We are given two inequalities and need to determine which point satisfies both.

1. The first inequality is: $y < -5x - 2$ This inequality represents the region below the line $y = -5x - 2$.

2. The second inequality is: $y > x + 5$ This inequality represents the region above the line $y = x + 5$.

Now, let’s evaluate the four points provided in the multiple-choice options to see which one satisfies both inequalities.

Checking the points:

Option 1: $(-9, -1)$

• For the first inequality $y < -5x - 2$: $-1 < -5(-9) - 2 \quad \Rightarrow \quad -1 < 45 - 2 \quad \Rightarrow \quad -1 < 43 \quad \text{(True)}$
• For the second inequality $y > x + 5$: $-1 > -9 + 5 \quad \Rightarrow \quad -1 > -4 \quad \text{(True)}$
• This point satisfies both inequalities.

Option 2: $(-5, -3)$

• For the first inequality $y < -5x - 2$: $-3 < -5(-5) - 2 \quad \Rightarrow \quad -3 < 25 - 2 \quad \Rightarrow \quad -3 < 23 \quad \text{(True)}$
• For the second inequality $y > x + 5$: $-3 > -5 + 5 \quad \Rightarrow \quad -3 > 0 \quad \text{(False)}$
• This point does not satisfy the second inequality.

Option 3: $(3, 10)$

• For the first inequality $y < -5x - 2$: $10 < -5(3) - 2 \quad \Rightarrow \quad 10 < -15 - 2 \quad \Rightarrow \quad 10 < -17 \quad \text{(False)}$
• This point does not satisfy the first inequality.

Option 4: $(2, -1)$

• For the first inequality $y < -5x - 2$: $-1 < -5(2) - 2 \quad \Rightarrow \quad -1 < -10 - 2 \quad \Rightarrow \quad -1 < -12 \quad \text{(False)}$
• This point does not satisfy the first inequality.

Conclusion:

The point that satisfies both inequalities is $(-9, -1)$.

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Would you like further explanations or have any questions about this? Here are some additional related questions you may find useful:

1. How do you graph inequalities on the coordinate plane?
2. How can we interpret shading for linear inequalities?
3. What happens when two inequalities are combined?
4. How can you test whether a point lies within the solution region of an inequality?
5. What is the difference between "greater than" and "less than" inequalities in terms of graphing?

Tip: When dealing with inequalities, always remember that the boundary lines represent "strict" inequalities when dashed and "inclusive" inequalities when solid.