Law of Trichotomy Applied to Equations and Inequalities in Two Variables

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A linear equation in two variables is an equation which may be written in the form
Ax + By= C where A, B, and C are real numbers and B is not zero.
A linear inequality in two variables x and y is an inequality which can be written as
Ax + By < C, or  Ax + By > C.
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 Law of Trichotomy affects how we consider equations and inequalities in two variables. When considering any one of the three possibilities

1. Ax + By= C
2. Ax + By< C
3. Ax + By > C

we consider all three.
The graph of a linear equation in two variables is a line in the coordinate plane which divides the coordinate plane into two half-planes. One of those half-planes is the graph of Ax + By< C, and the other half-plane is the graph of Ax + By> C.
When asked to analyze any of Ax + By= C, Ax + By< C, or Ax + By> C, we begin by graphing the equation by plotting two points (usually the x and y intercepts) and then testing one point (not on the boundary line) in one of the inequalities. Those few simple steps provide us with a complete analysis of the equation and both inequalities. The equation is called the boundary equation because its graph forms a boundary between the graphs of the two inequalities.

It is very helpful for understanding the big picture to examine the previous examples involving equations and inequalities in one variable.  Notice that in every case, the graph of the equation is the boundary between the graphs of the two sibling inequalities.  This is a very general principle.  Future examinations will permit us to generalize and extend this principle.