Subtracting Integers - Rules, Steps, Examples - Free Printable

The image you've uploaded is a table explaining the rules for subtracting integers, presented by Cuemath. Let's go through it step-by-step to understand and explain the rules clearly.

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🔍 Objective:

Understand how to subtract integers using sign rules, especially when dealing with positive and negative numbers.

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📊 Table Breakdown:

| Integer's Sign | Operation | Sign | Example |
|----------------|-----------|------|---------|
| + – + | Subtract | + | 7 - 3 = 4 |
| + – (–) | Add | + | 7 - (-3) = 10 |
| (–) – (–) | Add | + | 3 - (-7) = 10 |
| + – (–) | Subtract | – | 3 - 7 = -4 |
| (–) – (–) | Subtract | – | -7 - (-3) = -4 |
| (–) – (–) | Subtract | + | -3 - (-7) = 4 |
| (–) – + | Add | – | -3 - 7 = -10 |
| (–) – + | Add | – | -7 - 3 = -10 |

Wait! There seems to be a typo or inconsistency in the table — specifically in the "Integer's Sign" column. Let’s fix that and clarify.

Actually, let's re-analyze each row properly based on standard integer subtraction rules.

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## ✔ Correct Understanding of Subtracting Integers

The key rule is:

> Subtracting a number is the same as adding its opposite.

So:
> $ a - b = a + (-b) $

This means:
- Subtracting a positive number → add a negative
- Subtracting a negative number → add a positive

Let’s now interpret each row correctly.

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🔁 Revised Table with Explanation

| Integer's Sign (First Number) | Operation | Second Number | Rule Applied | Result Sign | Example | Explanation |
|-------------------------------|-----------|----------------|--------------|-------------|--------|-------------|
| + | – | + | Subtract | + | 7 - 3 = 4 | 7 minus 3 → just regular subtraction |
| + | – | – | Add | + | 7 - (-3) = 10 | Subtracting negative → add positive: 7 + 3 = 10 |
| + | – | – | Add | + | 3 - (-7) = 10 | Same: 3 + 7 = 10 |
| + | – | + | Subtract | – | 3 - 7 = -4 | 3 minus 7 → negative result: -(7 - 3) = -4 |
| – | – | – | Subtract | – | -7 - (-3) = -4 | Subtracting negative → add positive: -7 + 3 = -4 |
| – | – | – | Subtract | + | -3 - (-7) = 4 | -3 + 7 = 4 |
| – | – | + | Add | – | -3 - 7 = -10 | -3 + (-7) = -10 |
| – | – | + | Add | – | -7 - 3 = -10 | -7 + (-3) = -10 |

Wait — there are duplicate rows in your original table. Let’s correct and simplify.

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✔ Corrected & Simplified Rules

We can summarize the rules for subtracting integers as:

#### 🚀 Rule:
> To subtract an integer, add its opposite.

That is:
$$
a - b = a + (-b)
$$

Let’s apply this to common cases:

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💡 Case-by-Case Examples:

1. Positive – Positive
- $ 7 - 3 = 7 + (-3) = 4 $
- Result: Positive if first > second; negative otherwise.

2. Positive – Negative
- $ 7 - (-3) = 7 + 3 = 10 $
- Subtracting a negative → add a positive → result is larger.

3. Negative – Positive
- $ -7 - 3 = -7 + (-3) = -10 $
- Subtracting a positive → add a negative → more negative.

4. Negative – Negative
- $ -7 - (-3) = -7 + 3 = -4 $
- Subtracting a negative → add a positive → less negative.

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🧠 Key Takeaway:

> "Two negatives make a positive" only applies when you're subtracting a negative number.

So:
- $ -(-3) = +3 $
- So $ -7 - (-3) = -7 + 3 = -4 $

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✔ Summary of Rules:

| Expression | Equivalent | Result |
|----------|------------|--------|
| $ + - + $ | Subtract | e.g., $ 7 - 3 = 4 $ |
| $ + - (-) $ | Add | e.g., $ 7 - (-3) = 10 $ |
| $ (-) - (+) $ | Add negative | e.g., $ -3 - 7 = -10 $ |
| $ (-) - (-) $ | Add positive | e.g., $ -3 - (-7) = 4 $ |

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📌 Final Answer:

The table shows the correct rules, but some entries are repeated or mislabeled.

✔ The core rule is:
> Subtracting an integer is the same as adding its additive inverse.

So:
- $ a - b = a + (-b) $
- This helps avoid confusion.

For example:
- $ 7 - (-3) = 7 + 3 = 10 $
- $ -3 - (-7) = -3 + 7 = 4 $
- $ -7 - 3 = -7 + (-3) = -10 $

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✔ Conclusion:

The table is mostly correct, but it should be interpreted using the additive inverse rule:
> Change subtraction to addition and change the sign of the second number.

Then follow standard addition rules.

This method works for all integer subtractions.

Let me know if you'd like a practice worksheet or visual explanation!