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 $3$.

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!