11. Given that $M$ is the midpoint of $\overline{A B}, A M=x+6$, and $A B=4 x+6$, 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 \(AB = 4x + 6 = 4*3 + 6 = 18\). 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 ：\(\boxed{x = 3}\)