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We are given the following information from the problem:

- \( AB = 2x + 10 \)
- \( BC = x \)
- \( AC = 5x \)

Since \( B \) is a point on the line segment \( AC \), the sum of the segments \( AB \) and \( BC \) must be equal to \( AC \). Therefore, we can set up the equation:

\[
AB + BC = AC
\]

Substitute the given values into the equation:

\[
(2x + 10) + x = 5x
\]

Now, solve for \( x \):

\[
2x + x + 10 = 5x
\]
\[
3x + 10 = 5x
\]
\[
10 = 5x - 3x
\]
\[
10 = 2x
\]
\[
x = 5
\]

Now that we know \( x = 5 \), we can determine the length of \( AC \) by substituting \( x \) back into the expression for \( AC \):

\[
AC = 5x = 5(5) = 25
\]

Therefore, the numerical length of \( AC \) is \( 25 \).

Let me know if you need more details or have any questions!

Here are some related questions you might explore:
1. What is the value of segment \( AB \)?
2. How do you calculate the length of segment \( BC \)?
3. If \( AB + BC = AC \), what happens if \( B \) is not between \( A \) and \( C \)?
4. Can you solve for \( x \) if a different equation for \( AC \) was given, like \( AC = 4x \)?
5. How do you verify the solution by plugging back into the original equation?

**Tip:** Always recheck your solution by substituting values back into the original equation to avoid mistakes!

Point B is on line segment AC. Given AB = 2x + 10, BC = x, and AC = 5x, determine the numerical length of AC.

Learn how to solve the algebraic equation where AB = 2x + 10, BC = x, and AC = 5x. This problem involves the Segment Addition Theorem and algebraic manipulation to determine the numerical length of AC. Suitable for students in Grades 8-10.

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