remedies for common algebra errors - college of southern nevada

Remedies for

Common Algebra Mistakes

Tityik Wong Department of Mathematics College of Southern Nevada

Spring 2009

Remedies for Common Algebra Mistakes Wong

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Foreword Mathematics has its own unique characteristics that separate it from most of other subjects. It demands a very high degree of rigor and hence has a very low degree of tolerance for “innocent” mistakes. The “rules” in mathematics are many and they take years of practice and experience to fully grasp. That is the reason that even advanced mathematics students still make elementary mistakes from time to time. These mistakes are not severe enough to warrant a complete retake of previous courses, but their effect on the student can range from slight embarrassment to total discouragement. This project aims to provide a remedy for this situation. The most common algebra mistakes are identified and discussed. An extensive set of exercise problems then follow. Answers to all exercise problems are provided. This project would not have been possible without a one-semester sabbatical leave granted to me by the Nevada State Board of Regents in the spring of 2009. Sincere thanks also go to then CSN mathematics department chair Michelle Wyatt who gave me very cheerful support and encouragement.

Chapter 1 Philosophy Only pre-calculus level algebra will be discussed. Unless otherwise specified, all letters in the discussion represent real numbers. 1. Algebra as a language Algebra is a combination of logic and language. Some commonly committed mistakes in algebra can be attributed to either the logic component, or the language component, or both. What is conceived logically in the mind must be conveyed through writing, and writing by nature cannot be totally logical. This underlying conflict between logic and language is a major hurdle in the learning of mathematics for many students. One main characteristic that separates mathematics from a regular language is the precision it demands. In a language, occasional misspellings or grammatical errors may not prevent the readers from understanding the meaning of a sentence, although they may be annoyed or amused. For example, “I didn’t do nothing” is a silly version of “I didn’t do anything;” and everybody understands what the following sentence is saying “Me is a English major and mine


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Englis are gooder then you.” These kinds of mistakes will be disastrous in mathematics. The first learning experience most children have is that of learning a language, and since human languages are not as demanding in terms of precision as mathematics, it takes some extensive drilling for most students to form the habit of being precise. Despite the efforts by some good hearted people to make learning mathematics fun and interesting, drilling is not fun and interesting, but unfortunately necessary. In addition to trying to make learning mathematics as painless as possible, we also need to challenge the students, let them know early on that in order to become proficient in mathematics, they have to invest a lot of time and energy, and the feeling of discomfort and frustration is all part of the learning experience. 2. Forms are important The ability to perform common algebraic manipulations with a high degree of accuracy, even when the student does not fully understand the reasons behind those manipulations, is an important and useful asset. Examples of some of those manipulations are: operations of fractions, operations of rational exponents, and the binomial theorem. A one-year old child may not understand what “thank you” means, but if he can say it, then it is a good thing anyway. Some day he may fully understand, or he may not. But at the moment, the ability to say “thank you” should be praised and encouraged. This is the argument behind the “learn by rote” philosophy, which is not incorrect, but certainly might have been abused by some incompetent or impatient teachers. The other extreme, which is to expect that the student fully understands everything before the student moves on to the next topic, may seem logical at the first glance, but will eventually prove to be too ideological. Because as human beings, we first observe and then we imitate. We may eventually understand if we try hard but that is not a guarantee. We are capable of storing a large amount of information, even without a thorough comprehension, and making the information useful to various degrees. Mankind has taken advantage of experiences throughout history, and experiences are by definition knowledge without a full understanding. People had been dealing with lightning forever before some smart guy finally figured it out. Notice here that after we figured out lightning, we were better able to deal with it. This shows the superiority of a full understanding. But before that, people dealt with lightning anyway and accumulated vast experiences that later contributed to the breakthrough. As I mentioned earlier, for most children, their first learning experience is that of learning a language. How is a language learned? Observation, imitation, memorization, and practice! That is why we cannot give up “rote learning” altogether. Another reason this “move forward only if you understand everything” approach does not always produce the best results is that the approach is difficult to implement in a class environment. If one teacher teaches one student, then maybe it is a reasonable approach. But if one teacher teaches thirty students, the pace would have to be slowed to accommodate everyone, and that is if everyone is a willing participant and trying hard to learn. We all know that in a large class, there are always some students who for whatever reason just do not want to learn. Do we wait for them? We could, but that would be at the very costly expense of making the willing and capable students learn less. We do not consider such a trade-off a worthwhile and smart one. At the early stage of learning a new subject, high quality should be demanded for two reasons: (a) It is easier to demand it early than late, and (b) it is necessary. Because this is the habit-forming stage and if students are allowed to get by with low quality performance, they form a


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habit that will be very difficult to change later. Rote exercise is an integral part of early stage learning because it builds a good form that can be filled with substance later. Without a good form, even good substance may look bad and that would be a real shame. 3. Repetition Two people can commit the same algebraic mistake for different reasons. One person could have made a mistake because of a lack of understanding, another simply carelessness. For instance, the mistake 222)( baba +=+ can be a result of “not thinking at the moment,” or a more serious problem that is demonstrated by the counter-question from the offender “Why is that wrong?” Both crimes, however, can be remedied by extensive exercise. Algebra is mainly a tool that services other branches of mathematics and sciences. So repetition is the key – we want the student to be able to react to algebraic manipulations naturally and comfortably, the same way a good basketball player catches and shoots the ball without much hesitation. When a calculus student is unable to comfortably follow the algebra that is used as part of the new material being presented, confidence is shaken and learning is hindered, even if the student understands the “ideas” and can handle the algebra later with the help of an advanced calculator. At the moment, in class, something important is lost. The frustration the student feels would be equivalent to having the creativity of William Shakespeare, but without the complete command of a language. Hence we view algebra as essential to “mathematical living.” Algebra is to mathematics what walking and doing chores is to every-day life. These basic activities do not seem much when you can comfortably perform them, you feel the pain though when you cannot perform them. Learn the right ways, then repeat a lot to reinforce. 4. The way to master algebra How should algebra be learned? Ideally the student should first understand the “why” and then reinforce the understanding by doing a reasonable number of exercise problems. Explaining “why” is the teacher’s job, that is step one; understanding “why” and doing enough exercise are the student’s responsibility, they are steps two and three. If these three steps are followed without any compromise, a student good at algebra is made. Given that algebra is not the easiest subject on earth, and most mathematics classes nowadays are so large that it is impossible for the teacher to make sure that every student understands everything before the next topic is introduced, we end up with many students trying really hard at step three without properly completing the first two steps. Does that mean we are doomed? Not necessarily. Our argument is, if we wait until we fully grasp something before we go on to the next topic, we will have to slow the pace to such an extent that not much material can be learned in a reasonable amount of time. Hence our first motto: “Get the facts first, but remember to figure them out later.” Those who do only the first part, i.e., “get the facts,” will be decent students. Those who do both parts will be excellent students. If we over emphasize “understanding,” we produce a lot of students who can neither perform the basic operations nor understand the reasons behind these operations.


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Full understanding may not be required in certain fields that employ mathematics. If your job requires that you perform certain mathematical operations but there is no chance you will be required to perform any unexpected mathematical operations, then as long as you fully master what is required of you, even if you do not know exactly why those operations are correct mathematically, you are a competent worker anyway. For example, if you can add two fractions with different denominators together, and that’s all you do, then you don’t need to know the REASON 1/2 + 2/3 = 7/6. After all, how many of us know exactly how a car works? But most of us can drive a car. It is the same principle. A lot of times understanding can actually come only after extensive exercise. In algebra, understanding alone is not good enough anyway. It could be even more frustrating when a student understands a lot but cannot perform the basic operations correctly because of lack of practice. Hence our second motto: “Drill, baby, drill.” Mathematics, being a subject that accumulates the wisdom of hundreds of thousands of the smartest people over thousands of years, cannot be learned by pure “exploring.” If you let some children play and dance around, the probability that these children reinvent ballet is zero. If you do not teach children language, they may invent a new language but it will not be any one of the existing languages in the world. Any great art or science that was built upon the collective wisdom of many intelligent people over many centuries must be introduced and taught to the students. The learner has to go through some discomfort and make some sacrifice in order to learn the subject. Innovation can come only after the fundamentals have been fully mastered. There is no way to make learning any subject all “fun and interesting” to everyone. Even some activities that are generally considered fun and interesting could be boring or downright frightening to some people. For example, some people like but some people hate fishing; some people like but some people hate eating pizza. Allowing low level mathematics students to “explore and discover mathematics” is a fatal mistake. Even a seemingly straightforward activity like boxing has to be properly taught and learned, and I am quite sure that it takes more time and dedication to master decent boxing skills than what is needed to earn a four-year degree nowadays. There is a teaching approach called “guided discovery.” The key here is finding the right balance between “guide” and “discovery.” The lower the level, the more “guide” is needed. 100% “guide” is what we call “lecture.” If you are a teacher and you only lecture in your class, the students will still be “discovering” when they do their homework. So there is no need to feel guilty if lecture is your main instrument of instruction. 5. Calculator and other modern gadgets The notion that because we have more advanced technologies nowadays, students do not have to be as proficient at basic algebraic manipulations as before is false. Algebra is mostly the means, not the end. This is such a simple concept yet many people either fail or refuse to see. These people try to justify why we teach algebra by trying to find “real world” applications for every algebra topic in the book. If they could not find any “real world” applications for an algebra topic, they declare the topic “mathematical nonsense” and would not hesitate to drop the topic from the curriculum immediately. Take factoring as an example. How often is factoring performed in the “real, every day world?” The answer is “not often at all,” if not “never.” Yet


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factoring has been a standard topic in algebra for a long, long time. Why? Because factoring helps the presentation of other topics. For example, with the help of factoring, it is easy to find

the vertical asymptotes of the graph of the function 32

3)( 2 −−=

xxxf . Why is this important?

