Some Thoughts about College Algebra — Blog 3 — Law of Trichotomy – Linear Equations and Inequalties in One Variable

Some Thoughts about College Algebra

The textbook I am currently using for College Algebra is pretty much the same as every other College Algebra Textbook. The colors used on the cover vary from book to book but that is hardly a contributing factor to the book’s effectiveness. The book is much too large, contains a lot of useless information, trains the student to solve certain classical textbook problems rather than teaching mathematics, and has far too many exercises. These textbooks give students the false impression that you learn mathematics by repeating dozens of insipid textbook problems rather than teaching that one learns mathematics in order to solve problems. Students are not given even a hint that the “detached language” nature of mathematics is what makes it a powerful problem solving tool in every discipline. Students leave College Algebra believing it to be a collection of processes and procedures of no real value. In view of our current teaching practices they might be right.

The purpose of early college level algebra courses is to introduce the student to the use of abstraction, generalization, and deductive reasoning while exploring the patterns and relationships of a variety of algebraic entities including, but not limited to, equations, inequalities, algebraic fractions, polynomials, and functions. Along the way students should increase their ability to use critical thinking to solve the variety of problems they will surely encounter during a lifetime. No College Algebra book currently on the market (to my knowledge) addresses those goals.

My intention is to provide alternative methods through a series of blogs. As a mathematician I feel strongly that I should present a series of essays that are so arranged as to provide a complete and linearly ordered argument. I have struggled unsuccessfully with that for years and will not attempt it here, rather the order in which topics are presented follow no rhyme or reason.

Law of Trichotomy Linear Equations & Inequalities in One Variable

Law of Trichotomy: For any two expressions a and b which represent real numbers, exactly one of the following is true.

  1. a < b
  2. a = b
  3. a > b

The Law of Trichotomy is a formal statement of a property which most of us would consider to be quite obvious; when comparing two numbers; they are equal, the first is less than the second, or the first is greater than the second. The extension to expressions representing real numbers is obvious. The purpose of the formal statement here is to call attention to the obvious fact and to make it available for use with algebraic quantities which represent real numbers.

The following example illustrates what seems to me to be a better approach to teaching students about equations and inequalities. The example illustrates an important and useful consequence of The Law of Trichotomy. The example stresses deductive reasoning rather than rote training to solve equations and inequalities. The example also highlights the relation of an equation with its twin inequality siblings.

Law of Trichotomy Applied to Linear Equations and Inequalities in One Variable

During a consideration of the two linear algebraic expressions 3x + 5 and –2x + 7, the Law of Trichotomy reminds us that there exists three distinct possibilities;

  1. 3x + 5 < –2x + 7
  2. 3x + 5 = –2x + 7
  3. 3x + 5 > –2x + 7

This means is that if x is replaced (in these three statements) by any individual real number, exactly one of the expressions will be true.

Another more precise way of stating this is that the union of the three solution sets is R (the real numbers) and the intersection of any two of the solution sets is the empty set.

Another, possibly more understandable, way to state this is: The graph of a linear equation in one variable divides the real number line into a point and two rays. The point is the graph of the equation and one of the rays is the graph of one of the inequalities and the other ray is the graph of the other inequality.

The solution set for the above equation is . The graph of the equation 3x + 5 = –2x + 7 is shown in Fig. 1.

Blog3 Fig1

Clearly this graph of the equation divides the real number line into a point and two rays as shown in Fig. 2.

Blog3 Fig2

The blue ray is the graph of one of the inequalities and the red ray is the graph of the other inequality.

To determine if the blue ray is the solution set for the inequality 3x + 5 > –2x + 7 we need only test one number from the blue ray in 3x + 5 > –2x + 7. The number 0 is in the blue ray and is easy to test. Substituting 0 into 3x + 5 > –2x + 7 yields 5 > 7 which is false. This allows a number of conclusions;

  1. The blue ray is not the solution set for 3x + 5 > –2x + 7.
  2. The red ray (the other ray) is the solution set for 3x + 5 > –2x + 7
  3. The blue ray is the solution set for 3x + 5 < –2x + 7. Blog3 Fig3An examination of the graph in Fig. 3 quite clearly illustrates the following:
  1. The graph of a linear equation in one variable is a point on the real number line.
  2. The graph of a linear inequality in one variable is a ray on the real number line.
  3. The graph of the equation is the boundary between the graphs of the corresponding inequalities. For that reason, the equation is usually called the boundary equation for the inequalities.
  4. As predicted by The Law of Trichotomy, the union of the three solution sets is the set of all real numbers R and that the intersection of any two of the individual solution sets is the empty set ∅.
  5. The relationship between and equation and its two twin inequality siblings.
  6. An elegantly simple illustration of The Law of Trichotomy.

The deductive reasoning used in this example will be of more value to most college algebra students than the forgotten rote method of solving a linear equation and the completely divorced rote method of solving linear inequalities. The lessons learned about deductive reasoning can be applied to many situations other than mathematics.

In this example the student has also been exposed to the technique of examining a structure rather than concentrating solely on individual parts to exclusion of the structure. Perhaps the student will even learn that  a study of structure frequently leads to a better understanding of individual components.

Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s


%d bloggers like this: