The inequality is \( 15 \geq 3x - 3 > -12 \). Let's solve it step by step.

### Step 1: Break the inequality into two parts.

1. \( 15 \geq 3x - 3 \)
2. \( 3x - 3 > -12 \)

### Step 2: Solve each inequality separately.

#### Inequality 1: \( 15 \geq 3x - 3 \)
- Add 3 to both sides:
  \[
  15 + 3 \geq 3x \quad \text{or} \quad 18 \geq 3x
  \]
- Divide by 3:
  \[
  \frac{18}{3} \geq x \quad \text{or} \quad 6 \geq x \quad \text{(which is the same as } x \leq 6\text{)}
  \]

#### Inequality 2: \( 3x - 3 > -12 \)
- Add 3 to both sides:
  \[
  3x > -12 + 3 \quad \text{or} \quad 3x > -9
  \]
- Divide by 3:
  \[
  x > \frac{-9}{3} \quad \text{or} \quad x > -3
  \]

### Step 3: Combine the two results.

From both inequalities, we have:
\[
x > -3 \quad \text{and} \quad x \leq 6
\]

Thus, the solution is:
\[
-3 < x \leq 6
\]

This means that \( x \) lies in the interval \( (-3, 6] \).

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Would you like more details on any step or do you have any further questions?

Here are 5 related questions to explore:

1. How do we solve compound inequalities with different operations?
2. What happens when inequalities have variables on both sides?
3. How would the solution change if the inequality symbols were flipped?
4. How can inequalities be represented graphically on a number line?
5. What are the rules for multiplying or dividing inequalities by negative numbers?

**Tip:** When solving compound inequalities, treat each part independently before combining the results to avoid confusion.

This guide explains how to solve the compound inequality 15 ≥ 3x - 3 > -12, covering step-by-step inequality manipulation. Suitable for students in Grades 8-10, the problem involves algebraic concepts like addition, subtraction, and division of inequalities.

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