Because (1) the concept of vertical asymptotes is important and has important real world applications, (2) it is important to show the student that the graph of a function can have more than one vertical asymptote and hence the need of a second or higher degree polynomial as the denominator, and (3) given that a second or higher degree polynomial is desirable as the denominator of the function, it is much efficient to use a polynomial that is factorable because it demonstrates the idea without requiring the student to spend significantly more time to find the roots of some “realistic looking” polynomial such as 5.1157.231.1 2 −− xx . Calculators and other mathematics software are best used by people who already know mathematics. In general, using calculators as a learning tool compromises the building of a solid foundation. Calculators are over used and abused nowadays. One argument that supports the use of calculators as a learning tool is that if some tedious computations can be handled by machines, then why do we have to do those computations by hand? With the help of a calculator, the student can be freed to spend more time on concepts. This argument however does not answer the following question well: An algebraic software system can do almost all exercise problems in an algebra book, so why should we spend so much time in mathematics classes? Shouldn’t we be able to teach what used to be ten years of mathematics in three years? The answers to those questions are: (1) Tedious exercises are an integral part of the learning and maturing process. You take in more of the scenery when you walk than when you run, and (2) The exercise problems in a book are not important problems in the real world, solving them does not solve any real world problems. The exercise problems are themselves learning tools, they reinforce the concepts and build techniques, and those concepts and techniques will be useful when you take other classes or when you leave school. Solving a three by three linear system by hand using the Gauss-Jordan elimination method allows you to develop discipline, appreciate the thought process that can be applied to areas beyond solving a system of linear equations, and it trains you the habit of being precise and the ability to follow rules. Getting the answer immediately by inputting some numbers on a key pad lets you learn nothing other than how to operate that particular machine for that particular type of problems! And that is not good learning. Yet, this type of instructions is becoming more and more popular, pushed by people who have financial interests in selling more calculators, and people who are fooled by people who have financial interests in selling more calculators. There is one nice thing computers can offer though, besides being a good tool for people who already mastered mathematics, and that is programming using a high level language. It will be very helpful if mathematics students take one or two classes in a computer programming language such as FORTRAN, COBOL, or C. If you do not follow the rules when you write a program, the computer lets you know immediately. When a teacher says you are wrong, sometimes you suspect that is because the teacher does not like you. But when a computer says you are wrong, you know you are wrong and you go look for your mistakes in a hurry. Looking for mistakes in computer codes is not a pleasant experience, and to avoid that, you will have to be as right and as accurate as possible the first time. This after a while will instill in you the habit of being neat, precise, and economical.


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6. Real world applications Mathematics has many real world applications. It is wrong, however, to declare a mathematics topic not worth teaching or even worthless because it has no immediate applications in real life. Evolution and revolution are inevitable but mathematics is such a complex structure that judging its pieces by mere “real world applications” is elementary-school-student like–totally immature. What seems like a useless piece in this structure can be an important support to something else that is totally awesome, or it can someday find some important applications in the real world itself, or it can be just beautiful, which by itself is worth a lot. Mathematics is an art and a science. Sometimes you learn how to use mathematics, other times you may have to find ways to use mathematics. You can play basketball and learn nothing other than how to win games, or you can play basketball and learn how to live your life. It is not just about the subject, it is also about you. So ask not what mathematics can do for you, ask what you can do for mathematics. 7. How to use this workbook To obtain full benefit from this little workbook, the student needs to do all the exercise problems. There is truth in repetition. You cannot say I already know how to walk so I don’t need to walk. Walk you have to in order to receive the benefits, in order to walk without stumbling, in order to walk beautifully and powerfully. So, please do all the exercise problems if you want to walk with an air of confidence. This is like the boxer’s roadwork, you have to put in the mileage if you want to be good. It could be boring, tedious, and it may seem so trivial that you want to skip–that would be a huge mistake. It is a price you have to pay for your future success. The two components of algebra, logic and language, are intertwined in the exercise problems. It is helpful if you can differentiate the two, for the logic component can be tackled by reasoning, and the language component can only be tackled by memorization. Both components, however, must be reinforced by certain degree of repetition in order to achieve retention. So do not worry too much about which component you are dealing with, because repetition will normally take care of the matter. The good news is, once retained, algebra skills can be as permanent as the ability to ride a bicycle. Algebra is of course much harder than riding a bicycle, but persistence and perseverance will bring you the victory. If you are using this workbook to correct your tendency to make algebraic mistakes, you should follow the suggestions provided here. The workbook is arranged in the following manner: The important/difficult/interesting topics are discussed first. Exercise problems then follow. Some exercise problems are more straightforward, others involve several concepts in one single problem. This workbook is designed as an aid to students who are taking some classes that heavily employ algebra as a tool, and finding themselves constantly making “stupid” mistakes. It is not for students who have never taken algebra. In the case that you have forgotten some basic algebra facts that this workbook assumes you know, consult a mathematics teacher or algebra textbook.


