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The problem asks to find the length of \( AC \) given that \( B \) is the midpoint of \( AC \). The segment is divided into two parts: \( AB = 3x \) and \( BC = 4x - 6 \).

### Step-by-Step Solution:

1. **Understand the Problem:**
   Since \( B \) is the midpoint of \( AC \), the length of \( AB \) must be equal to the length of \( BC \).

   \[
   AB = BC
   \]

2. **Set Up the Equation:**
   Given \( AB = 3x \) and \( BC = 4x - 6 \), we set them equal to each other:

   \[
   3x = 4x - 6
   \]

3. **Solve for \( x \):**
   To find \( x \), subtract \( 3x \) from both sides:

   \[
   0 = x - 6
   \]

   Then, add 6 to both sides:

   \[
   x = 6
   \]

4. **Find \( AC \):**
   Since \( B \) is the midpoint, \( AC = AB + BC \).

   Substitute \( x = 6 \) back into the expressions for \( AB \) and \( BC \):

   \[
   AB = 3x = 3(6) = 18
   \]
   \[
   BC = 4x - 6 = 4(6) - 6 = 24 - 6 = 18
   \]

   Therefore, the length of \( AC \) is:

   \[
   AC = AB + BC = 18 + 18 = 36
   \]

### Final Answer:
The length of \( AC \) is 36 units.

Would you like further details or have any questions about this problem? Here are some related questions to consider:

1. What is the significance of a midpoint in geometry?
2. How would the problem change if \( B \) were not the midpoint?
3. Can this method be applied to any linear equation involving a midpoint?
4. What are other ways to verify that the solution is correct?
5. How would the answer change if \( AB = 2x \) and \( BC = 3x + 4 \)?

**Tip:** When solving equations, always double-check your arithmetic to ensure accuracy.

Find AC if B is the midpoint. AB = 3x and BC = 4x - 6.

Learn how to find the length of AC when B is the midpoint between A and C. This problem involves algebraic concepts and the midpoint theorem, suitable for students in Grades 8-10.

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