Intermediate Algebra MATH 1010
Weber State University
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MATH 1010 COURSE SUMMARY MIKE WILLS CONTENTS General Comments Study Guide The Real Numbers Linear Things The Plane Systems of linear equations Polynomials and Exponents Factorizing Rational Expressions Radicals Quadratics New Functions Turning Circles WNQP FWN E to HHHH WNEQ39 00NNNNOJOJU1gtgtgtgtMH 1 GENERAL COMMENTS Mathematics is everywhere In nature we observe patterns and try to under stand them In every day life we regularly use things that would not be around without mathematics For example automobiles buildings computers television and cartography maps all rely crucially on mathematical insights ln sports sta tistics plays a huge role in determining how good a player is In medicine we use charts and complicated machinery that require a certain amount of mathematics to use understand and interpret Figures numbers data samples voting results all appear in media that we are exposed to everyday Thus whether we like it or not mathematics plays a crucial role in how we Americans live our lives It is possible to argue that you personally donlt need to know mathematics to achieve success in your life There is of course a lot of truth to this but immediate utility is not the only reason to study an academic subject Other reasons to study an academic subject include 1 Increasing general knowledge the more you know the better armed you are for what life throws at you Ignorance is not necessarily bliss 2 Its inherent beauty and its ability to describe the human condition Although it may not be apparent mathematics can be very beautiful in much the same way that art or music can It takes a bit of training to appreciate it I suppose but the beauty is there 3 Many employers like people who have a good mathematical background 1 2 MIKE WILLS 4 This is a bad reason but 1711 mention it anyway for matriculation purposes a given subject may be required to nish a course This is the worst reason because if the only reason that you7re studying a subject is because you7re being forced to then you re pretty much wasting your own time With this in mind it is best to learn the real subject well rather than simply attempt to mimic the subject enough to fool the instructor into thinking that you know the material If nothing else the mimics tend not to fool the instructor A nonnegative attitude would therefore be highly desirable Mathematics is a living breathing subject which has fascinated many people for millennia It seems to me that a good question is What exactly is mathematics There is no easy answer to this but my thoughts are as follows 1 Mathematics is not just about numbers although most math involves num bers 2 Mathematics is not about memorizing formulae or equations any more than history is about memorizing dates 3 Mathematics is the study of ideas usually abstract Using logical reasoning mathematics considers the consequences of these ideas In that sense it is a game The fact that this abstract almost surreal approach leads to amazingly fruitful applications is an example oft e f 39 of 39 1 If you take a point of view along the above lines I think that you will nd studying mathematics more interesting less dif cult more enjoyable and more ef cient Moreover even if you forget the details of the present course you will retain the ability to think analytically which is an inevitable outcome if you study mathematics effective y One thing that is probably not emphasized enough at any level of mathematics is that the method matters far more than the outcome Stop asking the instructor Is that the right answer and start asking Is this a correct method That leads me to the next section where I will give my own study hints for the nal 2 STUDY GUIDE There is no panacea for passing a maths class We all learn a little differently and we all have different backgrounds I will tell you what has worked or me o Recopy lecture notes My lecture notes were always untidy when l was a student and so I made an early commitment to rewrite the notes when I got back to my of ce or study I did this for all classes that I took while in college 0 Prioritize As I mentioned in a class I spent my summers in Santa Barbara studying I did this because I decided that getting my degree was more important then drinking margaritas by the beach I stand by that l partied fairly hard as an undergraduate and as a direct result ended up with an inferior degree and no job prospects Worse l was in danger of becoming an alcoholic Naturally there are many things more important than getting a degree for example family and work However both family and work will likely bene t if and when you get the degree Keep that in mind 0 Do as many problems from the book as you can There is no substitute for practice ldeally doing mathematics should not be drudgery Unfortunately it often takes repetition to understand an idea which means that doing dozens of 1Bertrand Russell MATH 1010 COURSE SUMMARY 3 problems that are essentially the same is a useful exercise However don7t just take a cursory approach to the problems do them careful yr 0 Read the text carefully and take notes 0 Time yourself on practice exams without the aid of the book or your notes Do this frequently Repeat until you are con dent that you can get at least 90 on any exam in the subject 0 Look for the key ideas in the course Typically there really are not that many but if you understand those key ideas all the minutiae drop out as a natural consequence 0 Listen carefully to what your instructor says All but the very worst instructors know what they are talking about and are doing their best to pass that on to your 0 Write out your thoughts on paper using complete sentences in good English It is shocking how dif cult it is to persuade students to do t is For example it took me years to gure this out and yet once I started doing this my performance and understanding increased dramatically 0 Go through the solutions provided by your instructor They can give you an insight as to