Rules for multiplying integers with signs and examples.

The image you've uploaded presents a clear and helpful chart titled "Rules for Multiplying Integers", which outlines how to determine the sign of the product when multiplying two integers. Let's break down the rules and explain each one with reasoning.

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🔷 Understanding the Rules

The chart is divided into three main rows based on the signs of the integers being multiplied:

| Signs of Integers | Operation to Use | Answer Sign | Quick Example |
|-------------------|------------------|-------------|---------------|
| (+) × (+) | Multiply | Positive (+) | 5 × 7 = 35 |
| (−) × (−) | Multiply | Positive (+) | (−6) × (−4) = 24 |
| (+) × (−) | Multiply | Negative (−) | 3 × (−5) = −15 |

Let’s go through each rule in detail.

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✔ Rule 1: Positive × Positive = Positive

- When you multiply two positive integers, the result is positive.
- Example:
$$
5 \times 7 = 35
$$
Both numbers are positive, so the answer is positive.

> 💡 Why? Think of it as repeated addition: adding 5 seven times gives +35.

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✔ Rule 2: Negative × Negative = Positive

- When you multiply two negative integers, the result is positive.
- Example:
$$
(-6) \times (-4) = 24
$$
The product is positive.

> 💡 Why? This might seem counterintuitive at first, but think of it this way:
>
> - A negative times a negative represents "the opposite of a negative," which becomes positive.
> - In real-world terms: if you owe someone $6 and that debt is canceled four times, you gain $24.

> Mathematically, this follows from the distributive property and consistency in algebra.

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✔ Rule 3: Positive × Negative = Negative

- When you multiply a positive and a negative integer, the result is negative.
- Example:
$$
3 \times (-5) = -15
$$
The answer is negative.

> 💡 Why? You're adding a negative value multiple times. For example:
>
> $$
3 \times (-5) = (-5) + (-5) + (-5) = -15
> $$

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🧠 Summary of the Rules

You can remember the sign rules using this simple mnemonic:

> "Same signs give positive, different signs give negative."

- (+) × (+) → (+)
- (−) × (−) → (+)
- (+) × (−) or (−) × (+) → (−)

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📌 Final Note:

This chart is excellent for learning basic integer multiplication. It emphasizes that the operation is always multiplication, but the sign of the result depends on the signs of the numbers being multiplied.

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✔ Answer:
The rules are correctly presented in the image. The key takeaway is:
- Like signs → Positive result
- Unlike signs → Negative result

These rules apply universally to all integer multiplication problems.