Monitor Projector

DrDelMath

Intermediate Algebra
Elayn Martin-Gay

Chapter R&P: Review and Prerequisites

Section RP.0: Tips for Success in Mathematics

Instruction is much more than presentation of information. Instruction may include events that are generated by a page of print, a picture, a television program, a combination of physical objects, potentially many other stimuli, as well as activities of a teacher. Instruction is a deliberately arranged set of external events designed to support internal learning processes.

Teaching refers to the activities of the teacher. Therefore teaching is only one part (I think it is an important part) of instruction.

The purpose of early college level algebra courses is to introduce the student to the use of abstraction, generalization, deductive reasoning, and creative thinking while exploring the patterns and relationships of a variety of algebraic entities including, but not limited to, equations, inequalities, algebraic fractions, polynomials, and functions. [What Are We Studying]

Here are some terms that have frequently been forgotten but will be necessary in this course.

A very important and an extremely effective Study Skill is provided in an Intermediate Algebra text (formerly used at STLCC) and is repeated here.
"Many of the terms used in this text may be new to you. It will be helpful to make a list of new mathematical terms and symbols as you encounter them and to review them frequently."

I suggest that a second list of important properties also be constructed. Whether you use flash cards or a notebook list, it is imperative that you regularly review these definitions and important properties. You must know them precisely -- your own words will not be satisfactory.

In addition to the suggestions in the textbook, you should also read the following essay on this website.
Studying Mathematics

To realize the importance of definitions, read the essay: Mathematics Without Definitions.

A parallel concept that is important to the study of mathematics is the concept of binary relation. The most general definition of binary relation does not help understand algebra at this level, so the following definition is restricted to a particular kind of binary relation.

One of the things we do in mathematics is use deductive reasoning to study relations and operations on mathematical objects. It will help your understanding if you identify each "thing" as either:

Try to make that identification as soon as you are introduced to a new mathematical "thing".

Some familiar mathematical objects are numbers, variables, algebraic expressions, formulas, and geometric figures.

Examples of unary operations, binary operations, and binary relations that are of interest in this course are presented with the buttons below. You will be introduced to additional objects, relations, and operations as the course progresses.

       

Section RP.1: A Few Basic Algebra Concepts

Section RP.2: Fundamental Formulas

As we progress through Intermediate Algebra and College Algebra we will review, relearn, or learn for the first time quite a number of formulas. Most of the formulas encountered in Intermediate and College Algebra are formulas that every adult should know. We begin with the following list of formulas for areas of some familiar geometric shapes. Two very useful geometric terms are polygon and quadrilateral. For interactive information about polygons. Math Open Ref topic index about polygons
For interactive information about quadrilaterals. Math Open Ref interactive page about quadrilaterals

Math Open Ref interactive page about area of squares Math Open Ref interactive page about area of parallelogram Math Open Ref interactive page about area of rectangles Math Open Ref interactive page about area of triangles Math Open Ref interactive page about area of trapezoids Math Open Ref interactive page about area of circles Math Open Ref interactive page about circumference of a circle

 

About Variables, Algebraic Expressions, and Basic Formulas


Section RP.3: An Introduction to Sets

Now that we agree on a few common terms and formulas we are ready to begin the study of algebra. The concept of set is very important in mathematics. Consequently we begin the study of Algebra by examining some of the elementary ideas of sets.

 

About Sets, Elements, Subsets, and Set Notation

Section RP.4: Sets of Numbers

The previous introduction to sets permits a preliminary study of the sets of numbers used in algebra. It is important that you know the definitions of these sets as well as the symbols used when writing about them. The terminology of sets and these very important sets of numbers are the very foundation of any discussion of algebra.

Section RP.5: Introduction to Intervals and Rays

Your author discusses interval notation in Section 2.8. (If that is the wrong section for your book, look for the first section where she discusses solutions of inequalities.)

An interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set.

An interval corresponds to a contiguous set on the real number line.

The symbols (, [, ), and ] are used to describe intervals as defined in the following chart. The set builder notations in this chart are frequently used to define what is meant by the word interval as well as to define interval notation - a convenient notation which matches the graph of the interval. It is really pretty straightforward, parenthesis mean that end number is not part of the set and square brackets means that end number is part of the set. Always remember the left end must be less than the right end because numbers increase as you move from left to right on the number line.

 

About Sets of Numbers

Section RP.6: Opposites and Absolute Value

The algebraic form of the definition of absolute value will be required for all of algebra. The simpler number line version suffices for numbers and for concept formation, but not for algebraic work.

The concept of opposite will be generalized and extended during the next several algebra courses.

