1.3 Ordering Real Numbers Compare. Write $$ $$, or $$ = $$. $$ \begin{array}{ll} ext { 11. } \sqrt{8}+3 \ ext { Order the numbers from least to greatest. } \ ext { 13. } \sqrt{99}, \pi^{2}, 9 . \overline{8} & ext { 12. } \sqrt{5}+11 \ ext { (2) } & \ ext { ESSENTIAL QUESTION } & ext { 14. } \sqrt{\frac{1}{25}, \frac{1}{4}, 0 . \overline{2}} \ ext { 15. How are real numbers used to describe real-world situations? }\end{array} $$

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1.3 Ordering Real Numbers Compare. Write $$ <,> $$, or $$ = $$. $$ \begin{array}{ll}	ext { 11. } \sqrt{8}+3 \ 	ext { Order the numbers from least to greatest. } \ 	ext { 13. } \sqrt{99}, \pi^{2}, 9 . \overline{8} & 	ext { 12. } \sqrt{5}+11 \ 	ext { (2) } & \ 	ext { ESSENTIAL QUESTION } & 	ext { 14. } \sqrt{\frac{1}{25}, \frac{1}{4}, 0 . \overline{2}} \ 	ext { 15. How are real numbers used to describe real-world situations? }\end{array} $$

UpStudy AI · Accepted Answer

Certainly! Let's address each problem step by step to compare and order the given real numbers. Finally, we'll explore how real numbers are applied in real-world contexts.

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### **Problem 11: Compare and Write $$ <, >, $$ or $$ = $$**
**Expression:** $$ \sqrt{8} + 3 $$

Since there's no second value provided for comparison, we'll evaluate the expression:

$$
\sqrt{8} + 3 \approx 2.828 + 3 = 5.828
$$

*If you have another number to compare with $$ \sqrt{8} + 3 $$, please provide it, and I can help you determine the relationship.*

---

### **Problem 12: Evaluate $$ \sqrt{5} + 11 $$**
$$
\sqrt{5} + 11 \approx 2.236 + 11 = 13.236
$$

*This is a numerical evaluation. If you need to compare it with another number, please provide that number.*

---

### **Problem 13: Order the Numbers from Least to Greatest**
**Numbers to Order:** $$ \sqrt{99}, \pi^{2}, 9.\overline{8} $$

Let's calculate each value approximately:

1. $$ \sqrt{99} \approx 9.9499 $$
2. $$ \pi^{2} \approx 3.1416^2 \approx 9.8696 $$
3. $$ 9.\overline{8} = 9.8888\ldots $$

**Ordering:**
$$
\pi^{2} \approx 9.8696 < 9.\overline{8} = 9.8888\ldots < \sqrt{99} \approx 9.9499
$$

**From Least to Greatest:**
$$
\pi^{2} < 9.\overline{8} < \sqrt{99}
$$

---

### **Problem 14: Order the Numbers from Least to Greatest**
**Numbers to Order:** $$ \sqrt{\frac{1}{25}}, \frac{1}{4}, 0.\overline{2} $$

Let's evaluate each:

1. $$ \sqrt{\frac{1}{25}} = \frac{1}{5} = 0.2 $$
2. $$ \frac{1}{4} = 0.25 $$
3. $$ 0.\overline{2} = 0.2222\ldots $$

**Ordering:**
$$
0.2 < 0.\overline{2} < 0.25
$$

**From Least to Greatest:**
$$
\sqrt{\frac{1}{25}} < 0.\overline{2} < \frac{1}{4}
$$

---

### **Problem 15: How Are Real Numbers Used to Describe Real-World Situations?**

**Real-world applications of real numbers include:**

- **Measurement:** Quantifying lengths, weights, volumes, and temperatures using integers, fractions, and decimals.
- **Finance:** Calculating prices, interest rates, budgets, and financial forecasting with decimal numbers and percentages.
- **Engineering and Construction:** Designing structures requires precise measurements and calculations involving real numbers.
- **Science:** Real numbers are essential in experiments, data analysis, and representing scientific quantities like velocity, mass, and energy.
- **Everyday Activities:** Cooking recipes, timekeeping, and managing personal schedules involve the use of real numbers for accuracy and efficiency.

**Example:** When shopping, you use real numbers to compare prices, calculate discounts, and manage your budget to make informed purchasing decisions.

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If you have any further questions or need additional clarification on these problems, feel free to ask!
**Comparing and Ordering Real Numbers** 1. **Problem 11:** - **Expression:** $$ \sqrt{8} + 3 $$ - **Approximate Value:** $$ 5.828 $$ - **Note:** No second number provided for comparison. 2. **Problem 12:** - **Expression:** $$ \sqrt{5} + 11 $$ - **Approximate Value:** $$ 13.236 $$ - **Note:** No second number provided for comparison. 3. **Problem 13:** - **Numbers to Order:** $$ \sqrt{99}, \pi^{2}, 9.\overline{8} $$ - **Approximate Values:** - $$ \sqrt{99} \approx 9.9499 $$ - $$ \pi^{2} \approx 9.8696 $$ - $$ 9.\overline{8} = 9.8888\ldots $$ - **Order from Least to Greatest:** $$ \pi^{2} < 9.\overline{8} < \sqrt{99} $$ 4. **Problem 14:** - **Numbers to Order:** $$ \sqrt{\frac{1}{25}}, \frac{1}{4}, 0.\overline{2} $$ - **Approximate Values:** - $$ \sqrt{\frac{1}{25}} = 0.2 $$ - $$ \frac{1}{4} = 0.25 $$ - $$ 0.\overline{2} = 0.2222\ldots $$ - **Order from Least to Greatest:** $$ \sqrt{\frac{1}{25}} < 0.\overline{2} < \frac{1}{4} $$ 5. **Problem 15:** - **Real-World Applications of Real Numbers:** - **Measurement:** Quantifying lengths, weights, volumes, and temperatures. - **Finance:** Calculating prices, interest rates, and budgets. - **Engineering and Construction:** Designing structures with precise measurements. - **Science:** Representing quantities like velocity and mass. - **Everyday Activities:** Cooking, timekeeping, and managing personal schedules. **Summary:** - Calculated approximate values for expressions involving square roots and pi. - Ordered numbers from least to greatest based on their approximate values. - Explained the diverse applications of real numbers in various real-world scenarios. If you need further assistance or have additional questions, feel free to ask!

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