Question

question 14
solve and graph the following inequality:
|3x - 9| > 6

Explanation:

Step1: Recall absolute value inequality rule

For \(|A| > B\) (where \(B>0\)), it is equivalent to \(A > B\) or \(A < -B\). Here, \(A = 3x - 9\) and \(B = 6\). So we have two inequalities: \(3x - 9 > 6\) or \(3x - 9 < -6\).

Step2: Solve \(3x - 9 > 6\)

Add 9 to both sides: \(3x - 9 + 9 > 6 + 9\) → \(3x > 15\). Divide both sides by 3: \(\frac{3x}{3} > \frac{15}{3}\) → \(x > 5\).

Step3: Solve \(3x - 9 < -6\)

Add 9 to both sides: \(3x - 9 + 9 < -6 + 9\) → \(3x < 3\). Divide both sides by 3: \(\frac{3x}{3} < \frac{3}{3}\) → \(x < 1\).

Step4: Graph the solution

On the number line, we have an open circle at \(x = 1\) (since \(x < 1\), not including 1) and an open circle at \(x = 5\) (since \(x > 5\), not including 5). Then we shade the regions to the left of 1 and to the right of 5.

Answer:

The solution to the inequality \(|3x - 9| > 6\) is \(x < 1\) or \(x > 5\). The graph has open circles at \(x = 1\) and \(x = 5\), with shading to the left of \(x = 1\) and to the right of \(x = 5\) on the number line.