Let's solve the inequality:

$-4 \left| -1 - \frac{1}{2}(x - 8) \right| + 9 \geq 5$

Step 1: Simplify the inequality

Subtract 9 from both sides:

$-4 \left| -1 - \frac{1}{2}(x - 8) \right| \geq -4$

Step 2: Divide both sides by -4

Since we're dividing by a negative number, the inequality sign flips:

$\left| -1 - \frac{1}{2}(x - 8) \right| \leq 1$

Step 3: Solve the absolute value inequality

For $\left| A \right| \leq B$, it means:

$-1 \leq -1 - \frac{1}{2}(x - 8) \leq 1$

Step 4: Break into two inequalities and solve each

Inequality 1:

$-1 \leq -1 - \frac{1}{2}(x - 8)$

Add 1 to both sides:

$0 \leq -\frac{1}{2}(x - 8)$

Multiply both sides by -2 (flip the inequality):

$0 \geq x - 8$

Simplify:

$x \leq 8$

Inequality 2:

$-1 - \frac{1}{2}(x - 8) \leq 1$

Add 1 to both sides:

$0 \leq -\frac{1}{2}(x - 8)$

Multiply both sides by -2 (flip the inequality):

$2 \geq x - 8$

Simplify:

$x \geq 6$

Step 5: Combine the solutions

The solution is:

$6 \leq x \leq 8$

Final Answer

$x \in [6, 8]$

Would you like more details, or do you have any questions?

Here are some related questions:

1. How do you solve inequalities involving absolute values?
2. Why do you need to flip the inequality sign when multiplying by a negative number?
3. How can you graph the solution set of an inequality?
4. What are the differences between "and" and "or" in compound inequalities?
5. How do absolute values affect equations and inequalities differently?

Tip: When solving absolute value inequalities, always remember that $|A| \leq B$ means $-B \leq A \leq B$.