how mathematicians think I really do make an effort on the quiz solutions to give you enough detail so that you can follow the work while providing motivationi 0 Talk to the instructor frequently outside of class More than anyone else the instructor knows what is going on in the course what has been said in class and what to expect in the coming weeks 0 Attend every class Pay attention in class Leave discussions of your Halloween festivities in Majorca for another time Do not walk out in the middle of class just because you are bored or because the instructor said something you dislikedi Show some courtesy if not to the instructor who no doubt is an arrogant git who will be the rst against the wall when the revolution comes then at least to your fellow classmates Finally try to keep the workwhine ratio at least 5050 in favor of work It is remarkable the amount of correlation between people who pass math classes and people who follow the above tips I also nd it remarkable that I feel the need to bring some of these points up to an audience consisting of college students 0 Learn from your errors For example if you get marked wrong because you stated that a b2 a2 122 work out why you were marked wrong and x the issue It is human to make errors It is silly to repeatedly make the same error and refuse pointblank to learn from it Intermediate algebra may be tough but it is not by any stretch of the imagination insurmountable I am not going to write a practice na i There are two reasons for this the rst is that you have more than adequate practice problems from the tests the practice tests in the text and the practice tests on the developmental math webpagei The second is that if I wrote a practice test it would almost inevitably have my own personal stamp on it and thus likely give a misleading impression of what the nal is like I will now go through the individual chapters and highlight some of the crucial pointsi 4 MIKE WILLS 3 THE REAL NUMBERS We don7t cover Chapter 1 in this class but it is nevertheless a fundamental chapter Since I assume that you know the basics reasonably well I give here an advanced point of view An important point in mathematics is to distinguish between notation and the underlying concepts Notation is essentially convention for example the order of operations has the acronym PEMDAS This is a convention there7s no fundamental rule of the universe that says that this is the way the order of operations has to be However it has turned out that this is a useful order more useful than other orders A crucial concept in mathematics is the idea of a set In fact the vast majority of modern mathematics including the study of numbers is based on the socalled Peano axioms which are basic precepts about sets here are a number of different ways of representing sets of real numbers The most general way is setbuilder notation a key advantage of setbuilder notation over interval notation is that it generalizes to arbitrary classes of objects The real numbers are constructed as follows we start with the natural numbers N l23 We then throw in 0 and the negatives7 of the natural numbers to obtain the set of integers Z 0ili2i3 We then de ne the rational numbers as ratios of two integers that is Q l nm E Zm 0 Finally we de ne the real numbers R as all numbers that can be wellapproximated by a sequence of rational numbers Intuitively if we plot all the numbers in Q on a line not all the points on the line are lled but given any point on the line there are in nitely many rational numbers near it7 e ne addition and multiplication on the natural numbers in the way that we learned in primary school We extend the idea of these two operations to all real numbers in such a way that the extension when restricted to the natural numbers is what we started with Thus if abcd are integers with b and d nonzero E g ampl i Subtraction and division are inverses of addition and multiplication respectively Addition and multiplication on R are associative and commutative 0 is an ad ditive identity and l is a multiplicative identity Every real number has an additive inverse its negative and every nonzero real number has a multiplicative inverse its reciprocal The key property that ties together addition and multiplication is the distributive law inally the real numbers are totally ordered that is if ab E R then one and only one of the three following conditions holds a lt 12 ii I lt a iii a 12 Moreover if a lt b and c gt 0 then ac lt 120 This last statement together with the above remarks mean that R is what is called in the jargon a totally ordered eld 4 LINEAR THINGS Notation 41 For the rest of this handout z y represent real variables unless otherwise stated De nition 42 An equation is linear in one variable if it can be written in the form ax b c where a b c are real numbers and a f 0 2Technically a set is unde ned but the concept is one that we should all have an intuition for MATH 1010 COURSE SUMMARY 5 Using the basic properties of arithmetic it is clear that z 07 1quot Of course one may have an equation that looks linear but when written in the above form a turns out to be zero In that situation one either gets an identity 0 0 in which case I can be any real number or if I f 5 one gets a contradiction and the original statement has the empty set for a solution set Many formulae make use of linear equations For example the relationship between degrees Celsius and degrees Farenheit can be expressed as a linear equation Also many mixture problems involve linear equations It also makes sense to discuss linear inequalities that is inequalities that can be written in the form ax b lt c or am b S 5 again with a f 0 In that case the solution set will turn out to be an interval the exact nature of which will depend upon whether the inequality is strict and also upon the sign of a it is fundamental that multiplying an inequality by a negative number reverses the inequality This is a consequence of the fact that the real numbers are a totally ordered eld In various situations one may have to bound ax b