To determine which of the given points satisfies both inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \), we will check each point one by one.
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Point (-5, 5):
- For \( y > 12x + 5 \): \[ 5 > 12(-5) + 5 \implies 5 > -60 + 5 \implies 5 > -55 \quad \text{(True)} \]
- For \( y < -2x + 1 \): \[ 5 < -2(-5) + 1 \implies 5 < 10 + 1 \implies 5 < 11 \quad \text{(True)} \]
- This point satisfies both inequalities.
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Point (4, 8):
- For \( y > 12x + 5 \): \[ 8 > 12(4) + 5 \implies 8 > 48 + 5 \implies 8 > 53 \quad \text{(False)} \]
- This point does not satisfy the first inequality.
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Point (-1.6, 4.2):
- For \( y > 12x + 5 \): \[ 4.2 > 12(-1.6) + 5 \implies 4.2 > -19.2 + 5 \implies 4.2 > -14.2 \quad \text{(True)} \]
- For \( y < -2x + 1 \): \[ 4.2 < -2(-1.6) + 1 \implies 4.2 < 3.2 + 1 \implies 4.2 < 4.2 \quad \text{(False)} \]
- This point does not satisfy the second inequality.
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Point (1, 5.5):
- For \( y > 12x + 5 \): \[ 5.5 > 12(1) + 5 \implies 5.5 > 12 + 5 \implies 5.5 > 17 \quad \text{(False)} \]
- This point does not satisfy the first inequality.
After checking all the points, the only point that satisfies both inequalities is:
(−5, 5).