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Let's analyze the problem step by step.

### Given Information:
- Point \( A \) has a coordinate of \(-7\).
- Point \( B \) has a coordinate of \(28\).
- \( AM = 2x \) and \( MB = 5x \).
- \( M \) lies between \( A \) and \( B \).

We can use the total distance between \( A \) and \( B \), which is \( AB \), to set up an equation for \( x \).

#### Step-by-Step Solution:

1. **Find \( x \) (Part a):**
   The total distance from \( A \) to \( B \) is the sum of \( AM \) and \( MB \):
   \[
   AB = AM + MB
   \]
   Substituting the expressions for \( AM \) and \( MB \):
   \[
   AB = 2x + 5x = 7x
   \]
   The distance between \( A \) and \( B \) is:
   \[
   AB = 28 - (-7) = 35
   \]
   So, we have:
   \[
   7x = 35
   \]
   Solving for \( x \):
   \[
   x = \frac{35}{7} = 5
   \]

2. **Find \( AM \) (Part b):**
   \( AM = 2x \), and now that we know \( x = 5 \):
   \[
   AM = 2(5) = 10
   \]

3. **Find \( MB \) (Part c):**
   \( MB = 5x \), and since \( x = 5 \):
   \[
   MB = 5(5) = 25
   \]

4. **Find \( AB \) (Part d):**
   We have already calculated that the total distance from \( A \) to \( B \) is 35:
   \[
   AB = 35
   \]

5. **Find the coordinate of \( M \) (Part e):**
   The coordinate of \( M \) can be found by adding the distance \( AM \) to the coordinate of \( A \):
   \[
   M = A + AM = -7 + 10 = 3
   \]

### Final Answers:
- a. \( x = 5 \)
- b. \( AM = 10 \)
- c. \( MB = 25 \)
- d. \( AB = 35 \)
- e. The coordinate of \( M \) is \( 3 \).

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Do you want more details on any step? Here are some related questions:

1. How can you verify the position of \( M \) on the number line?
2. What would happen if the segments \( AM \) and \( MB \) were reversed?
3. How would the solution change if \( AM \) and \( MB \) had different ratios?
4. What if the total length of \( AB \) was different, how would you find \( x \)?
5. Can you calculate the midpoint of \( AB \) directly from the coordinates?

**Tip:** To solve problems with points on a number line, always use the relationship between total distance and segment distances!

The figure shows M between A and B with coordinates -7 and 28 respectively. Given AM = 2x and MB = 5x, find (a) x, (b) AM, (c) MB, (d) AB, and (e) the coordinate of M.

Learn how to find the value of x and the lengths of segments AM and MB using the distance formula. Also, calculate the coordinate of point M. Suitable for students in Grades 7-9.

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