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The problem provides the following conditions:

- \( B \) is between \( A \) and \( C \).
- \( AB = 3x + 6 \).
- \( BC = 15x - 2 \).
- \( \overline{AB} \cong \overline{BC} \), meaning that the segments \( AB \) and \( BC \) are congruent (they have the same length).

### Step-by-step solution:

1. Since \( \overline{AB} \cong \overline{BC} \), we know that:
   \[
   AB = BC
   \]
   This leads to the equation:
   \[
   3x + 6 = 15x - 2
   \]

2. Solve for \( x \):

   Subtract \( 3x \) from both sides:
   \[
   6 = 12x - 2
   \]

   Add 2 to both sides:
   \[
   8 = 12x
   \]

   Divide both sides by 12:
   \[
   x = \frac{8}{12} = \frac{2}{3}
   \]

Thus, the value of \( x \) is \( \frac{2}{3} \).

Would you like any further details or explanations?

Here are some additional questions to deepen your understanding:

1. How would you verify the solution by substituting \( x = \frac{2}{3} \) back into the expressions for \( AB \) and \( BC \)?
2. What does it mean for two line segments to be congruent in geometry?
3. How do we know that \( AB \) and \( BC \) are congruent based on the problem?
4. Could there be another way to represent this relationship using the distances between points \( A \), \( B \), and \( C \)?
5. How would the result change if the equation given was \( \overline{AB} = 2 \cdot \overline{BC} \)?

**Tip:** When solving equations involving congruence or equality of segments, it's helpful to clearly set up an equation based on the given relationships and solve step by step.

Use the following information to find x. Write the value of the variable. B is between A and C; AB = 3x + 6; BC = 15x - 2; AB ≅ BC.

Learn how to solve for x when given that AB ≅ BC, where AB = 3x + 6 and BC = 15x - 2. This problem covers concepts of algebra and congruent line segments. Suitable for students in Grades 7-9.

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