 

About absolute Value and Opposites of Numbers

Section RP.7: Operations on Real Numbers

With advanced mathematics, it can be proven that the points on a line are in one-to-one correspondence with the real numbers. Remember the real numbers include rationals, irrationals, integers, whole numbers, and natural numbers. To say there is a one-to-one correspondence between the points of the line and the real numbers means that:
(i) for every real number there corresponds exactly one point on the line
(ii) for every point on the line there corresponds exactly one real number.

Section RP.8: Addition of Real Numbers

This gives rise to the concept of "The Number Line" which is a geometric way of visualizing the real numbers.
It is common to draw the number line showing only the integer points. When that is the case it is your responsibility to recognize that all the rational numbers and all the irrational numbers are represented on that same line.

number line graphic

It is imperative that you observe the order of the real numbers as displayed by the number line.
At any point on the number line, the numbers increase to the right and decrease to the left.

This leads to a natural interpretation and visualization for addition of signed numbers. Now note that adding a positive number (to any number) means to increase and thus means move right. On the other hand, adding a negative number (to any number) means to decrease and thus means move left. This visualization will always yield the correct sign for the sum.

number line addition  number line addition

 

distance formula link

On the number line the symbol < (read as "less than") means "to the left of".
On the number line the symbol > (read as "greater than") means "to the right of".

Section RP.9: Subtraction of Real Numbers

distance formula link

Section RP.10: Multiplication of Real Numbers

Section RP.11: Division of Real Numbers

 

About Operations on Real Numbers

Section RP.12: Review of Fractions

Division of fractions is exactly like all divisions in mathematics


division graphic

Because the multiplicative inverse of a fraction is its reciprocal, the conversion to multiplication looks like:


fraction division graphic

After converting to multiplication and performing the multiplication according to the definition of multiplication we get the following rule.


division of fractions graphic

All of this boils down to a fairly simple rule for working with division of fractions as set forth in the following definition.

Subtraction of fractions is exactly like all subtractions in mathematics


subtraction graphic

Using one of the four forms for the opposite of a fraction makes the conversion to addition look like:


fractions subtraction graphic
From which we get the following definition of subtraction for fractions.

 

About Fractions

Section RP.13: Exponents and Roots

 

About Exponents and Roots

Section RP.14: Order of Operations

 

About Order of Operations

Minimal List of Exercises Page 26.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check: 1 - 14
Section 1.3 Exercise Set: 4, 9, 12, 18, 21, 22, 25, 28, 37, 38, 41, 42, 45, 47, 50, 51, 56, 61, 64, 71, 77, 79, 89, 90, 95, 96, 101, 102, 103, 107, 108.

Section RP.15: Properties of Real Numbers

A parallel concept that is important to the study of mathematics is the concept of binary relation. The most general definition of binary relation does not help understand algebra at this level, so the following definition is restricted to a particular kind of binary relation.









At this point in the course we are interested in understanding equality, inequality, their meaning on the Real Number line, and symbols used to represent these binary relations.

In the Real Number system, inequality is a statement about the relative size or order of two numbers. The Real Numbers are arranged on the Real Number line in such a manner that larger numbers are to the right. As you move from left to right, you move from smaller numbers to larger numbers. The Real Numbers are "ordered" on the number line according to their size.

 

About Equality and Inequality

Some very important properties of the Real numbers are presented below. Overt use of these properties will vary. Sometimes it will be necessary to explicitly reference one of these properties and on other occasions, the property will be used with hardly a notice.

The following properties of addition and multiplication are presented as properties of Real Numbers. Consequently they are properties of Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers.

Of all the following axioms and properties for the real numbers, the Distributive Property, The Zero Factor Property, the Transitive Properties, and the Law of Trichotomy are extremely important and are used more than the others in future algebraic work. Pay particular attention to these.

Axioms for the Real Number System





Section RP.16: Properties of Equality



Section RP.17: Properties of Inequality

Section RP.18: Law of Trichotomy



 

About Axioms

Section RP.19: Terms and Simplifying Expressions

The following are useful concepts for simplifying algebraic expressions. We will revisit these concepts in much greater detail in Chapter 5.

Section RP.20: Modeling with Algebra

Certain basic knowledge is required to do any mathematical modeling. For example to model a problem in mechanical engineering, you must know enough about mechanical engineering to understand the problem and convert it to mathematical statements. The very nature of the world in which we live requires that you must know and be able to use the basic formulas presented in the section of this page as well as many other formulas you will learn durning your study of mathematics.

 

About Modeling with Algebra

Minimal List of Exercises Page 37.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.

Vocabulary and Concept Check: 1 - 14
Exercise Set: 1, 3, 5, 7, 9, 12, 14, 15, 19, 21, 25 - 34, 35, 36, 54, 56, 59, 74, 76, 77, 110.

Additional worked out exercises.

Additional worked out examples are available. Some of them are interactive. Click HERE.