To solve the inequality:

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

we can follow these steps:

Step 1: Simplify the Inequality

Divide both sides by $-3$ (note that dividing by a negative reverses the inequality):

$|6 - x| \leq 1$

Step 2: Solve the Absolute Value Inequality

The inequality $|6 - x| \leq 1$ means that the expression inside the absolute value is within $1$ unit of $0$, so we can write:

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

Step 3: Solve the Compound Inequality

Now, we solve the compound inequality $-1 \leq 6 - x \leq 1$ by isolating $x$ in the middle.

1. Subtract $6$ from all parts:

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

   Simplifying, we get:

   $-7 \leq -x \leq -5$

2. Divide by $-1$ (which reverses the inequality signs):

   $7 \geq x \geq 5$

   or equivalently:

   $5 \leq x \leq 7$

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 further explanation on any step?

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Here are five related questions to consider:

1. How would you solve $-4|5 - x| \geq -8$?
2. What are the steps to graph the solution $5 \leq x \leq 7$ on a number line?
3. How do you solve inequalities involving absolute values without reversing the inequality?
4. What if the inequality was $-3|6 - x| > -3$ instead? How would the solution change?
5. How do you handle absolute value inequalities when there are variables on both sides?

Tip: When dealing with absolute value inequalities, remember that dividing or multiplying by a negative number reverses the inequality symbol.