We are tasked with determining whether the inequality \( \pi < 4 \Leftrightarrow \pi^2 < 16 \) is true.

### Step 1: Analyzing \( \pi < 4 \)
We know that \( \pi \approx 3.14159 \), so:
\[
\pi < 4
\]
This inequality is true because \( 3.14159 < 4 \).

### Step 2: Analyzing \( \pi^2 < 16 \)
Now, let's square \( \pi \) and compare it with 16:
\[
\pi^2 \approx (3.14159)^2 = 9.8696
\]
Since \( 9.8696 < 16 \), this inequality is also true.

### Conclusion:
Both \( \pi < 4 \) and \( \pi^2 < 16 \) are true, so the inequality \( \pi < 4 \Leftrightarrow \pi^2 < 16 \) holds true.

Would you like further details on this, or do you have any questions?

Here are 5 related questions to expand on this concept:
1. What is the numerical value of \( \pi^3 \), and how does it compare to 16?
2. Can \( \pi^2 \) ever be equal to 16 in any mathematical context?
3. How does \( \pi \)'s approximate value affect its use in inequalities?
4. What would happen if the inequalities were strict, such as \( \pi \leq 4 \)?
5. How do transcendental numbers like \( \pi \) differ from rational numbers in inequalities?

**Tip:** When working with approximations like \( \pi \), always consider how precise the approximation is and its impact on inequality results.

Explore the mathematical reasoning behind the inequality π < 4 and its relation to π² < 16. This analysis includes key concepts like transcendental numbers and inequalities, and explains the step-by-step solution. Suitable for high school students in Grades 9-12.

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