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The mathematical statement provided in the image is as follows:

"For every \( x > 0 \), there exists a \( y \) such that \( y > \frac{1}{x} \)."

### Explanation:
This is a mathematical assertion involving existential and universal quantifiers:
- **For every** \( x > 0 \): This means that the statement applies to all positive values of \( x \).
- **There exists a** \( y \): For each \( x \), we can find some \( y \).
- **Such that** \( y > \frac{1}{x} \): The value of \( y \) must be greater than \( \frac{1}{x} \).

### Meaning:
This implies that for any positive \( x \), no matter how small, we can always find a corresponding value of \( y \) that is greater than \( \frac{1}{x} \). The statement is true because you can always find such a value of \( y \), for instance, \( y = \frac{1}{x} + 1 \), which will always be greater than \( \frac{1}{x} \).

### Would you like further details or clarification on this?

Here are five related questions that can expand your understanding:
1. What are universal and existential quantifiers in mathematics?
2. Can you think of a real-world example where such inequalities arise?
3. What if the condition was changed to \( y \geq \frac{1}{x} \)? How would that affect the statement?
4. What is the geometric interpretation of \( y > \frac{1}{x} \) for positive \( x \)?
5. How does this statement relate to the concept of limits in calculus?

**Tip:** When working with inequalities, it's often useful to think about examples and how they behave at extreme values (e.g., when \( x \to 0^+ \) or \( x \to \infty \)).

For every x > 0, there exists a y such that y > 1/x.

Explore the mathematical statement 'For every x > 0, there exists a y such that y > 1/x' and understand how existential and universal quantifiers work in inequalities. Suitable for high school students in Grades 10-12.

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