above andor belowi In that case we end up with compound inequalities which must be handled with care Compound inequalities can be expressed succinctly using absolute values However when solving an inequality using an absolute value one must take into account the fact that if lal S lbl then 7b S a S bi Equivalently either I S 7a or b 2 a Again this follows from the fact that the real numbers are a totally ordered eld 5 THE PLANE It is often convenient to work with an in nite planer For convenience we pick one point call it the origin and then draw two perpendicular lines that intersect at the point By convention one line is horizontal and the other is vertical We impose the real numbers on each line with the 0 on each line at the origin and the numbers increasing from left to right on the horizontal line the zaxis and from down to up on the vertical axis the yaxis Having done this any point in the plane can be represented as an ordered pair of real numbers De nition 51 An equation is linear in two variables I and y if it can be written in the form ax by c with a2 b 0 The solution set of such an equation can be represented by a graph in the plane A fundamental result is that the graph will always be a straight line If I f 0 then y 791 f In this situation the slope of the line is 7 and the y intercept is 0 If I 0 the line is vertical and the slope is unde ned Two nonvertical lines are parallel if and only if they have the same slopei Two nonvertical lines are perpendicular if and only if the product of their slopes is 1i A nonvertical line has an equation that can be written in the form y mm b slopeintercept form i Often one knows the slope m and one of the points that the line passes through say 11 y1i In that case the equation of the line will be y 7 yl mz 7 11 This is the pointslope formi A linear inequality in two variables will typically have a solution set consisting of a halfplane whose boundary line is the graph of the corresponding linear equation Which of the two halfplanes formed by the boundary line actually gives the solution set is most easily discovered by plugging in suitable pointsi 6 MIKE WILLS De nition 52 A function is a rule that takes elements of one set and maps them to another set For each input there is only one output After sets functions are probably the most important concept you are likely to come across in mathematics In this course we are interested in real valued functions of real numbers The simplest function is the constant function 12 whose graph is a horizontal line y b in the planet The simplest reasonably interesting function is the linear plus constant function mm 12 whose graph is a straight line with slope m and y intercept 0 b A curve in the plane is the graph of a function if and only if the curve intersects each vertical line in the plane at most oncel One application of linear functions involves proportionality that is variation Proportionality comes up often in applications for example the rate of decay of a radioactive substance is proportional to the mass of that substance 6 SYSTEMS OF LINEAR EQUATIONS Sometimes we have two or three linear equations in two or three unknowns Since the graphs of linear equations in two variables are straight lines in the plane one immediately sees that such a system of equations has either one solution if the lines are not parallel no solutions if the lines are distinct and parallel or in nitely many if the lines are the same e can in principal solve systems like these graphicallyi However it is usually best to solve algebraicallyi To do that one uses substitution which is discussed in the box on page 230 A similar method will work for linear systems involving three variables Key applications involve net speeds of vehicles due to the speed of the medium for example the net speed of a river boat is affected by the speed of the river and which direction the boat is going in The simplest model assumes that the boat and river both go at constant speeds One thus gets a linear system in two unknowns 7i POLYNOMIALS AND EXPONENTS Multiplication can be thought of as repeated additioni Similarly exponentiation can be thought of as repeated multiplication Let a be a real number and n a positive integer Then a is de ned to be the product of n copies of at It is fundamenta that if m is a positive integer then amn amani If we want to extend the idea of exponents to other numbers this is the law that we want to preserve In doing so we discover that for a f 0 and n a natural number a0 an a alni If fx anx l l l alx a0 with an y 0 then f is a polynomial function of z of degree mi The function 0 is a polynomial with indeterminate degreei One performs arithmetic operations on polynomials so that the rules of arith metic and exponentiation are not violated For example I 22 I 2 z 2 12 4x 4 by the distributive lawi When one divides polynomials one ends up with rational expressions that is ratios of polynomials A rational expression is proper if the degree of the numerator is less than the degree of the denominator If a rational expression is improper it can be made proper by writing it as the sum of a polynomial with a proper rational expression The process used is long division which works exactly the same way t e long division of numbers wor s MATH 1010 COURSE SUMMARY 7 Special cases of polynomials include monomials one term binomials two terms and trinomials three terms We are already familiar with polynomials of degree 1 linear and zero constant 8 FACTORIZING With care multiplying two polynomials together is a straightforward application of the rules of arithmetic and exponents Factorizing a polynomial into a product of polynomials tends to be more difficult but is often essential in applications There are certain techniques that will work for example a degree 2 polynomial a quadratic can often be factored essentially by inspection eg I 7 61 8 z 7 4 z 7 2 Some basic techniques include factoring by grouping and hoping for the best A common factorization is 12 7 b2 z z b There are similar factorizations for 13 7 b3 and 13 123 