Law of Trichotomy Applied to Absolute Value Inequalities in One Variable Part VIII

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Example: Solve the inequality |3x – 5| > 6.

Discussion and Solution:  We begin by focusing not on the given inequality, but on its counterpart |3x – 5| < 6 which we immediately convert to the equivalent compact compound inequality.

 

 

The solution set for |3x – 5| < 6 is the interval .

The graph is shown in Fig. 16.

 

 

 

No further computation is required to deduce the solution sets for |3x – 5| = 6 and |3x – 5| > 6

The solution set for |3x – 5| = 6 is   whose graph is Fig. 17.

 

 

 

 

The solution set for the desired inequality |3x – 5| > 6 is

whose graph is Fig. 18. 

 

 

 

 

As before we can graph all three solution sets on the same number line to obtain Fig. 19 
Where it is easy to observe that the solution sets do not overlap and their union is the entire real number line.

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