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.