`|-x-4|ge8`

Answer

To solve the inequality \(|-x-4| \ge 8\), we will break it down into two cases based on the definition of absolute value.

### Step 1: Set up the two cases for the absolute value inequality

The expression \(|A| \ge B\) can be rewritten as:
1. \(A \ge B\)
2. \(A \le -B\)

In our case, \(A = -x - 4\) and \(B = 8\). Therefore, we have:
1. \(-x - 4 \ge 8\)
2. \(-x - 4 \le -8\)

### Step 2: Solve the first case \(-x - 4 \ge 8\)

Starting with the first inequality:
\[ -x - 4 \ge 8 \]

Add 4 to both sides:
\[ -x \ge 12 \]

Now, multiply both sides by -1 (remember to reverse the inequality sign):
\[ x \le -12 \]

### Step 3: Solve the second case \(-x - 4 \le -8\)

Now, let's solve the second inequality:
\[ -x - 4 \le -8 \]

Add 4 to both sides:
\[ -x \le -4 \]

Again, multiply both sides by -1 (reversing the inequality):
\[ x \ge 4 \]

### Step 4: Combine the results

From the two cases, we have:
1. \(x \le -12\)
2. \(x \ge 4\)

Thus, the solution to the inequality \(|-x - 4| \ge 8\) is:
\[ x \le -12 \quad \text{or} \quad x \ge 4 \]

### Final Answer
The final solution is:
\[ x \in (-\infty, -12] \cup [4, \infty) \]

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