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The problem states:

- \( AC = 3x + 19 \)
- \( AB = 8 \)
- \( BC = 4x - 1 \)

Given that points A, B, and C are collinear and B is between A and C, the equation to solve is based on the fact that:

\[
AC = AB + BC
\]

Now, substituting the given values:

\[
3x + 19 = 8 + (4x - 1)
\]

Let's solve this equation for \(x\):

\[
3x + 19 = 8 + 4x - 1
\]
\[
3x + 19 = 4x + 7
\]
\[
19 - 7 = 4x - 3x
\]
\[
12 = x
\]

Thus, \( x = 12 \).

Would you like to see a detailed explanation or have any further questions? Here are five additional related questions:

1. What are the lengths of \( AC \), \( AB \), and \( BC \) when \( x = 12 \)?
2. How would the problem change if point B were not between A and C?
3. Can you express the total length of \( AC \) in terms of a different variable?
4. How would you verify that the points are indeed collinear based on their coordinates?
5. If the distances were in a coordinate plane, how would you graph this problem?

**Tip:** When solving problems involving collinear points, always ensure that the segment lengths follow the sum property (i.e., one segment equals the sum of the two others if a point lies between).

AC = 3x + 19, AB = 8, and BC = 4x - 1. If points A, B, and C are collinear and B is between A and C, write an equation and solve for x.

In this problem, learn how to solve for x when given collinear points A, B, and C. Using the equation AC = AB + BC and the given expressions, solve the linear equation step by step. Suitable for middle school students in grades 6-8.

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