To find a pair that includes a rational number and an irrational number between \( 6 - \sqrt{2} \) and \( 8 - \sqrt{2} \), we first need to evaluate the approximate values of \( 6 - \sqrt{2} \) and \( 8 - \sqrt{2} \):
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Calculate \( 6 - \sqrt{2} \):
- \( \sqrt{2} \approx 1.414 \)
- \( 6 - \sqrt{2} \approx 6 - 1.414 \approx 4.586 \)
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Calculate \( 8 - \sqrt{2} \):
- \( 8 - \sqrt{2} \approx 8 - 1.414 \approx 6.586 \)
Thus, the range of interest is approximately from \( 4.586 \) to \( 6.586 \).
Now, let's examine the provided options to see if they contain one rational and one irrational number within this range:
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Option: 2.5, π
- 2.5 is rational.
- π is irrational.
- Both numbers do not lie within the range.
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Option: 2.7, \( \sqrt{9} \)
- 2.7 is rational.
- \( \sqrt{9} = 3 \) (rational).
- Both numbers do not lie within the range.
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Option: \( 2.6\overline{6} \), \( \sqrt{7} \)
- \( 2.6\overline{6} \approx 2.666... \) (rational).
- \( \sqrt{7} \approx 2.646 \) (irrational).
- Neither falls within the range.
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Option: 94, \( \sqrt{7} \)
- 94 is rational (but far above the range).
- \( \sqrt{7} \) is irrational.
- Only one number falls in the range.
Conclusion: None of the provided options contain a rational and an irrational number that both lie within \( 6 - \sqrt{2} \) and \( 8 - \sqrt{2} \) based on our calculations. None are correct in this context.