To determine which of the given points satisfies the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we will substitute each point into both inequalities.
- Inequality 1: \( y > \frac{1}{2}x + 5 \)
- Inequality 2: \( y < -2x + 1 \)
Let's evaluate each point:
A. (4, 8)
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For \( y > \frac{1}{2}x + 5 \): \( 8 > \frac{1}{2}(4) + 5 \) \( 8 > 2 + 5 \) \( 8 > 7 \) (True)
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For \( y < -2x + 1 \): \( 8 < -2(4) + 1 \) \( 8 < -8 + 1 \) \( 8 < -7 \) (False)
B. (-5, 5)
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For \( y > \frac{1}{2}x + 5 \): \( 5 > \frac{1}{2}(-5) + 5 \) \( 5 > -2.5 + 5 \) \( 5 > 2.5 \) (True)
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For \( y < -2x + 1 \): \( 5 < -2(-5) + 1 \) \( 5 < 10 + 1 \) \( 5 < 11 \) (True)
C. (1, 5.5)
- For \( y > \frac{1}{2}x + 5 \): \( 5.5 > \frac{1}{2}(1) + 5 \) \( 5.5 > 0.5 + 5 \) \( 5.5 > 5.5 \) (False)
D. (-1.6, 4.2)
- For \( y > \frac{1}{2}x + 5 \): \( 4.2 > \frac{1}{2}(-1.6) + 5 \) \( 4.2 > -0.8 + 5 \) \( 4.2 > 4.2 \) (False)
Now we can summarize the results:
- A. (4, 8): False for inequality 2
- B. (-5, 5): True for both inequalities
- C. (1, 5.5): False for inequality 1
- D. (-1.6, 4.2): False for inequality 1
The only point that satisfies both inequalities is:
B. (-5, 5)
1 answer