My primary recommendation is that we always simultaneously consider an equation and its two inequality siblings as a group and that we graph all three on the same number line.
Example 1: During a consideration of the two linear algebraic expressions 3x + 5 and –2x + 7, because these expressions represent real numbers, The Law of Trichotomy reminds us that there exists three distinct possibilities;
- 3x + 5 < –2x + 7
- 3x + 5 = –2x + 7
- 3x + 5 > –2x + 7
Consequently, we desire to solve the equation and both inequalities. Recall some of the observations made during the many practice exercises for solving linear equations and inequalities.
- The graph of a linear equation in one variable is a point on the real number line.
- The graph of a linear inequality in one variable is a ray on the real number line.
What may not have been observed is that the ray which is the graph of a linear inequality in one variable begins at the graph of the equation and extends infinitely far toward the right or the left and that the graph of the other inequality begins at the same point and extends infinitely far in the other direction.
Another, possibly more understandable, way to state this is:
Important: The graph of a linear equation in one variable divides the real number line into two rays, one of which is the graph of one of the corresponding inequalities and the other is the graph of the other inequality.
Refer to Example 1. The solution set (never mind how we determined the solution set) for the equation in Example 1 is .
The graph of the equation 3x + 5 = –2x + 7 is shown in Fig. 1.
Clearly this graph of the equation divides the real number line into a point and two rays as shown in Fig. 2.
1. The blue ray is the graph of one of the inequalities
2. The red ray is the graph of the other inequality.
In this example the blue ray is the graph of 3x + 5 < –2x + 7 and the red ray is the graph of 3x + 5 > –2x + 7.
We can determine if the blue ray is the solution set to one of the inequalities by testing just one number from the ray in the inequality. Refer to the above example. To determine if the blue ray is the solution to 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 several conclusions;
i. The blue ray is not the solution set for 3x + 5 > –2x + 7.
ii. The red ray (the other one) is the solution set for 3x + 5 > –2x + 7.
iii. The blue ray is the solution set for 3x + 5 < –2x + 7.
The following observations can be generalized to many other situations. They will apply to equations and inequalities involving nothing but polynomials.
1. When considering a conditional equation or inequality, The Law of Trichotomy dictates that we also consider the other two corresponding equations and/or inequalities.
2. The graph of the equation is a boundary between the graphs of the corresponding inequalities. For that reason, the equation is sometimes called the boundary equation for the inequalities.
3. Testing any single number from one of the rays in either inequality determines whether that ray is the solution set for that inequality.
4. Thereafter the solution sets may be deduced for each inequality.