11. Given that $M$ is the midpoint of $\overline{AB},AM=x+6$, and $AB=4x+6$, find $x$.

Step 1 ：Understand the problem: The problem states that M is the midpoint of line segment AB. This means that AM = MB. We are given that AM = x + 6 and AB = 4x + 6.

Step 2 ：Set up the equation: Since M is the midpoint, AM = MB. But we also know that AB = AM + MB. Therefore, we can set up the equation as follows: $AB=2AM$

Step 3 ：Substitute the given values into the equation: Substitute the given values into the equation: $4x+6=2(x+6)$

Step 4 ：Solve for x: Simplify the equation to solve for x: $4x-2x=12-6$ which simplifies to $2x=6$ and further simplifies to $x=6/2$ which gives $x=3$

Step 5 ：Check the solution: Substitute x = 3 into the original equations to check: $AM=x+6=3+6=9$ and $AB=4x+6=4\ast 3+6=18$. Since M is the midpoint, AM should be half of AB. 9 is indeed half of 18, so $x=3$ is the correct solution.

Step 6 ：$\boxed{{\displaystyle x=3}}$