https://file.aiquickdraw.com/mathbot/file/1724879069341vqxq6mt2.jpg

The image shows a line segment with points labeled \(A\), \(B\), and \(C\), with distances represented by expressions involving \(x\). Specifically:

- The distance between points \(A\) and \(B\) is given by \(3x + 2\).
- The distance between points \(B\) and \(C\) is given by \(4x + 5\).
- The total distance from \(A\) to \(C\) is given as \(17\).

To find \(x\), you can set up the equation by adding the distances between \(A\) and \(B\) and between \(B\) and \(C\) to equal the total distance from \(A\) to \(C\):

\[
3x + 2 + 4x + 5 = 17
\]

Now, combine like terms:

\[
7x + 7 = 17
\]

Next, subtract 7 from both sides:

\[
7x = 10
\]

Finally, divide by 7:

\[
x = \frac{10}{7}
\]

So, \(x = \frac{10}{7}\) or approximately \(1.43\).

Would you like any further details or have any questions? 

Here are some follow-up questions you might find helpful:
1. What is the total length of segment \(AB\) when \(x = \frac{10}{7}\)?
2. What is the total length of segment \(BC\) when \(x = \frac{10}{7}\)?
3. How do we check that the sum of segments \(AB\) and \(BC\) equals 17?
4. What would happen if the total distance \(AC\) were different?
5. Can you solve a similar problem where \(AC\) is given, and the expressions for \(AB\) and \(BC\) are different?

**Tip:** When solving linear equations, always combine like terms before isolating the variable to simplify the process.

Learn how to solve a linear equation involving distances between points A, B, and C, with detailed steps and solutions. This problem covers key concepts in linear equations and the distance formula. Suitable for students in Grades 9-12.

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