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The text in the image asks to demonstrate a particular logical statement regarding inequalities. It translates to the following problem in English:

**"Show that if \( y \geq z \), \( x \geq y \), and \( z \geq x \), then \( y = z = x \)."**

### Solution:
We are given three conditions:
1. \( y \geq z \)
2. \( x \geq y \)
3. \( z \geq x \)

We need to show that from these conditions, it follows that \( y = z = x \).

#### Step-by-step breakdown:
1. **From conditions (1) and (3):**
   - From \( y \geq z \) and \( z \geq x \), we can combine the inequalities:  
     \[ y \geq z \geq x. \]

2. **From condition (2):**
   - We are also given that \( x \geq y \).

3. **Combining (1), (2), and (3):**
   - From \( x \geq y \) and \( y \geq z \geq x \), we deduce:  
     \[ x \geq y \geq z \geq x. \]

4. **Concluding equality:**
   - Since \( x \geq y \geq z \geq x \), it follows that all three values must be equal. Hence,  
     \[ x = y = z. \]

Therefore, we have shown that if \( y \geq z \), \( x \geq y \), and \( z \geq x \), then \( y = z = x \).

Would you like any further details or have any questions?

Here are some related questions for further exploration:
1. What other inequalities can lead to equality under similar conditions?
2. How would this proof change if one of the inequalities was strict (e.g., \( y > z \))?
3. Can you generalize this result for more variables, such as \( x_1, x_2, \dots, x_n \)?
4. How does this concept apply in optimization problems?
5. How does the concept of transitivity play a role in this proof?

**Tip:** When working with inequalities, combining them systematically and checking for contradictions or redundant conditions can help clarify relationships between variables.

Show that if y ≥ z, x ≥ y, and z ≥ x, then y = z = x.

Explore how to prove equality when given inequalities y ≥ z, x ≥ y, and z ≥ x. This problem demonstrates the transitive property of inequalities, providing a step-by-step algebraic approach. Suitable for students in Grades 10-12.

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