To determine which points are part of the solution set for the inequality \( y < \frac{2}{5}x + 1 \), we need to test each point by substituting the \( x \) and \( y \) coordinates into the inequality.
The inequality can be rewritten as \( y < \frac{2}{5}x + 1 \).
-
For point (10, 4): \[ y = 4 \quad \text{and} \quad x = 10 \] Substitute into the inequality: \[ 4 < \frac{2}{5}(10) + 1 \] Calculate the right side: \[ 4 < 4 + 1 \quad \Rightarrow \quad 4 < 5 \quad \text{(True)} \]
-
For point (10, 6): \[ y = 6 \quad \text{and} \quad x = 10 \] Substitute into the inequality: \[ 6 < \frac{2}{5}(10) + 1 \] Calculate the right side: \[ 6 < 4 + 1 \quad \Rightarrow \quad 6 < 5 \quad \text{(False)} \]
-
For point (-10, 6): \[ y = 6 \quad \text{and} \quad x = -10 \] Substitute into the inequality: \[ 6 < \frac{2}{5}(-10) + 1 \] Calculate the right side: \[ 6 < -4 + 1 \quad \Rightarrow \quad 6 < -3 \quad \text{(False)} \]
-
For point (10, 5.5): \[ y = 5.5 \quad \text{and} \quad x = 10 \] Substitute into the inequality: \[ 5.5 < \frac{2}{5}(10) + 1 \] Calculate the right side: \[ 5.5 < 4 + 1 \quad \Rightarrow \quad 5.5 < 5 \quad \text{(False)} \]
Summary of results:
- (10, 4) is True
- (10, 6) is False
- (-10, 6) is False
- (10, 5.5) is False
The only point that is part of the solution set for the inequality \( y < \frac{2}{5}x + 1 \) is (10, 4).