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The problem you uploaded asks to find \( x \) given the following equations:

- \( AB = 2x - 14 \)
- \( AC = 9 \)
- \( BC = x - 7 \)

The relationship between these segments suggests that point \( C \) is between points \( A \) and \( B \), so:

\[
AB = AC + BC
\]

Substitute the given expressions for \( AB \), \( AC \), and \( BC \):

\[
2x - 14 = 9 + (x - 7)
\]

Now, simplify and solve for \( x \):

\[
2x - 14 = 9 + x - 7
\]
\[
2x - 14 = x + 2
\]
\[
2x - x = 2 + 14
\]
\[
x = 16
\]

Now that we have \( x = 16 \), let’s verify by plugging it back into the original segment equations to check if everything adds up:

- \( AB = 2(16) - 14 = 32 - 14 = 18 \)
- \( BC = 16 - 7 = 9 \)
- \( AC = 9 \)

So, \( AB = AC + BC = 9 + 9 = 18 \), which is correct.

However, the choices provided were A) 2, B) 9, C) 10, D) 11, none of which is correct. The solution to the equation yields \( x = 16 \).

Would you like further clarification or details on this solution?

Here are 5 related questions you might explore:

1. What if the relationship between segments was different, e.g., \( AC = AB + BC \)?
2. How would you solve if both \( AB \) and \( BC \) were quadratic expressions?
3. Can you solve for other unknowns in triangle problems using segment relationships?
4. How does changing the position of point \( C \) on the segment affect the result for \( x \)?
5. What geometric theorems can help in problems involving collinear points?

**Tip:** Always verify your solutions by substituting the result back into the original equations to check for consistency.

Find x if AB = 2x - 14, AC = 9, and BC = x - 7.

Learn how to solve for x using segment relationships in the equation AB = 2x - 14, AC = 9, BC = x - 7. This problem covers key algebraic concepts and the Segment Addition Theorem. Suitable for students in Grades 7-9.

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