We are given the system of inequalities:

1. $y < -2x + 1$
2. $x + 2y \geq -4$

We need to test which of the points provided satisfies both inequalities.

Option A: $(-5, 4)$

1. First inequality: $y < -2x + 1$ $4 < -2(-5) + 1$ $4 < 10 + 1$ $4 < 11 \quad \text{(True)}$

2. Second inequality: $x + 2y \geq -4$ $-5 + 2(4) \geq -4$ $-5 + 8 \geq -4$ $3 \geq -4 \quad \text{(True)}$

Since both inequalities are satisfied, $(-5, 4)$ is in the solution set.

Option B: $(-9, 1)$

1. First inequality: $y < -2x + 1$ $1 < -2(-9) + 1$ $1 < 18 + 1$ $1 < 19 \quad \text{(True)}$

2. Second inequality: $x + 2y \geq -4$ $-9 + 2(1) \geq -4$ $-9 + 2 \geq -4$ $-7 \geq -4 \quad \text{(False)}$

Since the second inequality is not satisfied, $(-9, 1)$ is not in the solution set.

Option C: $(6, -8)$

1. First inequality: $y < -2x + 1$ $-8 < -2(6) + 1$ $-8 < -12 + 1$ $-8 < -11 \quad \text{(False)}$

Since the first inequality is not satisfied, $(6, -8)$ is not in the solution set.

Option D: $(2, -3)$

1. First inequality: $y < -2x + 1$ $-3 < -2(2) + 1$ $-3 < -4 + 1$ $-3 < -3 \quad \text{(False)}$

Since the first inequality is not satisfied, $(2, -3)$ is not in the solution set.

Conclusion:

The point that satisfies both inequalities is Option A: $(-5, 4)$.

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Would you like more details or have any questions?

Here are 5 related questions to extend your understanding:

1. How can you graph the system of inequalities?
2. What happens when a point lies on the boundary line of an inequality?
3. How do you solve a system of inequalities with more than two constraints?
4. How can you determine the feasible region for a system of inequalities?
5. Can the solution set of a system of inequalities be empty?

Tip: Always test inequalities by substituting points back into the system to verify their validity!