To determine which of the provided points satisfies the simultaneous inequalities \(y > 12x + 5\) and \(y < -2x + 1\), we need to check each point against both inequalities.
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For the point (1, 5.5):
- Check \(y > 12x + 5\): \[ 5.5 > 12(1) + 5 \implies 5.5 > 12 + 5 \implies 5.5 > 17 \quad (\text{False}) \]
- Since the first inequality is not satisfied, this point does not work.
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For the point (-1.6, 4.2):
- Check \(y > 12x + 5\): \[ 4.2 > 12(-1.6) + 5 \implies 4.2 > -19.2 + 5 \implies 4.2 > -14.2 \quad (\text{True}) \]
- Check \(y < -2x + 1\): \[ 4.2 < -2(-1.6) + 1 \implies 4.2 < 3.2 + 1 \implies 4.2 < 4.2 \quad (\text{False}) \]
- Since the second inequality is not satisfied, this point does not work.
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For the point (-5, 5):
- Check \(y > 12x + 5\): \[ 5 > 12(-5) + 5 \implies 5 > -60 + 5 \implies 5 > -55 \quad (\text{True}) \]
- Check \(y < -2x + 1\): \[ 5 < -2(-5) + 1 \implies 5 < 10 + 1 \implies 5 < 11 \quad (\text{True}) \]
- Both inequalities are satisfied, so this point works.
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For the point (4, 8):
- Check \(y > 12x + 5\): \[ 8 > 12(4) + 5 \implies 8 > 48 + 5 \implies 8 > 53 \quad (\text{False}) \]
- Since the first inequality is not satisfied, this point does not work.
After checking all points, the only point that satisfies both inequalities is \((-5, 5)\).
Thus, the solution is: \[ \boxed{(-5, 5)} \]
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