To determine which option includes a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \):
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Calculate \( \sqrt{6} \) and \( \sqrt{8} \):
- \( \sqrt{6} \) is approximately \( 2.45 \).
- \( \sqrt{8} \) is approximately \( 2.83 \).
Therefore, we are looking for one rational number and one irrational number that lie within the interval \( (2.45, 2.83) \).
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Evaluate the options:
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Option A: \( \frac{9}{4} \) and \( \sqrt{7} \)
- \( \frac{9}{4} = 2.25 \) (rational, outside the interval)
- \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)
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Option B: \( 2.7 \) and \( \sqrt{9} \)
- \( 2.7 \) (rational, inside the interval)
- \( \sqrt{9} = 3 \) (rational, outside the interval)
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Option C: \( \overline{6} = 6.\overline{6} \) (6.666...) and \( \sqrt{7} \)
- \( 6.\overline{6} \) (rational, outside the interval)
- \( \sqrt{7} \approx 2.645 \) (irrational, inside the interval)
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Option D: \( 2.5 \) and \( \pi \)
- \( 2.5 \) (rational, inside the interval)
- \( \pi \approx 3.14159 \) (irrational, outside the interval)
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Conclusion: Only Option A \( \frac{9}{4} , and , \sqrt{7} \) contains one rational number (\(\sqrt{7}\)) and one irrational number. However, \(\frac{9}{4}\) is outside the specified range. All other options mix rational and irrational numbers but also fall outside the range.
Thus, the best choice is A, where only one number is irrational within the range.