Answer
**2. Inequalities and Their Solutions**
**a. Illustrate the Inequalities on a Number Line**
- **ii. \( x \geqslant 3 \):**
- Draw a number line.
- Place a closed circle at 3.
- Shade to the right of 3.
- **iii. \( 0 < x < 4 \):**
- Draw a number line.
- Place open circles at 0 and 4.
- Shade between 0 and 4.
**b. Smallest Integer Values That Satisfy Each Inequality**
- **ii. \( x \geqslant 3 \):** 3
- **iii. \( 0 < x < 4 \):** 1
**3. Ordering Fractions by Magnitude**
The fractions ordered from smallest to largest are:
\[ \frac{3}{14}, \frac{2}{5}, \frac{1}{2}, \frac{4}{7}, \frac{9}{10} \]
Solution
Sure, let's tackle each part of your questions step by step.
---
### **2. Inequalities and Their Solutions**
#### **a. Illustrate the Inequalities on a Number Line**
**ii. \( x \geqslant 3 \)**
- **Description:**
- Draw a horizontal line representing the number line.
- Place a **closed circle** at **3** to indicate that **3** is included in the solution.
- Shade the line **to the right** of **3** to represent all numbers greater than or equal to **3**.
- **Visualization:**
```
---|----●=================>
3
```
- The closed circle at **3** (●) shows that **3** is part of the solution.
- The arrow (=======>) indicates all numbers greater than **3**.
---
**iii. \( 0 < x < 4 \)**
- **Description:**
- Draw a horizontal line representing the number line.
- Place **open circles** at **0** and **4** to indicate that **0** and **4** are **not** included in the solution.
- Shade the line **between 0 and 4** to represent all numbers greater than **0** and less than **4**.
- **Visualization:**
```
<=================>
0 4
(○) (○)
```
- The open circles at **0** and **4** (○) show that these endpoints are excluded.
- The shading (======) between **0** and **4** represents the range of valid solutions.
---
#### **b. Smallest Integer Values That Satisfy Each Inequality**
- **ii. \( x \geqslant 3 \)**
- **Smallest Integer:** **3**
- Since **x** can be **3** or greater, the smallest integer satisfying the inequality is **3**.
- **iii. \( 0 < x < 4 \)**
- **Smallest Integer:** **1**
- The integers between **0** and **4** are **1**, **2**, and **3**. The smallest among these is **1**.
---
### **3. Ordering Fractions by Magnitude**
Let's order the following fractions from smallest to largest:
\[ \frac{4}{7}, \frac{3}{14}, \frac{9}{10}, \frac{1}{2}, \frac{2}{5} \]
**Step 1: Convert Fractions to Decimal Form**
- \( \frac{3}{14} \approx 0.214 \)
- \( \frac{2}{5} = 0.4 \)
- \( \frac{1}{2} = 0.5 \)
- \( \frac{4}{7} \approx 0.571 \)
- \( \frac{9}{10} = 0.9 \)
**Step 2: Order the Decimals from Smallest to Largest**
\[ 0.214 < 0.4 < 0.5 < 0.571 < 0.9 \]
**Step 3: Convert Back to Original Fractions**
\[ \frac{3}{14} < \frac{2}{5} < \frac{1}{2} < \frac{4}{7} < \frac{9}{10} \]
**Final Ordered List:**
\[ \boxed{ \frac{3}{14} < \frac{2}{5} < \frac{1}{2} < \frac{4}{7} < \frac{9}{10} } \]
---
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