The inequality presented is:

$4 |4 - x| - 4 \geq 32$

Let’s solve this step-by-step:

Step 1: Simplify the Inequality

Start by adding 4 to both sides to eliminate the -4:

$4 |4 - x| \geq 36$

Step 2: Divide by 4

Divide both sides by 4:

$|4 - x| \geq 9$

Step 3: Remove the Absolute Value

The inequality $|4 - x| \geq 9$ translates into two separate inequalities:

$4 - x \geq 9 \quad \text{or} \quad 4 - x \leq -9$

Step 4: Solve Each Inequality

1. For $4 - x \geq 9$:

   $-x \geq 5 \implies x \leq -5$

2. For $4 - x \leq -9$:

   $-x \leq -13 \implies x \geq 13$

Step 5: Combine the Solutions

The solution is:

$x \leq -5 \quad \text{or} \quad x \geq 13$

So, the inequality holds for values of $x$ that are less than or equal to -5, or greater than or equal to 13.

Final Answer

$(-\infty, -5] \cup [13, \infty)$

Would you like more details on any of the steps, or have any questions about this solution?

Here are 5 related questions to further your understanding:

1. How do you solve absolute value equations that result in compound inequalities?
2. What are the differences between solving inequalities with absolute values versus those without?
3. How do you graph the solution set for absolute value inequalities?
4. What does it mean for a solution to include both "and" and "or" conditions in absolute value problems?
5. How do absolute values affect the domain of a function?

Tip: When solving absolute value inequalities, always remember that they represent two scenarios—one where the expression inside the absolute value is positive, and another where it's negative.