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

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Possibility 2:   k < 0.

Case 1: (The boundary equation |M| = k) If M is an algebraic expression and k is negative, then the solution set for |M| = k is the null set. The absolute value cannot be negative!

Example: The solution |–2x – 6| = –3 is the null set because the absolute value is never negative.

Case 2: (The “less than” inequality |M| < k) If M is an algebraic expression and k is negative, then the solution set for |M| < k is the null set. The absolute value cannot be negative!

Example: The solution |–2x + 1| < –5 is the null set because the absolute value is never negative.

Case 3: (The “greater than” inequality |M| > k) If M is an algebraic expression, then the solution set for |M| > k is the set of all real numbers.

Example: The solution |5x – 20| > –3 is the all real numbers because the absolute value of 5x – 20 is non-negative for all values of x.

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