To determine which points satisfy the simultaneous inequalities \( y > \frac{1}{2}x + 5 \) and \( y < -2x + 1 \), we need to analyze the inequalities individually and find a solution that fulfills both conditions.
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Graph the inequalities:
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For \( y > \frac{1}{2}x + 5 \):
- This inequality represents the area above the line \( y = \frac{1}{2}x + 5 \). The line has a slope of \( \frac{1}{2} \) and a y-intercept of \( 5 \).
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For \( y < -2x + 1 \):
- This inequality represents the area below the line \( y = -2x + 1 \). The line has a slope of \( -2 \) and a y-intercept of \( 1 \).
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Finding the intersection:
- The solution to the simultaneous inequalities would be where the area above the first line and below the second line intercept. To find the solution region, we can graph both lines and observe where the shaded regions overlap.
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Testing points:
- To confirm which points satisfy both inequalities, you can test specific (x, y) coordinates in both inequalities. If a point lies in the overlapping shaded region, it satisfies both inequalities.
If you have specific points to test, please provide them, and I can help you evaluate which of those points are solutions to the simultaneous inequalities.
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