We are given the inequality involving absolute value:

$|x - 2| + 4 \geq 11$

Step 1: Isolate the absolute value expression.

Subtract 4 from both sides:

$|x - 2| \geq 11 - 4$

$|x - 2| \geq 7$

Step 2: Solve the inequality for the absolute value.

The expression $|x - 2| \geq 7$ means that the distance between $x$ and 2 is at least 7. This gives two cases to consider:

Case 1:

$x - 2 \geq 7$

Solve for $x$:

$x \geq 7 + 2$ $x \geq 9$

Case 2:

$x - 2 \leq -7$

Solve for $x$:

$x \leq -7 + 2$ $x \leq -5$

Step 3: Combine the solutions.

The solution is:

$x \geq 9 \quad \text{or} \quad x \leq -5$

In interval notation, the solution is:

$(-\infty, -5] \cup [9, \infty)$

Would you like further details or have any questions?

Here are 5 related questions to expand on this:

1. How do we solve inequalities involving more complex absolute value functions?
2. What happens if the inequality sign is strict (i.e., $>$ instead of $\geq$)?
3. How do we graph the solution to an absolute value inequality?
4. How do absolute value inequalities compare to quadratic inequalities?
5. What is the geometric interpretation of absolute value inequalities?

Tip: Always check both cases when solving absolute value inequalities, as they represent the two possible directions from the central point.