1. The first problem asks to find the irrational number between 5 and 6 from the options given: a) 5.5, b) \(\sqrt{5}\), c) \(\sqrt{30}\), d) \(\sqrt{10}\). 2. Recall that an irrational number cannot be expressed as a simple fraction and its decimal form is non-terminating and non-repeating. 3. Check each option: - a) 5.5 is a rational number (a decimal). - b) \(\sqrt{5} \approx 2.236\), which is less than 5, so not between 5 and 6. - c) \(\sqrt{30} \approx 5.477\), which lies between 5 and 6 and is irrational. - d) \(\sqrt{10} \approx 3.162\), which is less than 5. 4. Therefore, the irrational number between 5 and 6 is \(\sqrt{30}\). 5. The second problem involves interval subtraction: Given \(x = ]-2, 2[\) and \(y = [1, 4[\), find \(x - y\). 6. Interval subtraction \(x - y\) means the set of all numbers \(a - b\) where \(a \in x\) and \(b \in y\). 7. The smallest value of \(x - y\) is \(-2 - 4 = -6\) (since \(-2\) is not included and \(4\) is not included, the interval is open at these ends). 8. The largest value of \(x - y\) is \(2 - 1 = 1\) (\(2\) not included, \(1\) included). 9. So the interval for \(x - y\) is \(]-6, 1]\), but since \(-2\) and \(4\) are not included, the subtraction interval is \(]-6, 1[\). 10. However, the options given are: - a) ] - 2, 1[ - b) [1, 2[ - c) [1, 2] - d) ]2, 4] 11. The correct interval from the options that matches the subtraction is a) ] - 2, 1[. Final answers: - Problem 1: \(\sqrt{30}\) - Problem 2: ] - 2, 1[