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Chapter 2 Mathematics Grammar Writing is important. If a rock is as smart as Albert Einstein, we have no way of knowing because the rock has no means to communicate with us. So to us, the rock is as dumb as… a rock! There are some conventions in mathematical writing that have been getting ignored more and more often. The result is sloppy writing that at the harmless level, reflects poorly on the writer, and at the harmful level, causes confusion and inevitably eventual mistake. To make matters worse, many calculators actually allow the use of grammatically incorrect mathematics expressions. For example, the calculator I have in my hand right now allows me to type in: 5+–2, and sin(–30, both are terrible writings, and yet I get away with correct answers in both cases. These kinds of violations of mathematics grammar by powerful calculator makers are certainly not helping the cause of mathematics education. Mathematical writings are mostly, though not 100%, logical. They are man-made rules that have to be learned, memorized, and practiced much the same way a regular language is mastered. Sometimes you see some mathematical notations or ways of writing that are not exactly the most logical or convenient, that is because the first person who invented a notation might not have given enough thought to the notation, or later developments made the notation difficult to work with. But because the notation has been widely used for a long time, revision is not realistic and we just have to live with it. The same situation exists in all languages. The following are some important writing rules.

1. When a term involves a number and some letters, always write the number first. For example, x2y is not exactly a wrong way to write “2 times x times y,” but 2xy is what should be written. In a term, always write the number first. When there are two or more numbers, those numbers should be multiplied together. So 3x(–2)y should be rewritten as –6xy.

2. When adding a negative number to another number, enclose the negative number in a

pair of parentheses. For example, 5−+x is unacceptable. The correct expression is )5(−+x , which must then be simplified to 5−x .

3. When multiplying a number by a negative number, please enclose the negative number in

a pair of parentheses. For example, 5−⋅x is totally unacceptable. Even when you “know” what you are doing. That little dot could have easily been missed and the next step you will be doing x – 5. The correct way to do this is xx 5)5( −=−⋅ . Notice the minus sign should be moved to the very front of the term.

4. Parentheses come in pairs. When there is an open parenthesis, there must also be a close

parenthesis. For example, xf −( is an incorrect way of writing )( xf − . If your calculator allows you to do 5log( , resist this little convenience of not having to type in the


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complete, CORRECT expression of )5log( . Because once you accept this untidy

mathematical grammar, the next time you want to do 6log5log , you will be typing on your

calculator 6log(/5log( and then wondering why you are getting the wrong answer. The correct typing should be )6log(/)5log( .

5. Be precise. When a pair of parentheses is needed, write it explicitly, not just have it in

your mind. For example, )2(1 +⋅+ xx and )2)(1( ++ xx are totally different animals. The former equals 22 +x , and the latter 232 ++ xx .

6. When writing a fraction, pay attention to the dividing line. For example, 5

6+x

does not

really look like 6

5+x , if the latter is what you want.

7. In a fraction, the dividing line automatically acts as a grouping symbol like a pair of

parentheses. For example, 21

−+

xx is the same as

)2()1(

−+

xx . However, parentheses must be

added when two fractions are combined together for the operations to be carried out

correctly. For example, 43

21

+−

⋅−+

xx

xx is a grammatically correct expression that shows the

multiplication of two rational functions. If we are to actually multiply the two together,

then parentheses must be added in the next step. So 4231

43

21

+⋅−−⋅+

=+−

⋅−+

xxxx

xx

xx is

incorrect. The correct expression is )4)(2()3)(1(

43

21

+−−+

=+−

⋅−+

xxxx

xx

xx .

8. Avoid using a slash when writing a fraction. For example, does 1 2/3 represent 321 , or

12/3? Does x3/1 mean x3

1 , or x31 ?

9. To avoid potential misreading, mixed numbers should be used only as a final answer.

Form the habit of converting all mixed numbers to improper fractions before you start

performing any operations. For example, to perform the multiplication:

+ x

5123 , we

first convert 512 to

511 then do xx 3

533

5113 +=

+ . Knowing that you don’t have to

deal with mixed numbers reduces the chance of making silly mistakes.


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10. Pay attention to subscripts and superscripts. For example, 5logb is not the same as 5logb , and 5e does not equal 5e .

11. In general, an expression should be simplified as far as possible, especially if the

expression is the final answer of some problem. For example, all like terms must be combined, all fractions must be reduced to the lowest terms, and of course everything must be written in grammatically correct forms.

12. The exponent –1 should be treated carefully. When applied to a number, or an algebraic expression, it means “the reciprocal.” When applied to the notation of a function, it

means “the inverse.” For example, 313 1 =− , and

121)12( 2

12

++=++ −

xxxx . But

xx

sin1sin 1 ≠− , and in general

)(1)(1

xfxf ≠− . x1sin − stands for “the inverse sine

function” and )(1 xf − stands for the inverse function of )(xf .

