Point $M$ is the midpoint of $\overline{AB}$   where $AM=3x+8$ and $MB=6x-4$ .

Given that point $ M $ is the midpoint of $ \overline{AB} $, we know that the distances from $ A $ to $ M $ and from $ M $ to $ B $ are equal. Thus, we have the equation:

\[
AM = MB
\]

Substituting the expressions for $ AM $ and $ MB $:

\[
3x + 8 = 6x - 4
\]

Now, solve for $ x $:

1. Subtract $ 3x $ from both sides:
   \[
   8 = 3x - 4
   \]

2. Add 4 to both sides:
   \[
   12 = 3x
   \]

3. Divide both sides by 3:
   \[
   x = 4
   \]

Now that we have $ x = 4 $, we can substitute this back into the expressions for $ AM $ and $ MB $:

\[
AM = 3(4) + 8 = 12 + 8 = 20
\]
\[
MB = 6(4) - 4 = 24 - 4 = 20
\]

Thus, $ AM = MB = 20 $, confirming that $ M $ is indeed the midpoint of $ \overline{AB} $.

### Do you want more details or have any questions?

### Related Questions:
1. How do you find the midpoint of a line segment if you are given coordinates?
2. What is the general formula for the midpoint of two points in a coordinate plane?
3. How would the problem change if $ AM $ was not equal to $ MB $?
4. Can you determine the length of $ AB $ given just the value of $ AM $?
5. How do you solve for $ x $ if both $ AM $ and $ MB $ involve quadratic expressions?

### Tip:
Always check if a point is the midpoint by ensuring the two segments it divides are equal in length.

Point M is the midpoint of $\overline{AB}$ where $AM = 3x + 8$ and $MB = 6x - 4$.

Learn how to solve a problem where point M is the midpoint of segment AB, with given expressions for AM and MB. This step-by-step solution involves algebraic manipulation, solving linear equations, and applying the midpoint theorem. Suitable for students in Grades 8-10.

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