In general 12 122 does not factor over the real numbers Frankly the best approach to factorization is experience 9 RATIONAL EXPRESSIONS Let p and L be polynomials Let fx Then f is a rational function with domain all real numbers I such that 41 0 One usually wants to write f in lowest terms that is if one has a choice between or I one usually picks the latter There are some technical issues here which we gloss over Graphing rational functions tends to be dif cult Common techniques to aid in graphing rational functions is to write the function in lowest terms note that the points not in the domain of the function give rise to vertical asymptotes nd the z and y intercepts and then plot points away from the intercepts Unfortunately nding the zintercepts when the numerator is zero and the vertical asymptotes when the denominator is zero explicitly is usually nontrivial and sometimes impossible Many formulae involve rational expressions for example Newtonls Laws of physics 10 RADICALS De nition 101 Let n be a natural number We say that y is an nth root of I if y I If Ly gt 0 and n is even then we write y If I y are real and n is odd then we write y Again using the fundamental fact about exponents namely the sum rule we can now extend the idea of exponents to rational numbers suppose a is a real number with a real nth root Then a 5 Hence a It is customary to write expressions involving ratios of radicals so that only the numerator contains radicals One does this by rationalizing the denominator 11 QUADRATICS Quadratic polynomials come up in a variety of applications such as distance problems area problems and others Not all quadratics can be factored over the real numbers For example 12 l is irreducible 8 MIKE WILLS For a variety of reasons it became useful to invent the symbol 239 and de ne i2 71 After several centuries of work this symbol was given a rigorous mathe matically satisfying framework The set of complex numbers is given by C a bi 1 ab E R The normal laws of multiplication and addition apply to the complex numbers However C is not totally ordered in particular the number i is not positive zero or negative Note that if a gt 0 then H i The conjugate of a 12239 is a 7 12239 If divide one complex number by another 1 can write the resulting ratio in standard form by multiplying and dividing the ratio by the conjugate of the divisor thus abi abicidi acbdibciad cdi7abiaibii a2b2 111 We can now find the roots of any quadratic equation a12 121 c 0 by completing the square that is by observing that for a f 0 112 az2bzcialtz2kzgtcialtltzigt27i 7 a 7 2a 4a2 Once having completed the square standard algebraic manipulations yield the qua M gtc dratic formula I The graph of y 12 is a parabola in the plane which is concave up and has vertex the origin By completing the square one can write any quadratic function as az 7 h2 k The graph of y is the graph of y 12 shifted to the right by h units then scaled by a factor of a and then shifted vertically up by k units Notice that the vertex is h k This method of obtaining new graphs from known ones is quite general 12 NEW FUNCTIONS We say that a function f is onetoone injective if fa forces a b A function is onetoone if and only if its graph intersects each horizontal line at most once If f is onetoone it has an inverse function f l whose graph is a reflection of the graph of f through the line y 1 Finding an inverse function is usually nontrivial but in principal the method is straightforward switch the roles of y and z and then solve for y Notice that 1 Functions and inverse functions have the same duality that multiplication and division do one operation undoes the other The single most important class of functions in mathematics are the exponen tial functions a where a gt 0 and a f 1 Exactly how one computes a for an irrational number I is best left for a more advanced course The graph of f is concave up and has the zaxis as a horizontal asymptote The y intercept is 01 and the function is strictly positive If a gt 1 the function is increasing otherwise it is decreasing Of particular interest is e where e 11 i It turns out that the slope of the graph of e at z a is 6 This is not immediately obvious and does not generalize to more arbitrary exponential functions If a with 1 f a gt 0 then f is onetoone and its inverse function is f 1z log I that is f 1 is a logarithmic function The domain of this function is the positive real numbers There are a number of fundamental properties We list a few here MATH 1010 COURSE SUMMARY 9 i alogax I ii logauv loga u log 1 for uv gt 0 iii log a 1 iv logal 0i Logarithmic functions come up with respect to questions about sound decibels and earthquakesRichter scalei One reason is the second property noted above which converts multiplication into additioni In fact slide rules made crucial use of 11i Exponential functions are important in nance notably continuous compound interest Standard scienti c calculators have buttons for a 10ei By convention log10 is simply denoted log while loge is denoted ln Notice that a 6 mar Hence there is essentially only one exponential function One can show that log u igi zi By taking 5 10 or e we can use calculators to obtain good approximations ofaloga u fun fact is that exponential and logarithmic functions can be extended to complex numbers with care and one obtains the equation 6 1 0 when one does so Most mathematicians consider this to be amazing and beautiful 13 TURNING CIRCLES One of the most fundamental class of curves in our experience are circlesi A Circle is the set of all points in the plane equidistant from a xed point the center The distance between a point on the circle and the center is called the radius By the distance formula Pythagoras the circle with center h k and radius 7 2 0 has the formula I 7 h2 y 7 k2 T If one is given an arbitrary secondorder equation in the variables I and y and one is told in advance it is a circle one can write the equation in the above form by completing the square in both variablesi
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