Which of the following options includes a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \) ? (1 point) \( 2 . \overline{6}, \sqrt{7} \) \( 2.7, \sqrt{9} \) \( \frac{9}{4}, \sqrt{7} \) \( 2.5, \pi \)

To determine which option includes a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \), we first need to find the approximate values of \( \sqrt{6} \) and \( \sqrt{8} \): - \( \sqrt{6} \approx 2.45 \) - \( \sqrt{8} = 2\sqrt{2} \approx 2.83 \) Now, we need to check each option to see if it contains one rational number and one irrational number that lies between \( 2.45 \) and \( 2.83 \). 1. **Option 1: \( 2.\overline{6}, \sqrt{7} \)** - \( 2.\overline{6} = 2.666... \) (rational) - \( \sqrt{7} \approx 2.6457513110645906 \) (irrational) - Both numbers are between \( \sqrt{6} \) and \( \sqrt{8} \). 2. **Option 2: \( 2.7, \sqrt{9} \)** - \( 2.7 \) (rational) - \( \sqrt{9} = 3 \) (irrational) - \( 3 \) is not between \( \sqrt{6} \) and \( \sqrt{8} \). 3. **Option 3: \( \frac{9}{4}, \sqrt{7} \)** - \( \frac{9}{4} = 2.25 \) (rational) - \( \sqrt{7} \approx 2.6457513110645906 \) (irrational) - \( 2.25 \) is not between \( \sqrt{6} \) and \( \sqrt{8} \). 4. **Option 4: \( 2.5, \pi \)** - \( 2.5 \) (rational) - \( \pi \approx 3.14159 \) (irrational) - \( 3.14159 \) is not between \( \sqrt{6} \) and \( \sqrt{8} \). The only option that includes both a rational number and an irrational number between \( \sqrt{6} \) and \( \sqrt{8} \) is: **Option 1: \( 2.\overline{6}, \sqrt{7} \)**.