13. When a function )(xf is raised to an exponent k, 1−≠k , it is written as )(xf k . In other words, kk xfxf )]([)( = . This notation is especially common in trigonometry, e.g.,

22 ][sinsin xx = , xx tantan 2/1 = , etc.

14. When dealing with long expressions, make sure you carry the whole expression with you at every step to prevent losing some “parts” in the process. For example, let’s say we

want to simplify the expression 5

)10)(3()5( 2 −+−+ xxx . The numerator is much more

complicated than the denominator, so it is natural to pay more attention to the numerator, and in the process you may forget the denominator altogether. It may go like this:

5)10)(5()5( 2 +−−+ xxx

)505(2510 22 −+−++= xxxx 755 += x

The correct answer is 155

755+=

+ xx .

15. Do not put an equal sign between two equivalent equations. For example, the following

equation is solved correctly, but the writing is inappropriate.

6215 +=+ xx 53 == x

35

== x


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By putting an “=” between the first equation and the second equation, you are saying that

536215 ==+=+ xxx . The second “=” signifies xx 362 =+ , which we know is not true. The proper way to write the steps is

6215 +=+ xx 53 =x

35

=x

16. When manipulating an equation, both sides of the equation must be treated the same even

if one side is already “nice.” (After all, mathematics is supposed to be the fairest game in the universe.) That means we have to multiply EVERY term by the SAME number or

expression. For example, we want to solve the equation 8)7(45)2(

31

=+−− xx . One

way to start is to eliminate all the denominators first so we don’t have to deal with fractions. So we multiply both sides by 12 and it goes like this:

8)7(4512)2(

3112 =+⋅−−⋅ xx

8)7(15)2(4 =+−− xx 81051584 =−−− xx

11−=x

Do you see the mistake? The number 8 on the right hand side of the equation should have been multiplied by a 12 also. The correct answer is 9/209−=x

17. In some notations there are “invisible” numbers that become visible only when you need to perform certain operations. The most notorious invisible number is of course the number 1. For example, x may have to be viewed as 1x if you are combining it with another like term, as in xxxxx 2.12.012.0 =+=+ ; or x may have to be viewed as 1x if you are multiplying it with another term that has x as the base, as in

23

211

21

121

xxxxxx ==⋅=⋅+

. 1 is not the only number that is sometimes invisible. For example, 3 x is the cube root of x, 4 x is the fourth root of x. Following this pattern, the second, or square, root of x should be written as 2 x , except, it is not. The square root of x is written as x , the 2 becomes invisible, and illogical it may be, we just have to live with it because tons of books have used this notation and it is not a good idea to change the notation now lest the great dead mathematicians should be unable to read our textbooks in their graves. You will see some more of such illogical notations if you venture into calculus.

Are we done with invisible numbers? Not quite. 10 can also become invisible as in

xx 10loglog = , and so can the natural number e as in xx elogln = . It would certainly be


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better for mathematics students in terms of learning if x10log and xelog were used in place of xlog and xln , respectively. What we can learn from such a small matter is that history matters and the most logical or most correct does not always run the show. I hope you find some of those politicians more tolerable now.

Chapter 3 Functions The concept of functions is very important in mathematics. All books in the market provide sufficient definition and basic operations. Here we discuss a few topics that are important and can be made easier when approached from different angles.

1. A lot of times, a function is written in the function notation =)(xf some mathematical expression that involves x. When a function is evaluated at a certain number, we simply replace the letter x in the mathematical expression by the given number. For example, if

xxxf 3)( 2 −= , then 434)4( 2 ⋅−=f . Most students have no problem with this substitution. Problems arise when a function is evaluated at another expression or another function, because now letters are involved, and more “parts” are also involved. This is a good opportunity for learning to see things in “blocks,” which is a very important skill in mathematics.

The letter x in a function can be viewed as a part, and the function itself as a whole that relies on the part to perform certain task. For example, wheels are parts, a car is a whole. So we can think of a car as a function of wheels. Using function notation, we can write

f( ) =

Now suppose you decide to replace the tires with some new ones that have fancy rims, then in function notation, we can write

f( ) = We see that only the tires get changed, the car otherwise remains the same. In mathematical notation, say we express the original tires as x, and the new tires as x + y, then the pictures above can be rewritten as

f(x) = x x – before


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f(x+y) = (x+y) (x+y) – after Which is exactly the same as the following substitution if we have function

xxxf 3)( 2 −= . xxxf 3)( 2 −= – before )(3)()( 2 yxyxyxf +−+=+ – after x, the original tires, have been replaced by x + y, the new tires with fancy rims.

