To determine which point lies within the solution set of this system of inequalities, we’ll evaluate each of the given points for both inequalities in the system:

System of inequalities:

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

Testing each option:

Option A: $(-9, 1)$

1. Substitute $(-9, 1)$ into the first inequality: $y < -2x + 1 \Rightarrow 1 < -2(-9) + 1 \Rightarrow 1 < 18 + 1 \Rightarrow 1 < 19 \quad (\text{True})$
2. Substitute $(-9, 1)$ into the second inequality: $x + 2y \geq -4 \Rightarrow -9 + 2(1) \geq -4 \Rightarrow -9 + 2 \geq -4 \Rightarrow -7 \geq -4 \quad (\text{False})$

Since the second inequality fails, Option A is not in the solution set.

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Option B: $(-5, 4)$

1. Substitute $(-5, 4)$ into the first inequality: $y < -2x + 1 \Rightarrow 4 < -2(-5) + 1 \Rightarrow 4 < 10 + 1 \Rightarrow 4 < 11 \quad (\text{True})$
2. Substitute $(-5, 4)$ into the second inequality: $x + 2y \geq -4 \Rightarrow -5 + 2(4) \geq -4 \Rightarrow -5 + 8 \geq -4 \Rightarrow 3 \geq -4 \quad (\text{True})$

Since both inequalities are satisfied, Option B is in the solution set.

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Option C: $(2, -3)$

1. Substitute $(2, -3)$ into the first inequality: $y < -2x + 1 \Rightarrow -3 < -2(2) + 1 \Rightarrow -3 < -4 + 1 \Rightarrow -3 < -3 \quad (\text{False})$

Since the first inequality fails, Option C is not in the solution set.

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Option D: $(6, -8)$

1. Substitute $(6, -8)$ into the first inequality: $y < -2x + 1 \Rightarrow -8 < -2(6) + 1 \Rightarrow -8 < -12 + 1 \Rightarrow -8 < -11 \quad (\text{False})$

Since the first inequality fails, Option D is not in the solution set.

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Conclusion:

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

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Would you like more details on how each inequality was solved, or do you have any questions?

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1. How can we solve a system of linear inequalities graphically?
2. What methods can verify a point's inclusion in multiple inequalities?
3. Why does the combination of “<” and “≥” limit solution sets for inequalities?
4. How does changing inequality direction affect solution boundaries?
5. What strategies help quickly determine valid points in a solution set?

Tip: Always check each inequality separately to verify all conditions for a solution point.