To determine which point represents a solution to the system of inequalities, we must analyze each inequality and check if the given points satisfy all the conditions:
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For \( y > -\frac{5}{4}x + 125 \):
- This means we are looking for points that lie above the line defined by this inequality.
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For \( y \leq 40 \):
- This means we are looking for points that are on or below the horizontal line where \( y = 40 \).
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For \( y \geq -x + 120 \):
- This means we are looking for points that lie above the line defined by \( y = -x + 120 \).
Now, let's check the candidate points:
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Point (100, 0):
- \( y > -\frac{5}{4}(100) + 125 \) → \( 0 > -125 + 125 \) → \( 0 > 0 \) (not satisfied)
- \( y \leq 40 \) → \( 0 \leq 40 \) (satisfied)
- \( y \geq -100 + 120 \) → \( 0 \geq 20 \) (not satisfied)
This point does not satisfy all inequalities.
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Point (110, -5):
- \( y > -\frac{5}{4}(110) + 125 \) → \( -5 > -137.5 + 125 \) → \( -5 > -12.5 \) (satisfied)
- \( y \leq 40 \) → \( -5 \leq 40 \) (satisfied)
- \( y \geq -110 + 120 \) → \( -5 \geq 10 \) (not satisfied)
This point does not satisfy all inequalities.
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Point (150, 20):
- \( y > -\frac{5}{4}(150) + 125 \) → \( 20 > -187.5 + 125 \) → \( 20 > -62.5 \) (satisfied)
- \( y \leq 40 \) → \( 20 \leq 40 \) (satisfied)
- \( y \geq -150 + 120 \) → \( 20 \geq -30 \) (satisfied)
This point satisfies all inequalities.
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Point (120, 50):
- \( y > -\frac{5}{4}(120) + 125 \) → \( 50 > -150 + 125 \) → \( 50 > -25 \) (satisfied)
- \( y \leq 40 \) → \( 50 \leq 40 \) (not satisfied)
- \( y \geq -120 + 120 \) → \( 50 \geq 0 \) (satisfied)
This point does not satisfy all inequalities.
Based on this analysis, the point \( (150, 20) \) is a solution to the system of inequalities.