We are given the equation:

$3 + 2|x - 6| = 11$

Step 1: Isolate the absolute value expression.

Subtract 3 from both sides:

$2|x - 6| = 8$

Step 2: Divide both sides by 2.

$|x - 6| = 4$

Step 3: Solve for $x$.

Since $x - 6$ is inside the absolute value, we must consider two cases:

Case 1: $x - 6 = 4$

$x = 4 + 6 = 10$

Case 2: $x - 6 = -4$

$x = -4 + 6 = 2$

Final Solution:

The solutions are $x = 10$ and $x = 2$.

Would you like more details or have any questions?

Here are 5 related questions for you:

1. How do you solve equations with absolute values?
2. What happens when the right-hand side of the equation is negative?
3. Can absolute value expressions have two solutions in all cases?
4. How do we graph absolute value equations?
5. What are the properties of absolute value functions?

Tip: Always isolate the absolute value before splitting into two cases.