Bot GPT 4-o mini To determine which ordered pairs are solutions to the system of inequalities \(2x + 2y \geq 7\) and \(y < 3x - 5\), we will test each pair one by one.
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For the pair (3, 4):
- Check \(2x + 2y \geq 7\): \[ 2(3) + 2(4) = 6 + 8 = 14 \geq 7 \quad \text{(True)} \]
- Check \(y < 3x - 5\): \[ 4 < 3(3) - 5 \implies 4 < 9 - 5 \implies 4 < 4 \quad \text{(False)} \]
- Conclusion: (3, 4) is not a solution.
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For the pair (2, -5):
- Check \(2x + 2y \geq 7\): \[ 2(2) + 2(-5) = 4 - 10 = -6 \geq 7 \quad \text{(False)} \]
- Conclusion: (2, -5) is not a solution.
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For the pair (9, -3):
- Check \(2x + 2y \geq 7\): \[ 2(9) + 2(-3) = 18 - 6 = 12 \geq 7 \quad \text{(True)} \]
- Check \(y < 3x - 5\): \[ -3 < 3(9) - 5 \implies -3 < 27 - 5 \implies -3 < 22 \quad \text{(True)} \]
- Conclusion: (9, -3) is a solution.
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For the pair (7, 0):
- Check \(2x + 2y \geq 7\): \[ 2(7) + 2(0) = 14 + 0 = 14 \geq 7 \quad \text{(True)} \]
- Check \(y < 3x - 5\): \[ 0 < 3(7) - 5 \implies 0 < 21 - 5 \implies 0 < 16 \quad \text{(True)} \]
- Conclusion: (7, 0) is a solution.
Final Results The ordered pairs that are solutions to the system of inequalities are:
- c. (9, -3)
- d. (7, 0)
1 answer