Let's solve the inequality step by step:

Given the inequality:

$-3|6 - x| \geq -3$

Step 1: Divide both sides by -3

Since we are dividing by a negative number, the inequality's direction will change:

$|6 - x| \leq 1$

Step 2: Solve the absolute value inequality

The inequality $|6 - x| \leq 1$ implies that:

$-1 \leq 6 - x \leq 1$

Step 3: Solve for $x$

Now, solve the compound inequality for $x$:

1. For the left part: $-1 \leq 6 - x$ Subtract 6 from both sides: $-7 \leq -x$ Multiply both sides by -1 (and reverse the inequality): $7 \geq x \quad \text{or} \quad x \leq 7$

2. For the right part: $6 - x \leq 1$ Subtract 6 from both sides: $-x \leq -5$ Multiply both sides by -1 (and reverse the inequality): $x \geq 5$

Step 4: Combine the results

We now have the inequality:

$5 \leq x \leq 7$

Final Solution:

The solution to the inequality is:

$x \in [5, 7]$

This means that $x$ can take any value between 5 and 7, inclusive.

Would you like a more detailed explanation or have any specific questions about this solution?

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Here are 5 related questions to explore further:

1. How do you handle absolute value inequalities with different signs?
2. What happens if the inequality sign is reversed when dividing by a negative number?
3. How would the solution change if the inequality was $-3|6 - x| < -3$?
4. What are the properties of absolute value functions and their inequalities?
5. How do you solve absolute value equations compared to inequalities?

Tip: Always remember to reverse the inequality sign when multiplying or dividing by a negative number in inequalities!