To determine which point satisfies both inequalities $y > \frac{1}{2}x + 5$ and $y < -2x + 1$, let's test each option.

1. Inequality 1: $y > \frac{1}{2}x + 5$
2. Inequality 2: $y < -2x + 1$

Testing Each Point:

Option A: (4, 8)

1. Substitute $x = 4$ and $y = 8$ into the first inequality: $8 > \frac{1}{2}(4) + 5 \Rightarrow 8 > 2 + 5 \Rightarrow 8 > 7$ This is true.

2. Substitute $x = 4$ and $y = 8$ into the second inequality: $8 < -2(4) + 1 \Rightarrow 8 < -8 + 1 \Rightarrow 8 < -7$ This is false.

Since (4, 8) does not satisfy both inequalities, it is not a solution.

Option B: (-1.6, 4.2)

1. Substitute $x = -1.6$ and $y = 4.2$ into the first inequality: $4.2 > \frac{1}{2}(-1.6) + 5 \Rightarrow 4.2 > -0.8 + 5 \Rightarrow 4.2 > 4.2$ This is false because 4.2 is not greater than 4.2.

Since it does not satisfy the first inequality, (−1.6, 4.2) is not a solution.

Option C: (1, 5.5)

1. Substitute $x = 1$ and $y = 5.5$ into the first inequality: $5.5 > \frac{1}{2}(1) + 5 \Rightarrow 5.5 > 0.5 + 5 \Rightarrow 5.5 > 5.5$ This is false because 5.5 is not greater than 5.5.

Since it does not satisfy the first inequality, (1, 5.5) is not a solution.

Option D: (-5.5, y)

This option is missing a $y$-value, so we cannot test it.

Conclusion:

None of the given options satisfy both inequalities. It seems that there is no point among the choices provided that is a solution to the system of inequalities.

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Would you like further details on how these calculations were made or have additional questions?

Here are related questions you might explore:

1. How do you find the solution region for two inequalities graphically?
2. What methods can verify solutions to a system of inequalities?
3. How does changing coefficients in linear inequalities affect their graphs?
4. How can simultaneous inequalities be solved with substitution?
5. What are real-world applications of systems of inequalities?

Tip: When testing points for inequalities, ensure each inequality holds true individually for a point to be a solution to the system.