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The exercise in the image contains the following problem:

**Problem 3:**

"Montrer que : ∀(x, y, z) ∈ ℝ³ ; (x+1)(y+1)(z+1) = 8 ⟹ x + y + z ≥ 3."

This is a mathematical proof problem. The goal is to show that for all real numbers \(x\), \(y\), and \(z\) in \(ℝ³\), if the product \((x+1)(y+1)(z+1) = 8\), then the sum \(x + y + z \geq 3\).

### Solution Outline:

1. **Step 1: Set new variables**  
   Let’s introduce new variables:  
   \[
   a = x + 1,\quad b = y + 1,\quad c = z + 1
   \]
   This converts the equation \((x+1)(y+1)(z+1) = 8\) into:  
   \[
   abc = 8
   \]
   Now, we aim to prove that:  
   \[
   (a - 1) + (b - 1) + (c - 1) \geq 3
   \]
   This simplifies to:  
   \[
   a + b + c \geq 6
   \]

2. **Step 2: Apply the Arithmetic-Geometric Mean Inequality (AM-GM inequality)**  
   By the AM-GM inequality, for any three non-negative numbers \(a\), \(b\), and \(c\):  
   \[
   \frac{a + b + c}{3} \geq \sqrt[3]{abc}
   \]
   Substituting \(abc = 8\) into this inequality:  
   \[
   \frac{a + b + c}{3} \geq \sqrt[3]{8} = 2
   \]
   Multiplying both sides by 3:  
   \[
   a + b + c \geq 6
   \]

3. **Step 3: Conclusion**  
   Since \(a + b + c \geq 6\), this implies that:  
   \[
   (x+1) + (y+1) + (z+1) \geq 6
   \]
   Therefore, \(x + y + z \geq 3\), which is what we wanted to prove.

Thus, the inequality is proven.

Would you like further details or have any questions?

Here are 5 related questions to further explore this topic:
1. How does the AM-GM inequality work for more than three variables?
2. Can you generalize the proof for \(n\) variables?
3. What are other inequalities related to sums and products, such as Hölder's inequality?
4. Can you solve the inequality using a different method, like direct substitution or factorization?
5. How does this proof change if the domain is restricted to positive real numbers?

**Tip:** When dealing with products or sums, the AM-GM inequality is a powerful tool to compare them efficiently.

Montrer que : ∀(x, y, z) ∈ ℝ³ ; (x+1)(y+1)(z+1) = 8 ⟹ x + y + z ≥ 3.

Explore how to prove the algebraic inequality (x+1)(y+1)(z+1) = 8 ⟹ x + y + z ≥ 3 using the Arithmetic-Geometric Mean Inequality. This guide includes a step-by-step approach to transform the problem and apply AM-GM, suitable for university-level students.

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