Final answer:
To determine the solutions to the system of inequalities, we substitute the values and check if the inequalities hold true. Using this method, points A, B, and D are solutions, while point C is not.
Step-by-step explanation:
To determine which points are solutions to the given system of inequalities, we can substitute the values of x and y into each inequality and check if the inequality holds true.
Let's go through each point:
A) (-1, 1)
Substituting x = -1 and y = 1 into the system of inequalities:
x ≥ -4 is true (-1 ≥ -4),
x + 2y < 10 is true (-1 + 2(1) < 10),
3x + 6y ≥ 9 is true (3(-1) + 6(1) ≥ 9).
Therefore, point (-1, 1) is a solution to the system of inequalities.
B) (3, 0)
Substituting x = 3 and y = 0 into the system of inequalities:
x ≥ -4 is true (3 ≥ -4),
x + 2y < 10 is true (3 + 2(0) < 10),
3x + 6y ≥ 9 is true (3(3) + 6(0) ≥ 9).
Therefore, point (3, 0) is a solution to the system of inequalities.
C) (-5, -5)
Substituting x = -5 and y = -5 into the system of inequalities:
x ≥ -4 is false (-5 ≥ -4 is not true),
x + 2y < 10 is true (-5 + 2(-5) < 10),
3x + 6y ≥ 9 is true (3(-5) + 6(-5) ≥ 9).
Therefore, point (-5, -5) is not a solution to the system of inequalities.
D) (0, 5)
Substituting x = 0 and y = 5 into the system of inequalities:
x ≥ -4 is true (0 ≥ -4),
x + 2y < 10 is true (0 + 2(5) < 10),
3x + 6y ≥ 9 is true (3(0) + 6(5) ≥ 9).
Therefore, point (0, 5) is a solution to the system of inequalities.
In conclusion, points A (-1, 1), B (3, 0), and D (0, 5) are solutions to the given system of inequalities, but point C (-5, -5) is not a solution.