To determine which point is a solution to the system of linear inequalities:

1. $y < 2x + 2$
2. $y \geq -\frac{3}{4}x + 3$

We need to check each given point by substituting its $x$ and $y$ values into both inequalities. A point is a solution if it satisfies both inequalities.

Checking each point:

1. Point $(-4, 2)$

   • Substitute $x = -4$ and $y = 2$ into both inequalities:

   $y < 2x + 2 \implies 2 < 2(-4) + 2 \implies 2 < -8 + 2 \implies 2 < -6 \quad \text{(False)}$

   $y \geq -\frac{3}{4}x + 3 \implies 2 \geq -\frac{3}{4}(-4) + 3 \implies 2 \geq 3 + 3 \implies 2 \geq 6 \quad \text{(False)}$

   • The first inequality is not satisfied, so $(-4, 2)$ is not a solution.

2. Point $(-8, -1)$

   • Substitute $x = -8$ and $y = -1$:

   $y < 2x + 2 \implies -1 < 2(-8) + 2 \implies -1 < -16 + 2 \implies -1 < -14 \quad \text{(False)}$

   $y \geq -\frac{3}{4}x + 3 \implies -1 \geq -\frac{3}{4}(-8) + 3 \implies -1 \geq 6 + 3 \implies -1 \geq 9 \quad \text{(False)}$

   • Neither inequality is satisfied, so $(-8, -1)$ is not a solution.

3. Point $(4, -8)$

   • Substitute $x = 4$ and $y = -8$:

   $y < 2x + 2 \implies -8 < 2(4) + 2 \implies -8 < 8 + 2 \implies -8 < 10 \quad \text{(True)}$

   $y \geq -\frac{3}{4}x + 3 \implies -8 \geq -\frac{3}{4}(4) + 3 \implies -8 \geq -3 + 3 \implies -8 \geq 0 \quad \text{(False)}$

   • The second inequality is not satisfied, so $(4, -8)$ is not a solution.

4. Point $(8, -2)$

   • Substitute $x = 8$ and $y = -2$:

   $y < 2x + 2 \implies -2 < 2(8) + 2 \implies -2 < 16 + 2 \implies -2 < 18 \quad \text{(True)}$

   $y \geq -\frac{3}{4}x + 3 \implies -2 \geq -\frac{3}{4}(8) + 3 \implies -2 \geq -6 + 3 \implies -2 \geq -3 \quad \text{(True)}$

   • Both inequalities are satisfied, so $(8, -2)$ is a solution.

Conclusion

The point $(8, -2)$ satisfies both inequalities and is a solution to the system.

Correct answer: $(8, -2)$

Would you like more details or have any questions?

Here are 5 related questions for further exploration:

1. How do you determine the solution set for a system of linear inequalities graphically?
2. What is the difference between strict and non-strict inequalities in the context of graphing?
3. How can a system of inequalities have infinitely many solutions?
4. What methods can be used to solve systems of linear inequalities with three or more inequalities?
5. What are the implications of changing the inequality signs in a system of inequalities?

Tip: Always check the solutions by substituting the points into each inequality of the system to ensure they meet all the conditions.