https://file.aiquickdraw.com/mathbot/file/1729540981587df9om46i.jpg

Let's solve each of the inequalities step-by-step based on the image provided.

### 1. \( 3\pi \) versus \( \sqrt{80} \)

- Approximate values:
  - \( 3\pi \approx 3 \times 3.1416 = 9.4248 \)
  - \( \sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5} \approx 4 \times 2.236 = 8.944 \)
  
  So, \( 3\pi > \sqrt{80} \).

### 2. \( \sqrt{10} \) versus \( 2 \times \sqrt{5} \)

- Approximate values:
  - \( \sqrt{10} \approx 3.162 \)
  - \( 2 \times \sqrt{5} = 2 \times 2.236 \approx 4.472 \)
  
  So, \( \sqrt{10} < 2 \times \sqrt{5} \).

### 3. \( \sqrt{\frac{1}{10}} \) versus \( \sqrt{\frac{1}{8}} \)

- Approximate values:
  - \( \sqrt{\frac{1}{10}} = \frac{1}{\sqrt{10}} \approx 0.316 \)
  - \( \sqrt{\frac{1}{8}} = \frac{1}{\sqrt{8}} \approx 0.354 \)
  
  So, \( \sqrt{\frac{1}{10}} < \sqrt{\frac{1}{8}} \).

### 4. \( \pi \) versus \( \sqrt{8} \)

- Approximate values:
  - \( \pi \approx 3.1416 \)
  - \( \sqrt{8} = 2\sqrt{2} \approx 2 \times 1.414 \approx 2.828 \)
  
  So, \( \pi > \sqrt{8} \).

### Summary:

- \( 3\pi > \sqrt{80} \)
- \( \sqrt{10} < 2 \times \sqrt{5} \)
- \( \sqrt{\frac{1}{10}} < \sqrt{\frac{1}{8}} \)
- \( \pi > \sqrt{8} \)

Would you like a more detailed explanation for any of these comparisons?

Here are 5 related questions you might find helpful:
1. How do you approximate square roots without a calculator?
2. What is the relationship between powers of 10 and square roots?
3. How do you compare irrational numbers like \( \pi \) and \( e \)?
4. What are common approximations used for \( \pi \) in geometry problems?
5. How can we simplify square roots of fractions?

**Tip**: Always check for simplifications when comparing square roots, as this can make mental math easier!

Compare each pair of real numbers using the appropriate inequality sign.

Learn how to compare irrational numbers like π, square roots, and other real numbers. This step-by-step guide demonstrates comparisons using approximations and properties of inequalities, suitable for Grades 9-12.

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