https://file.aiquickdraw.com/mathbot/file/1729348599745l58tqppp.jpg

The image contains a system of inequalities:

\[
-27 > -4x - 7 \quad \text{and} \quad -35 \geq -4x - 7
\]

### Solving the first inequality:
\[
-27 > -4x - 7
\]
1. Add 7 to both sides:
\[
-27 + 7 > -4x
\]
\[
-20 > -4x
\]
2. Divide by -4, remembering to reverse the inequality because we are dividing by a negative number:
\[
\frac{-20}{-4} < x
\]
\[
5 < x \quad \text{or} \quad x > 5
\]

### Solving the second inequality:
\[
-35 \geq -4x - 7
\]
1. Add 7 to both sides:
\[
-35 + 7 \geq -4x
\]
\[
-28 \geq -4x
\]
2. Divide by -4, again reversing the inequality:
\[
\frac{-28}{-4} \leq x
\]
\[
7 \leq x \quad \text{or} \quad x \geq 7
\]

### Solution:
Now combine both inequalities:
\[
x > 5 \quad \text{and} \quad x \geq 7
\]
This simplifies to:
\[
x \geq 7
\]

Thus, the solution to the system of inequalities is \( x \geq 7 \).

Would you like more details or clarifications?

Here are 5 related questions to further explore this topic:
1. How do you handle inequalities when multiplying or dividing by negative numbers?
2. Can you explain how to represent this solution on a number line?
3. How would the solution change if the inequalities were reversed?
4. What if the inequality signs were strict (e.g., \( > \) instead of \( \geq \))?
5. How can you check if a solution satisfies both inequalities?

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

Learn how to solve the system of inequalities -27 > -4x - 7 and -35 ≥ -4x - 7. This algebraic problem involves solving linear inequalities and understanding the concept of inequality reversal when dividing by negative numbers. Suitable for students in Grades 8-10.

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