Which of the following is the solution set of `|2/3x-5|gt8`?

Options

A. `{x:-39/2ltxlt9/2}`
B. `{x:-9/2ltxle39/2}`
C. `{x:xgt39/2` or `xlt-9/2}`
D. `{x:xgt9/2` or `xlt (-39)/2}`

Answer

To solve the inequality \( | \frac{2}{3}x - 5 | > 8 \), we will follow these steps:

### Step 1: Understand the Absolute Value Inequality
The expression \( |A| > B \) implies two cases:
1. \( A > B \)
2. \( A < -B \)

In our case, \( A = \frac{2}{3}x - 5 \) and \( B = 8 \).

### Step 2: Set Up the Two Inequalities
From the absolute value inequality, we can write:
1. \( \frac{2}{3}x - 5 > 8 \)
2. \( \frac{2}{3}x - 5 < -8 \)

### Step 3: Solve the First Inequality
**Inequality 1:**
\[ \frac{2}{3}x - 5 > 8 \]
Add 5 to both sides:
\[ \frac{2}{3}x > 8 + 5 \]
\[ \frac{2}{3}x > 13 \]
Now, multiply both sides by \( \frac{3}{2} \) to isolate \( x \):
\[ x > 13 \times \frac{3}{2} \]
\[ x > \frac{39}{2} \]

### Step 4: Solve the Second Inequality
**Inequality 2:**
\[ \frac{2}{3}x - 5 < -8 \]
Add 5 to both sides:
\[ \frac{2}{3}x < -8 + 5 \]
\[ \frac{2}{3}x < -3 \]
Now, multiply both sides by \( \frac{3}{2} \):
\[ x < -3 \times \frac{3}{2} \]
\[ x < -\frac{9}{2} \]

### Step 5: Combine the Results
From the two inequalities, we have:
1. \( x < -\frac{9}{2} \)
2. \( x > \frac{39}{2} \)

Thus, the solution set is:
\[ x < -\frac{9}{2} \quad \text{or} \quad x > \frac{39}{2} \]

### Final Answer
The solution set of the inequality \( | \frac{2}{3}x - 5 | > 8 \) is:
\[ \{ x | x < -\frac{9}{2} \text{ or } x > \frac{39}{2} \} \]

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