Solve the inequality: (- 3|6 - x| >= - 3) (2)

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!

Learn how to solve the inequality -3|6 - x| >= -3 step by step. This problem involves absolute value inequalities, the reversal of inequality signs when dividing by negative numbers, and solving compound inequalities. Suitable for students in Grades 8-10.

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