∑∫ax²+bx+c=0π√x

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By **MathPal Editorial Team**·Published Apr 10, 2026·Updated Apr 14, 2026

> **Grade:** 6-7 \| **Topic:** Arithmetic

## What You Will Learn

By the end of this page you will be able to add any two integers -- whether both are positive, both are negative, or one of each -- using clear sign rules. You will also master subtracting integers by converting every subtraction into an addition problem, and you will feel confident applying these skills to multi-step calculations.

## Theory

### Why integer addition needs special rules

When you first learned addition, every number was positive. Adding 3 + 5 just meant "move 5 steps to the right on a number line." Negative numbers add a new direction -- moving to the **left**. That is why we need rules that account for the **signs** of the numbers.

### Rule 1 -- Adding integers with the same sign

When both numbers have the **same sign**, add their absolute values and keep the common sign.

Same signs: (+a) + (+b) = +(a+b)

Same signs: (−a) + (−b) = −(a+b)

**Example:** (−7) + (−5) = −(7+5) = −12

### Rule 2 -- Adding integers with different signs

When the signs are **different**, subtract the smaller absolute value from the larger one, then take the sign of the number whose absolute value is larger.

**Example:** (−9) + 4 = −(9−4) = −5

**Example:** (−3) + 10 = +(10−3) = 7

### Rule 3 -- Subtracting integers (add the opposite)

Every subtraction problem can be rewritten as an addition problem. Change the subtraction sign to addition and flip the sign of the number being subtracted:

a − b = a + (−b)  
a − (−b) = a + (+b) = a + b

### Quick reference table

| Expression type | Rewrite as         | Then apply                                                         |
| ---------------- | ------------------- | ------------------------------------------------------------------ |
| a + b (same signs) | --                   | Rule 1: add absolute values, keep sign                              |
| a + b (different signs) | --                   | Rule 2: subtract absolute values, keep sign of larger              |
| a − b          | a + (−b)            | Rule 1 or 2                                                       |
| a − (−b)        | a + b               | Rule 1 or 2                                                       |

### Number line approach

You can always verify on a number line:
- **Adding a positive** number: move **right**.
- **Adding a negative** number: move **left**.
- **Subtracting** is the same as adding the opposite direction.

Starting at −2 and adding +5 means moving 5 steps right, landing on 3: (−2) + 5 = 3.

## Worked Examples

### Example 1: Adding two negative integers (easy)

**Problem:** Calculate (−12) + (−8).

**Step 1:** Both signs are negative (same sign), so add the absolute values. 12 + 8 = 20.

**Step 2:** Keep the common sign (negative).

**Answer:** (−12) + (−8) = −20.

### Example 2: Adding integers with different signs (easy)

**Problem:** Calculate (−15) + 9.

**Step 1:** The signs are different, so subtract the smaller absolute value from the larger one. 15 − 9 = 6.

**Step 2:** The larger absolute value belongs to −15 (negative), so the result is negative.

**Answer:** (−15) + 9 = −6.

### Example 3: Subtracting a positive integer (medium)

**Problem:** Calculate 4 − 13.

**Step 1:** Rewrite as addition. 4 − 13 = 4 + (−13).

**Step 2:** Different signs -- subtract absolute values. 13 − 4 = 9.

**Step 3:** The larger absolute value belongs to −13 (negative), so the result is negative.

**Answer:** 4 − 13 = −9.

## Common Mistakes

**Mistake 1: Forgetting to flip the sign when subtracting a negative**

❌ 8 − (−5) = 8 − 5 = 3

✅ 8 − (−5) = 8 + 5 = 13

**Mistake 2: Applying multiplication sign rules to addition**

❌ (−4) + (−6) = +10

✅ (−4) + (−6) = −10

**Mistake 3: Losing track of signs in multi-step problems**

❌ Solving 5 − 8 + 3 − (−2) left-to-right without rewriting:  
5 − 8 = −3, −3 + 3 = 0, 0 − (−2) ... confusion

✅ Rewrite all subtractions first: 5 + (−8) + 3 + 2. Group: positives = 10, negatives = −8. Total = 2.

## Practice Problems

Try these on your own before checking the answers:

1. (−9) + (−7)
2. (−11) + 18
3. 15 − 23
4. (−4) − (−12)
5. (−6) + 10 − (−3) − 8

## Summary

- **Same signs:** add absolute values, keep the common sign.
- **Different signs:** subtract absolute values, take the sign of the number with the larger absolute value.
- **Subtraction rule:** always rewrite a − b as a + (−b) before computing.
- **Double negative:** a − (−b) = a + b -- subtracting a negative is the same as adding.
- For multi-step problems, **rewrite all subtractions as additions first**, then group positives and negatives separately.