Clear educational chart outlining the rules for arithmetic with rational numbers, including subtraction and sign rules for multiplication and division.

Actually, you haven’t uploaded an image with a *task* — what you’ve shared is an educational reference sheet titled “Rules for Rational Numbers.” It explains foundational rules for arithmetic with rational numbers, particularly subtraction and multiplication/division involving positive and negative numbers.

Since there’s no specific problem or question attached (like “Solve 5 - (-3)” or “What is the sign of (-4) × 7?”), I’ll instead explain the content clearly and provide examples so you can apply these rules to any problem you encounter.

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## ✔ Explanation of the Rules

1. Subtraction Rule:

> In general: `a - b = a + (-b)`

This means subtracting a number is the same as adding its opposite (additive inverse).

🔹 Example:
`7 - 4 = 7 + (-4) = 3`
`-2 - 5 = -2 + (-5) = -7`
`8 - (-3) = 8 + 3 = 11` ← *Subtracting a negative becomes addition!*

📌 Important Note:
> *“The distance between two numbers is independent of the order, but the difference depends on the order.”*

- Distance: Always positive. Example: Distance between 3 and 7 is |7 - 3| = 4, same as |3 - 7| = 4.
- Difference: Order matters. 7 - 3 = 4, but 3 - 7 = -4.

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2. Multiplication & Division Sign Rules:

These rules tell you the sign of the result when multiplying or dividing positive and negative numbers.

#### ➤ Rule 1:
> Positive × Negative = Negative
> Positive ÷ Negative = Negative

🔹 Examples:
`6 × (-2) = -12`
`10 ÷ (-5) = -2`

#### ➤ Rule 2:
> Negative × Positive = Negative
> Negative ÷ Positive = Negative

🔹 Examples:
`(-3) × 4 = -12`
`(-15) ÷ 3 = -5`

#### ➤ Rule 3:
> Negative × Negative = Positive
> Negative ÷ Negative = Positive

🔹 Examples:
`(-4) × (-5) = 20`
`(-20) ÷ (-4) = 5`

💡 Mnemonic:
> “Same signs → Positive; Different signs → Negative”

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## 🧠 How to Apply These Rules

Let’s say you’re given a problem like:

> Problem: Solve `(-8) ÷ 2 + 3 × (-4)`

Step-by-step:

1. Follow Order of Operations (PEMDAS):
→ Do multiplication and division left to right first.

2. `(-8) ÷ 2 = -4` (Negative ÷ Positive = Negative)

3. `3 × (-4) = -12` (Positive × Negative = Negative)

4. Now add: `-4 + (-12) = -16`

✔ Final Answer: -16

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## 📌 Summary Cheat Sheet

| Operation | Signs | Result Sign |
|------------------|-------------------|-------------|
| Addition | (+) + (+) | + |
| | (-) + (-) | - |
| | (+) + (-) | Depends on magnitude |
| Subtraction | a - b = a + (-b) | Use addition rules |
| Multiplication | (+) × (+) | + |
| | (+) × (-) | - |
| | (-) × (+) | - |
| | (-) × (-) | + |
| Division | (+) ÷ (+) | + |
| | (+) ÷ (-) | - |
| | (-) ÷ (+) | - |
| | (-) ÷ (-) | + |

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If you have a specific problem you’d like solved using these rules, feel free to paste it here — I’ll walk you through it step by step!

Otherwise, this guide should help you confidently tackle any rational number arithmetic problem. 😊