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The problem is asking to solve the equation:

\[
|3a + 4| = |4a - 10|
\]

Since both sides of the equation are absolute values, we need to consider the cases where the expressions inside the absolute values can be positive or negative.

### Case 1: Both expressions are equal
\[
3a + 4 = 4a - 10
\]
Solving for \( a \):
\[
3a + 4 = 4a - 10
\]
Subtract \( 3a \) from both sides:
\[
4 = a - 10
\]
Add 10 to both sides:
\[
a = 14
\]

### Case 2: The expressions are negatives of each other
\[
3a + 4 = -(4a - 10)
\]
This simplifies to:
\[
3a + 4 = -4a + 10
\]
Now solve for \( a \):
\[
3a + 4a = 10 - 4
\]
\[
7a = 6
\]
\[
a = \frac{6}{7}
\]

### Verifying the solutions:
- For \( a = 14 \):
  \[
  |3(14) + 4| = |42 + 4| = 46
  \]
  \[
  |4(14) - 10| = |56 - 10| = 46
  \]
  This is true, so \( a = 14 \) is a solution.

- For \( a = \frac{6}{7} \):
  \[
  |3 \times \frac{6}{7} + 4| = | \frac{18}{7} + \frac{28}{7} | = \frac{46}{7}
  \]
  \[
  |4 \times \frac{6}{7} - 10| = | \frac{24}{7} - \frac{70}{7} | = \frac{46}{7}
  \]
  This is also true, so \( a = \frac{6}{7} \) is a solution.

### Conclusion:
The solution set is \( a = 14, \frac{6}{7} \), so the correct answer is option **A**.

Would you like more details or explanations on any step? Here are 5 additional questions to consider:

1. How do absolute value equations generally work?
2. What are the conditions for the absolute values to be equal?
3. Why is it necessary to check both cases (positive and negative)?
4. Could you encounter a case where there are no solutions?
5. How do we ensure the solution set is correct by verification?

**Tip:** Always check your solutions by substituting them back into the original equation, especially for absolute value problems!

Solve |3a + 4| = |4a - 10|. Select the correct solution set.

Learn how to solve the absolute value equation |3a + 4| = |4a - 10| with a detailed step-by-step solution. This algebra problem involves absolute values and linear equations, suitable for students in Grades 9-10.

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