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To solve the absolute value equation given:

\[
|6x - 8| - 5 = 9
\]

we first isolate the absolute value expression:

1. Add 5 to both sides:
   \[
   |6x - 8| = 14
   \]

2. Now that we have \(|6x - 8| = 14\), we consider the two cases for absolute values:

   **Case 1:** \(6x - 8 = 14\)
   - Solve for \(x\):
     \[
     6x = 14 + 8
     \]
     \[
     6x = 22
     \]
     \[
     x = \frac{22}{6} = \frac{11}{3}
     \]

   **Case 2:** \(6x - 8 = -14\)
   - Solve for \(x\):
     \[
     6x = -14 + 8
     \]
     \[
     6x = -6
     \]
     \[
     x = \frac{-6}{6} = -1
     \]

3. So the solutions are:
   \[
   x = \frac{11}{3} \quad \text{and} \quad x = -1
   \]

### Answer:
The correct choice is **A** with solutions:
\[
x = \frac{11}{3}, -1
\]

Would you like more details on any step, or have any questions?

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Here are 5 related questions to explore further:

1. How do you solve absolute value equations in general?
2. What happens when there is no solution to an absolute value equation?
3. How would the process change if the equation had a variable outside the absolute value?
4. Can you graph absolute value functions to visualize solutions?
5. How do absolute values affect inequality solutions?

**Tip:** Always isolate the absolute value expression before solving absolute value equations.

Solve the absolute value equation |6x - 8| - 5 = 9.

Learn how to solve the absolute value equation |6x - 8| - 5 = 9 with a detailed step-by-step solution. This problem covers key algebraic concepts, including isolating the absolute value and solving cases. Suitable for students in Grades 8-10.

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