2. Many common mistakes committed by students can be generalized as one: )()()( yfxfyxf +=+ . Some such examples are: 222)( baba +=+ ,

baba +=+ , and in trigonometry, yxyx sinsin)sin( +=+ . It is therefore important to know that there is only one function that has the property

)()()( yfxfyxf +=+ . Once you remember that, you are less likely to make the mistake again. So which function has this property? It is the function kxxf =)( where k is a constant. For all other functions, )()()( yfxfyxf +≠+ . kxxf =)( is also the only function that has the property )()( xcfcxf = for any constant c.

3. Piece-wise functions are a class of functions that involve more mathematical grammar. The concept of a piece-wise function is easy to understand but a lot of students have problems with the notation. There are ample every day examples of piece-wise functions. Tax brackets is one; how a 10-yard penalty in NFL is imposed when the guilty team is less than 10 yards from its own end zone is another; if you live in a place where you pay a higher rate for water once your monthly water usage passes certain limit is yet another.

Example 1. Suppose water is $2 per 1,000 gallons for the first 12,000 gallons, and $3 per 1,000 gallons for the amount over 12,000 gallons. So if you use 8,000 gallons of water, you pay $2 × 8 = $16. If you use 14,000 gallons of water, you pay $2 × 12 + $3 × 2 = $30. The general formula for this calculation must be stored in a computer so the computer can calculate the dollar amount every customer has to pay. The formula is a piece-wise function. Here let x represent the amount of water used in 1,000 gallons, and

)(xf the total dollar amount due.

>−+≤≤

=12 if ),12(324

120 if ,2)(

xxxx

xf


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This is a mathematical translation of the first two sentences in this example. How do we use it to evaluate )(xf ? For every x value, we use the “if” statements to determine which formula to use to do the calculation. For example, to evaluate )7(f , we first check that

1270 ≤≤ so we use xxf 2)( = to calculate 1472)7( =×=f . To evaluate )15(f , we check that 1215 > and therefore we calculate 33)1215(324)15( =−+=f . The common mistakes students make are (a) use both formulas and so

=−+=×

=9)127(324

1472)7(f and (b) after using both formulas to arrive at two numbers,

proceed to check the numbers against the given inequalities and get really confused, like

>=−+≤≤=×

=12 if ,9)124(324

120 if ,1472)7(

xx

f but the first answer 14 obviously does not

satisfy 12140 ≤≤ and also the second answer 9 does not satisfy 129 > , now what? Here the mistake is trying to compare the function values with the independent variable values. I will bet that most people are capable of calculating this problem correctly when it is written in plain English. The mathematical notation really messes things up. So why don’t we just abolish the mathematical notations? Well, they are a necessary evil we have to live with because higher mathematics will be impossible without the succinct mathematical notations.

Chapter 4 Division by Zero Any number divided by zero is undefined. It is one sentence and if you decide to just memorize it, it is perfectly ok. In mathematics, a simple rule may not be as simple as it seems when you have to always watch out for possible violations. Stop when you see a stop sign, right? Well, thousands of people run a stop sign every day. Anyway, back to division by zero. There are many ways to explain this rule, and you can find one or two in any textbook. Here we offer an explanation hoping it will reinforce the rule and prevent you from breaking the rule again. Suppose there are 12 apples and 4 monkeys. The monkeys decide to divide the apples evenly among them, how many apples does each monkey get? To answer the question, we divide the

number of apples by the number of monkeys: 34

12= . So each monkey gets three apples.

If there are 12 apples and NO monkeys, then the question “How many apples does each monkey get?” makes no sense because there exists no logical answer. There are no monkeys to divide the apples! However if we stick to our calculation, we still divide the number of apples by the

number of monkeys, we get 0

12 , which makes no sense, or as we call it in mathematics,

“undefined.”


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Alright, now that’s out of the way. Let’s have a quiz.

1. =50

2. =05

The first one equals zero, the second one undefined. It becomes much less obvious when letters are introduced.

3. Solve the equation xx 32 = . We divide both sides by x to get 3=x . Done! …? But wait a minute, it seems that 0=x is also a solution. How did we miss that? Well we missed it because we violated the no-division-by-zero rule without realizing it. When we divide both sides of the equation by x, the rule is violated if 0=x . Since we did not know what x would be yet, we could not just assume it was not zero. For this reason the operation of dividing both sides of an equation by an unknown expression is to be avoided. The better way to solve the equation is by first subtracting x3 from both sides to get 032 =− xx , then factor, 0)3( =−xx . Now we see either 0=x or 03 =−x , i.e., 0=x , or 3.

4. Find all real number solutions for the equation 0)6)(5()2)(5( 222 =+++−++ xxxxxxx .

Hint: (a) )5( 2 xx + is a common factor. (b) The solutions are 0 and –5.

