Let's solve the inequality step by step:

We are given:

$| -3u | - 9 \geq -6$

Step 1: Isolate the absolute value expression.

Add 9 to both sides to get:

$| -3u | \geq -6 + 9$

$| -3u | \geq 3$

Step 2: Solve the absolute value inequality.

For any absolute value inequality $|x| \geq a$, where $a \geq 0$, the solution is:

$x \leq -a \quad \text{or} \quad x \geq a$

Applying this to $| -3u | \geq 3$, we have:

$-3u \leq -3 \quad \text{or} \quad -3u \geq 3$

Step 3: Solve each inequality.

• For $-3u \leq -3$:

  Divide both sides by -3 (remember to flip the inequality sign when dividing by a negative number):

  $u \geq 1$

• For $-3u \geq 3$:

  Divide both sides by -3 (again, flip the inequality sign):

  $u \leq -1$

Step 4: Combine the solutions.

The solution is:

$u \geq 1 \quad \text{or} \quad u \leq -1$

Thus, the final solution in interval notation is:

$(-\infty, -1] \cup [1, \infty)$

Let me know if you'd like more details or have any questions.

Here are 5 related questions:

1. How do you solve absolute value equations with two sides?
2. Can you solve $|x + 2| \geq 5$?
3. What happens if the inequality is $|x| < a$ instead?
4. What does it mean to flip the inequality when dividing by a negative number?
5. How do you solve compound inequalities?

Tip: When dealing with absolute value inequalities, always isolate the absolute value expression before solving.