1. The first problem asks to identify an irrational number between 5 and 6. 2. Let's analyze each option: - a) 5.5 is a rational number because it can be expressed as a fraction $\frac{11}{2}$. - b) $\sqrt{5}$ is approximately 2.236, which is less than 5, so it is not between 5 and 6. - c) $\sqrt{30}$ is approximately 5.477, which lies between 5 and 6 and is irrational. - d) $\sqrt{10}$ is approximately 3.162, which is less than 5. 3. Therefore, the irrational number between 5 and 6 is $\sqrt{30}$. 4. The second problem involves intervals: $x = ]-2, 2[$ and $y = [1, 4[$. 5. The subtraction of intervals $x - y$ is defined as $\{a - b \mid a \in x, b \in y\}$. 6. The smallest value of $x - y$ is $-2 - 4 = -6$ (since $x$ approaches but does not include $-2$, and $y$ includes 1 but not 4). 7. The largest value of $x - y$ is $2 - 1 = 1$ (since $x$ approaches but does not include 2, and $y$ includes 1). 8. Since $x$ is an open interval and $y$ is half-open, the resulting interval is $]-6, 1[$. 9. However, the options given do not include $]-6, 1[$, so let's re-examine the problem carefully. 10. The problem states $x = ]-2, 2[$ and $y = [1, 4[$. 11. The subtraction $x - y$ is $\{a - b \mid a \in ]-2, 2[, b \in [1, 4[\}$. 12. The minimum is $\min(x) - \max(y) = -2 - 4 = -6$ (not included because $x$ and $y$ are open or half-open). 13. The maximum is $\max(x) - \min(y) = 2 - 1 = 1$ (not included because $x$ is open). 14. So the interval is $]-6, 1[$. 15. None of the options match $]-6, 1[$, so check if the problem expects $x - y$ as interval subtraction or set subtraction. 16. If the problem means $x - y = \{z \mid z = x - y\}$ as interval subtraction, the answer is $]-6, 1[$. 17. Since option a) is $]-2, 1[$, which is a subset of $]-6, 1[$, and the problem likely expects the difference of intervals as $]-2, 1[$. 18. Therefore, the best matching answer is a) $]-2, 1[$. Final answers: - Problem 1: $\sqrt{30}$ - Problem 2: $]-2, 1[$