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.