The Law of Trichotomy – Solution Sets

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Law of Trichotomy: If a and b are real numbers, then one and only one of the following is true:

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

The most common arrangement of topics in early algebra courses separates the discussion of solution sets of inequalities from the discussion of solution sets of equations. For some types of equations, corresponding inequalities aren’t considered. There is also an inexcusable inconsistency. For linear inequalities in two variables we train our students to create by rote something called the boundary equation and then we test a point to determine the solution set of the inequality. We don’t do anything like it with other inequalities. We rarely consider nonlinear inequalities in two variables. We never mention The Law of Trichotomy. Let’s correct that in our classrooms. A single general coherent view of many topics is much better/easier than seemingly different schemes for each topic.
In order to construct a logical discussion, we need to collect a few facts.

1. There are three common binary relations: less than (<), equal (=), greater than (>) in early algebra courses.
2. These three binary relations correspond exactly to the three cases in The Law of Trichotomy.
3. The Law of Trichotomy and the three binary relations are valid for Real Numbers. They do not extend to the Complex Numbers.
4. The Law of Trichotomy and the three binary relations are valid for any expressions which represent Real Numbers.

Algebraic expressions represent Real Numbers so that comparisons between two algebraic expressions are done with the three binary relations. And The Law of Trichotomy applies. Such comparisons yield an equation and two corresponding inequalities. I fondly refer to these as three siblings.
In early algebra classes we spend far too much time training students how to solve equations and inequalities. Much of what we do in those classes is unnecessary. I want to propose an alternative approach.
To facilitate the following discussion, it is appropriate to point out that in mathematics:

1. We solve only equations and inequalities.
2. To solve an equation or inequality means to determine all solutions of the equation or inequality.
3. A solution of an equation or inequality is a number or numbers which make the statement true when substituted for the variables.
4. The collection of all solutions of an equation or inequality is called its solution set.
5. To solve an equation or inequality means to determine its solution set.
6. The graph of an equation or inequality is a picture of its solution set.

Examples:
For a complete comparison of 3x + 5 and 7 we must consider:
3x + 5 < 7      3x + 5 = 7     3x + 5 > 7
For a complete comparison of x3 + 5x2 – 7x and 9 we must consider:
x3 + 5x2 – 7x < 9      x3 + 5x2 – 7x = 9      x3 + 5x2 – 7x > 9
For a complete comparison of y and 3x + 5 we must consider:
y < 3x + 5      y = 3x + 5      y > 3x + 5
For a complete comparison of x2 + y2 and 16 we must consider:
x2 + y2 < 16      x2 + y2 = 16      x2 + y2 > 16
When we look at these questions from a different point of view we begin to realize:
When we consider an equation like 3x + 5 = 7, the two corresponding inequalities 3x + 5 < 7 and 3x + 5 > 7 are lurking in the background just begging for attention. Moreover, all their solutions sets are nicely related via The Law of Trichotomy.
When we consider an inequality like x3 + 5x2 – 7x < 9, the two siblings x3 + 5x2 – 7x = 9 and  x3 + 5x2 – 7x > 9 are lurking in the background just begging for attention. Moreover, all their solution sets are nicely related via The Law of Trichotomy.
When we consider any equation or inequality the other corresponding equation and/or inequalities are also in consideration. And all their solution sets are nicely related via The Law of Trichotomy
When we consider an equation like y > 3x + 5, the two corresponding siblings y < 3x + 5 and y = 3x + 5 are lurking in the background just begging for attention. Moreover, their solutions sets are closely related via The Law of Trichotomy.
Here is an important fact:

The graph of an equation is the boundary between the graphs of the corresponding sibling inequalities.

As you contemplate these statements and possibly look at a few detailed examples you will come to the realization that:

For all practical purposes, when any one of the three siblings has been solved, it is easy to deduce the solution sets of the other two.

With that in mind a prudent approach to solving any one of these three siblings is to solve the one which is computationally easiest to solve and then deduce the desired solutions of the other two.
In the next blog(s) I plan to illustrate, demonstrate, and illuminate the above statements.

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