https://file.aiquickdraw.com/mathbot/file/1726449759895os8wapbv.jpg

The problem in the image asks to solve the following compound inequality:

\[
2w - 6 \leq -12 \quad \text{and} \quad -3w \leq -9
\]

We will solve each inequality separately.

### Solving the first inequality:

\[
2w - 6 \leq -12
\]

1. Add 6 to both sides:
   \[
   2w \leq -6
   \]

2. Divide both sides by 2:
   \[
   w \leq -3
   \]

### Solving the second inequality:

\[
-3w \leq -9
\]

1. Divide both sides by -3. (Remember to reverse the inequality when dividing by a negative number):
   \[
   w \geq 3
   \]

### Compound inequality:
We now have:
\[
w \leq -3 \quad \text{and} \quad w \geq 3
\]

This implies there is no value of \( w \) that satisfies both inequalities simultaneously. Therefore, the solution is the empty set \( \varnothing \).

### Conclusion:
The solution is \( \varnothing \).

Would you like more details, or have any questions about this?

Here are 5 related questions to deepen your understanding:
1. What happens if we encounter a compound inequality using "or" instead of "and"?
2. How do you represent an interval when the inequality includes a strict inequality, like \( < \)?
3. Can you explain what happens when dividing by a negative number in an inequality?
4. What is the meaning of the empty set in the context of inequalities?
5. How would the solution change if the inequalities were \( w \geq -3 \) and \( w \leq 3 \)?

**Tip:** When solving inequalities, always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Solve the compound inequality: 2w - 6 ≤ -12 and -3w ≤ -9.

Learn how to solve the compound inequality 2w - 6 ≤ -12 and -3w ≤ -9. This problem covers key algebra concepts, including properties of inequalities and how to reverse an inequality when dividing by a negative number. Suitable for students in Grades 8-10.

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