Chapter 5 Absolute Value The absolute value of a real number is another simple concept that can cause some confusion when letters are involved. But of course letters are what algebra is all about. The absolute value of a number x is written as x . Roughly speaking, the absolute value operation takes away the minus sign of a number if the number is negative. It cannot be any simpler. If x is positive or zero, xx = . For example, 55 = . If x is a negative number, xx −= . This takes some meditation to realize. Here we assume x is a negative number, therefore x− is actually a positive number, despite the misleading minus sign in front of x. For example, suppose

5−=x , then 5)5(5 =−−=−=x . Use the piece-wise function notation we just discussed, we can write

<−≥

=0 if ,0 if ,

xxxx

x


15

Geometrically, the absolute value of a number can be interpreted as the distance between the number and the origin on a real number line. For example, both 5 and –5 have a distance 5 to the origin, therefore both 5 and –5 have an absolute value 5: 555 =−= . We will summarize the three typical equations/inequalities that involve the absolute value. Study them carefully and you will save yourself from a lot of future troubles.

1. An equation of the type: =x involving expression some non-negative number.

Example. Solve 512 =−x . Since 55 =− and 55 = , we see that 12 −x can be either 5− or 5, i.e., 512 −=−x or 512 =−x , and we obtain two solutions: 2−=x or 3=x .

2. An inequality of the type: <x involving expression some non-negative number.

Example. Solve 512 <−x . Since an absolute value is a distance, we want the distance to be “small” as signaled by the direction of the inequality “less than” <. The numbers that have a distance less than 5 from the origin on the real number line are those between – 5 and 5. So the inequality is equivalent to two inequalities:

5125 <−<− x Solve the inequalities we get 32 <<− x .

3. An inequality of the type: >x involving expression some non-negative number.

Example. Solve 512 >−x . Since an absolute value is a distance, we want the distance to be “large” as signaled by the direction of the inequality “greater than” >. The numbers that have a distance more than 5 from the origin on the real number line are those less then – 5 or greater than 5. So the inequality is equivalent to two inequalities:

512 −<−x or 512 >−x Note here that a number can be either less than – 5 or greater than 5, but cannot be both at the same time. That’s why the connective word “or” is a must and cannot be omitted, and the two inequalities cannot be put together in some nice way as those in the last example. Solve the inequalities we get 2−<x or 3>x .

Equations and inequalities that involve absolute values are one area where many mistakes are made. Here are the typical ones:

1. Ignore the two vertical lines altogether. For example, when solving 93 =+x , simply solve 93 =+x to obtain 6=x . The correct answers are 6,12−=x .


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2. Do not know that the directions of inequalities need to be changed when the two vertical

lines are removed. For example, when solving 31 ≤−x , simply solve 31 −≤−x or/and 31≤−x , and obtain 2−≤x or/and 4≤x . The correct answer is 42 ≤≤− x .

3. Know the directions of inequalities must be changed but unable to perform the changes

correctly. For example, when solving 72 >+x , proceed to solve 727 >+>− x . The correct step is: 72 −<+x OR 72 >+x .

4. Combine two contradictory inequalities together. For example, the inequalities: 2−<x

OR 3>x , cannot be combined into 23 −<< x because the latter means x is greater than 3 AND less than – 2, which, we know, is impossible.

One way to avoid getting tangled up in all the inequality direction changes is to memorize the form of the solution an absolute value equation or inequality has. Before you actually start solving the equation or inequality, you write down the form of the solution and proceed to solve the absolute value equation/inequality the way you solve an absolute value equation, and finally plug the solutions into the form of the solution to obtain the final answer. Here is how:

a. If you have an absolute value equation of the type |some expression involving x| = non-negative number, the solution will be 1#=x , or 2#=x . For consistency, we assume

21 ## ≤ here and in the next two cases. b. If you have an absolute value inequality of the type |some expression involving x| < non-

negative number, the solution will be 21 ## << x . c. If you have an absolute value inequality of the type |some expression involving x| > non-

negative number, the solution will be 1#<x or 2#>x . For example, suppose we want to solve 523 ≥−x . First recognize that by the direction of the inequality the solution will look like 1#≤x , or 2#≥x . How do we find the two numbers? We simply solve 523 −=−x or 523 =−x , i.e., 1−=x or 3/7=x . Now plug these two numbers into the solution 1#≤x , or 2#≥x and we have 1−≤x or 3/7≥x .

Chapter 6 Average Everyone does the average. We might as well say the average person can calculate the average. This is true to some extend but there are situations that can lead to wrong answers if one is not careful enough. A straightforward problem is easy. What is the average of three numbers 4, 7, and 13? The average is (4 + 7 + 13)/3 = 8. Now try the next few problems.


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Example 1. Suppose I walked to K-Mart from home at a speed of 3 mile per hour. On the way back I increased my speed to 5 miles per hour because I wanted to catch a TV show. What was the average speed of my entire trip (to and from K-Mart)? If you answer 4 miles per hour, you have just fallen into a common trap. The answer is 3.75 miles per hour. Here is why: Suppose the distance between home and K-Mart is D, the time it

took to go from home to K-Mart 3rate

distance D== . Similarly, the time the return trip took

5D

= .

