDs/DVDs Now

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There is one of the most important subjects in mathematics, known as algebra. Algebra studies about constants, variables, expressions and equations. It also deals with abstractions such as – groups, rings etc. Algebra is an essential part for any fields related to mathematics, physics, science, engineering or technology, as well as economics and medicine too.

2) Pre Algebra – A course of algebra learnt in middle school, i.e. from 5 to 7 standards.

3) Abstract or Modern Algebra – Algebra about abstract mathematics, such as – groups and rings.

4) Linear Algebra – It deals with the study of linear equations, matrices, vector spaces etc.

5) Algebraic geometry – The branch of geometry that deals with curves and surfaces with polynomial equations.

6) Computer Algebra – Study of algebraic methods that are used in computer programming and algorithms.

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## Pre Algebra Word Problems

With pre algebra word problem solver free explanations and examples help you get used to the terms used to describe the relation between variables. The Internet is already used extensively for research and extra information. The emergence of tutoring services has taken the classroom teaching experience online with the launch of several free algebra solvers with steps.