To determine which point is a solution to the simultaneous inequalities \( y < 12x \) and \( y > 13x + 3 \), we will test each point in both inequalities.
-
Point (2, 2):
- For \( y < 12x \): \[ 2 < 12(2) \quad \Rightarrow \quad 2 < 24 \quad \text{(True)} \]
- For \( y > 13x + 3 \): \[ 2 > 13(2) + 3 \quad \Rightarrow \quad 2 > 26 + 3 \quad \Rightarrow \quad 2 > 29 \quad \text{(False)} \]
-
Point (25, 12):
- For \( y < 12x \): \[ 12 < 12(25) \quad \Rightarrow \quad 12 < 300 \quad \text{(True)} \]
- For \( y > 13x + 3 \): \[ 12 > 13(25) + 3 \quad \Rightarrow \quad 12 > 325 + 3 \quad \Rightarrow \quad 12 > 328 \quad \text{(False)} \]
-
Point (3, 6):
- For \( y < 12x \): \[ 6 < 12(3) \quad \Rightarrow \quad 6 < 36 \quad \text{(True)} \]
- For \( y > 13x + 3 \): \[ 6 > 13(3) + 3 \quad \Rightarrow \quad 6 > 39 + 3 \quad \Rightarrow \quad 6 > 42 \quad \text{(False)} \]
-
Point (10, 5):
- For \( y < 12x \): \[ 5 < 12(10) \quad \Rightarrow \quad 5 < 120 \quad \text{(True)} \]
- For \( y > 13x + 3 \): \[ 5 > 13(10) + 3 \quad \Rightarrow \quad 5 > 130 + 3 \quad \Rightarrow \quad 5 > 133 \quad \text{(False)} \]
After testing all points, we see that none of the points meet both conditions simultaneously:
- Point (2, 2): True for \( y < 12x \), False for \( y > 13x + 3 \)
- Point (25, 12): True for \( y < 12x \), False for \( y > 13x + 3 \)
- Point (3, 6): True for \( y < 12x \), False for \( y > 13x + 3 \)
- Point (10, 5): True for \( y < 12x \), False for \( y > 13x + 3 \)
None of the provided points are a solution to the simultaneous inequalities.