https://r2.erweima.ai/mathbot/image/1726086474175eli5svmm.jpg

Let's solve the exercises shown in the uploaded image.

**Exercice 11:**
We are asked to calculate the distance between the given real numbers. The distance between two numbers \( a \) and \( b \) is given by:
\[
\text{Distance} = |a - b|
\]

Let's solve each case:

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

2. **b) -2\(\sqrt{2}\) and -\(\sqrt{12}\)**  
   We simplify \(\sqrt{12}\):
   \[
   \sqrt{12} = 2\sqrt{3}
   \]
   Now calculate the distance:
   \[
   \text{Distance} = |-2\sqrt{2} - (-2\sqrt{3})| = |-2\sqrt{2} + 2\sqrt{3}|
   \]
   This is the simplest form, so we leave the expression as:
   \[
   \boxed{|2\sqrt{3} - 2\sqrt{2}|}
   \]
   
3. **c) -π and 4π**  
   \[
   \text{Distance} = |-π - 4π| = |-5π| = 5π
   \]

---

**Exercice 12:**
We are asked to interpret each expression as the distance between real numbers:

1. **a) |x - 5|**  
   This represents the distance between \( x \) and 5.

2. **b) |x - (-5)|** or **|x + 5|**  
   This represents the distance between \( x \) and -5.

3. **c) |x + 6|**  
   This represents the distance between \( x \) and -6.

4. **d) |7 - x|**  
   This is equivalent to \( |x - 7| \), which represents the distance between \( x \) and 7.

5. **e) |9 + x|**  
   This represents the distance between \( x \) and -9.

---

Would you like more details on any of these solutions or have any questions?

Here are 5 related questions for deeper understanding:
1. How would you calculate the distance between two complex numbers?
2. What is the distance between \(-2\sqrt{2}\) and \(2\sqrt{2}\)?
3. How can you apply the distance formula to points in 2D space?
4. What is the difference between absolute value and distance?
5. What happens to the distance formula if both numbers are negative?

**Tip:** When calculating distances, always remember that the order of subtraction doesn't matter because of the absolute value.

In each case, calculate the distance between the real numbers shown in Exercice 11 and interpret the expressions as distances in Exercice 12.

Learn how to calculate the distance between real numbers and interpret absolute value expressions through exercises. This includes problems involving pi and square roots, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation