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.