https://file.aiquickdraw.com/mathbot/file/1730774909619hhjr14qb.jpg

To solve for \( c \), we need to address each inequality separately and then combine the solutions as a compound inequality.

### Step 1: Solve the first inequality
\[
\frac{-c + 7}{4} > 3
\]
1. Multiply both sides by 4 to clear the denominator:
   \[
   -c + 7 > 12
   \]
2. Subtract 7 from both sides:
   \[
   -c > 5
   \]
3. Divide by -1, remembering to reverse the inequality:
   \[
   c < -5
   \]

### Step 2: Solve the second inequality
\[
\frac{-3c + 16}{2} \leq 8
\]
1. Multiply both sides by 2 to clear the denominator:
   \[
   -3c + 16 \leq 16
   \]
2. Subtract 16 from both sides:
   \[
   -3c \leq 0
   \]
3. Divide by -3, remembering to reverse the inequality:
   \[
   c \geq 0
   \]

### Combine the solutions
The solution to the compound inequality is:
\[
c < -5 \quad \text{or} \quad c \geq 0
\]

Would you like a further breakdown or have any questions?

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**Questions for further exploration:**
1. What if the inequalities were connected by "and" instead of "or"?
2. How would the solution change if the inequalities had different coefficients?
3. Can you graph the solution on a number line?
4. How do you handle inequalities when dividing by a negative number?
5. What are other examples of compound inequalities?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Solve for c: \( \frac{-c + 7}{4} > 3 \) or \( \frac{-3c + 16}{2} \leq 8 \). Write your answer as a compound inequality with integers.

Learn how to solve the compound inequality \( \frac{-c + 7}{4} > 3 \) or \( \frac{-3c + 16}{2} \leq 8 \) and express the solution as a compound inequality with integers. This example covers important algebraic concepts, including solving inequalities with integer solutions and properties of inequalities. Suitable for students in Grades 8-10.

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