https://r2.erweima.ai/mathbot/image/17260865838449byj04gq.jpg

The exercise in the image is about calculating the distance between real numbers. Let's break down the problems from **Exercice 11** one by one and provide solutions:

### Exercice 11:
In each case, the distance between two numbers \( a \) and \( b \) is calculated using the formula:

\[
\text{Distance} = |a - b|
\]

---

**a) \(-5\) and \(10\)**  
\[
\text{Distance} = | -5 - 10 | = |-15| = 15
\]

---

**b) \(-2\sqrt{2}\) and \(-\sqrt{12}\)**  
First, simplify \(\sqrt{12}\) as:
\[
\sqrt{12} = 2\sqrt{3}
\]
Now calculate the distance:
\[
\text{Distance} = |-2\sqrt{2} - (-2\sqrt{3})| = |-2\sqrt{2} + 2\sqrt{3}|
\]
Since this cannot be simplified easily, the distance remains \( |-2\sqrt{2} + 2\sqrt{3}| \), which is approximately \( 1.034 \).

---

**c) \(-\pi\) and \(4\pi\)**  
\[
\text{Distance} = |-\pi - 4\pi| = |-5\pi| = 5\pi
\]

This completes **Exercice 11**. Would you like a breakdown for **Exercice 12**, or any further details on these calculations?

### Follow-up Questions:
1. Can you simplify square roots of non-perfect squares in similar problems?
2. How would you generalize the distance formula for more than two points on a number line?
3. What happens if both numbers are negative in the distance formula?
4. How can we express the distance between irrational numbers more precisely?
5. What is the importance of absolute values in calculating distances between real numbers?

**Tip:** Remember that the distance formula always gives a non-negative result due to the absolute value.

Calculate the distance between the given real numbers in the image.

This exercise focuses on calculating the distance between real numbers, using absolute values. Key mathematical concepts include the formula Distance = |a - b|. Suitable for high school students in Grades 9-11.

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