So the total time for the round trip was 53DD

+ . Finally, the average speed for the entire trip

75.3830

51

31

2

53

2 timetotal

distance total==

+=

+==

DDD miles per hour.

Example 2. Suppose there were three tests and each is worth 100 points. Before taking the third test my test average on the first two tests were 77. If I got a 95 on the third test, what was my average over the three tests? The quick calculation, (77 + 95)/2 = 86, is wrong. 83 is the correct answer. The reason? 77 is not one score, but the average of two scores. So the sum of the first two scores must be 154277 =× .

The average of three scores is then 833

951543

scores threeof sum=

+== .

Chapter 7 Percent Change The average person can also calculate percentage, or can he? If the price of tuna fish has increased from $100 per pound to $120 per pound, the percent

increase is 20%, because there is a $20 price change, and %2020.010020

===priceoriginal

change In

general, we use the following formula:

%100numberolder

numberolder number newer change Percentage ×−

=

The formula has an equivalent form: changepercent number older number older number Newer ×+= Because percent change is such a common calculation in everyday life, everyone thinks it is easy–and that’s exactly what gets us into trouble!


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Example 1. In city K the sales tax rate is 8%. If you see a TV set in a store on sale for $500, what would you actually have to pay when you buy the TV set? Using the second formula, we calculate Price after tax 54040500%8500500 =+=×+= . So you pay $540. Example 2. Kim was making $50,000 per year two years ago. Last year she received a 20% pay raise, but this year her pay was cut 20% due to bad economy. How much will she make this year? 20% up and 20% down, back to the starting point, $50,000 is what she will make! Are you sure? Kim would love that! But her actual pay this year is only $48,000. Here is how the calculation is done: Last year’s pay 60000%205000050000 =×+= This year’s pay 48000%)20(6000060000 =−×+= The minus sign in the percent change for this year’s pay reflects the fact that the change is a decrease. Some times a problem does not agree with our “common sense.” It trains us to have the discipline to follow the prescribed steps even when the answer seems to be obvious. Example 3. A 100 pound pumpkin is 99% water. How much weight does the pumpkin have to lose so it contains 98% water? If you think one pound maybe, you are in for a big surprise. It has to lose 50 pounds! The original pumpkin had 99%99100 =× pounds of water. After x pounds of water evaporated, it now has x−99 pounds of water. The total weight of the pumpkin is x−100 pounds. The

percentage of water in the pumpkin is xx

−−

=10099 , this has to be 98%. Solve %98

10099

=−−

xx we

get 50=x , i.e., the pumpkin has to lose 50 pounds to become a new pumpkin that’s 98% water. Example 4. You paid $4000 tax to the county last year. Later you found out you paid 20% more than you had to. What was the amount you actually had to pay? Well, 20% of $4000 is $800, take $800 away from $4000 and we arrive at $3200. So the actual tax was $3200, right? You wish! NO, you had to pay $3333.33. Suppose the actual tax was x. We know that 4000 is 20% more than x, that means

%204000 ⋅+= xx . Convert 20% to 0.20 and combine like terms on the right hand side of the equation we get x20.14000 = , or 33.3333=x .


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Chapter 8 Logarithm This is one of the topics that the notations make a major contribution to the subject’s degree of difficulty. A quick summary is provided here.

1. xy blog= is equivalent to ybx = , here 1,0 ≠>b , and 0>x .

2. Some user-unfriendly formulas need to be memorized and practiced. Given that 1,0 ≠>b .

a) xb x

b =log , for any real number x. b) xb xb =log , for any real positive number x. c) yxxy bbb logloglog += , for any real positive numbers x and y.

d) yxyx

bbb logloglog −= , for any real positive numbers x and y.

e) xnx bn

b loglog = , for any real positive number x and any real number n.

f) bx

xa

ab log

loglog = , for any real positive numbers x and a, with 1≠a .

3. To make matters worse, there are two special cases with unique notations. The notations

have some minor variations. Here are the two most commonly used ones:

a) The common logarithm, xx 10loglog = . b) The natural logarithm, xx elogln = .

All the formulas can be rather easily proved from the definition. This, however, does not change the fact that these are some of the most mistake-prone formulas. What do you do? There is no silver bullet. Read the proofs if you want, but DO memorize the formulas and do a lot of exercise problems until the formulas become your second nature. That is the only way to avoid making “silly” mistakes. Common mistakes involving logarithms:

1. bxxb loglog = 2. n

bn

b xx )(loglog = 3. yxyx bbb loglog)(log +=+ 4. yxyx bbb loglog)(log −=−

5. yx

yxb

bbb log

logloglog =−

6. yxxy bbb logloglog =


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7. yx

yx

b

bb log

loglog =

8. )(loglog yxyx

bb −=

Make sure you see the wrong in each one of them.

remedies for common algebra errors - college of southern nevada

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