Determine whether the ordered pair is a solution to the system of inequalities. \( y<-x^{2}+8 \) \( x+2 y \leq 2 \) (a) \( (-2,4) \) (b) \( (0,3) \) (c) \( (0,2) \) (d) \( (-1,3) \)

To determine if the ordered pairs are solutions to the system of inequalities, we need to evaluate each pair by substituting the x-value into both inequalities. 1. **For \( (-2, 4) \)**: - First inequality: \( 4 < -(-2)^{2} + 8 \) → \( 4 < -4 + 8 \) → \( 4 < 4 \) (False) - Second inequality: \( -2 + 2(4) \leq 2 \) → \( -2 + 8 \leq 2 \) → \( 6 \leq 2 \) (False) - Conclusion: Not a solution. 2. **For \( (0, 3) \)**: - First inequality: \( 3 < -0^{2} + 8 \) → \( 3 < 8 \) (True) - Second inequality: \( 0 + 2(3) \leq 2 \) → \( 0 + 6 \leq 2 \) → \( 6 \leq 2 \) (False) - Conclusion: Not a solution. 3. **For \( (0, 2) \)**: - First inequality: \( 2 < -0^{2} + 8 \) → \( 2 < 8 \) (True) - Second inequality: \( 0 + 2(2) \leq 2 \) → \( 0 + 4 \leq 2 \) → \( 4 \leq 2 \) (False) - Conclusion: Not a solution. 4. **For \( (-1, 3) \)**: - First inequality: \( 3 < -(-1)^{2} + 8 \) → \( 3 < -1 + 8 \) → \( 3 < 7 \) (True) - Second inequality: \( -1 + 2(3) \leq 2 \) → \( -1 + 6 \leq 2 \) → \( 5 \leq 2 \) (False) - Conclusion: Not a solution. Final results are that none of the ordered pairs \( (-2,4) \), \( (0,3) \), \( (0,2) \), or \( (-1,3) \) satisfy both inequalities.