https://file.aiquickdraw.com/mathbot/file/1726712500852jwvhyjyi.jpg

Let's solve the compound inequality step by step:

The compound inequality is:
\[
-3y \geq 9 \quad \text{and} \quad 2y + 2 \geq 4
\]

### Step 1: Solve the first inequality \(-3y \geq 9\)
- Divide both sides by \(-3\). When dividing or multiplying by a negative number, the inequality sign flips:
  \[
  y \leq -3
  \]

### Step 2: Solve the second inequality \(2y + 2 \geq 4\)
- Subtract 2 from both sides:
  \[
  2y \geq 2
  \]
- Divide both sides by 2:
  \[
  y \geq 1
  \]

### Step 3: Combine the results
The compound inequality uses "and," meaning both conditions must be satisfied simultaneously. The solution is:
\[
y \leq -3 \quad \text{and} \quad y \geq 1
\]
However, there is no overlap between these two intervals since \(y \leq -3\) and \(y \geq 1\) describe disjoint sets.

### Step 4: Final Solution in Interval Notation
Since there is no overlap between the two intervals, the solution is **no solution**, which can be written as:
\[
\varnothing
\]

Let me know if you need further explanation!

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Here are some related questions for further practice:
1. How do you solve compound inequalities with "or" instead of "and"?
2. What happens if both inequalities have overlapping solutions?
3. How would you graph the solution to a compound inequality?
4. How do you express inequalities involving absolute values?
5. How can inequalities with fractions be solved?

**Tip**: Always be careful to reverse the inequality sign when multiplying or dividing by a negative number!

Solve the compound inequality: -3y ≥ 9 and 2y + 2 ≥ 4. Write the solution in interval notation.

Learn how to solve the compound inequality -3y ≥ 9 and 2y + 2 ≥ 4 step-by-step. This problem explores solving inequalities, flipping inequality signs, and writing solutions in interval notation. Suitable for Grades 8-10.

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