DrDelMath

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1. Discuss the inequality |3x - 6| > 9.
Because |3x - 6| < 9 is equivalent to the compact compound inequality -9 < 3x - 6 < 9 we recognize it as the easy one and will use computational methods to solve it.
-9 < 3x - 6 < 9
-3 < 3x < 15
-1 < x < 5
The solution set for |3x - 6| < 9 is the interval (-1, 5) = {x|-1 < x < 5}.
The graph of |3x - 6| < 9 is



The solution set for the equality |3x - 6| = 9 is the set {-1, 5}. The graph of the equality is



The solution set for the inequality |3x - 6| > 9 is (-∞, -1) ∪ (5, ∞). The graph of the greater than inequality |3x - 6| > 9 is



Finally all three are graphed on the same number line.

2. NOT READY Discuss the equation |5x + 7| = 3.
Because |5x + 7| < 3 is equivalent to the compact compound inequality -3 < 5x + 7 < 3 we recognize it as the easy one and will use computational methods to solve it.
-3 < 5x + 7 < 3
-10 < 5x < -4
-2 < x < (-4)/5
The solution set for |5x + 7| < 3 is the interval (-1, 5) = {x|-1 < x < 5}.
The graph of |5x + 7| < 3 is



The solution set for the equality |5x + 7| < 3 is the set {-1, 5}. The graph of the equality is



The solution set for the inequality |5x + 7| > 3 is (-∞, -1) ∪ (5, ∞). The graph of the greater than inequality |5x + 7| > 3 is



Finally all three are graphed on the same number line.