To determine which of the given points satisfies the simultaneous inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \), we will evaluate each point one by one against both inequalities.
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Point \((-5, 5)\):
- For \( y > 12x + 5 \): \[ 5 > 12(-5) + 5 \] \[ 5 > -60 + 5 \] \[ 5 > -55 \text{ (True)} \]
- For \( y < -2x + 1 \): \[ 5 < -2(-5) + 1 \] \[ 5 < 10 + 1 \] \[ 5 < 11 \text{ (True)} \]
- Result: Satisfies both inequalities.
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Point \((-1.6, 4.2)\):
- For \( y > 12x + 5 \): \[ 4.2 > 12(-1.6) + 5 \] \[ 4.2 > -19.2 + 5 \] \[ 4.2 > -14.2 \text{ (True)} \]
- For \( y < -2x + 1 \): \[ 4.2 < -2(-1.6) + 1 \] \[ 4.2 < 3.2 + 1 \] \[ 4.2 < 4.2 \text{ (False)} \]
- Result: Does not satisfy the inequalities.
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Point \((4, 8)\):
- For \( y > 12x + 5 \): \[ 8 > 12(4) + 5 \] \[ 8 > 48 + 5 \] \[ 8 > 53 \text{ (False)} \]
- Result: Does not satisfy the inequalities.
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Point \((1, 5.5)\):
- For \( y > 12x + 5 \): \[ 5.5 > 12(1) + 5 \] \[ 5.5 > 12 + 5 \] \[ 5.5 > 17 \text{ (False)} \]
- Result: Does not satisfy the inequalities.
Final Conclusion: The only point that satisfies both inequalities \( y > 12x + 5 \) and \( y < -2x + 1 \) is \((-5, 5)\).