Inverse Relationships in Arithmetic
An inverse relationship exists when two operations undo each other. In arithmetic, there are two pairs of inverse operations.
Addition and Subtraction
Adding and subtracting are inverses. Adding 7 and then subtracting 7 brings you back to where you started.
15 + 7 = 22 → 22 − 7 = 15 ✓
Multiplication and Division
8 × 6 = 48 → 48 ÷ 6 = 8 ✓
Using Inverses to Solve Equations
| Equation | Inverse used | Solution |
|---|---|---|
| x + 9 = 17 | Subtract 9 from both sides | x = 8 |
| x − 5 = 12 | Add 5 to both sides | x = 17 |
| 4x = 28 | Divide both sides by 4 | x = 7 |
| x ÷ 6 = 9 | Multiply both sides by 6 | x = 54 |
Using Inverses to Check Answers
After any calculation, apply the inverse to verify your answer.
Check 345 + 278 = 623: 623 − 278 = 345 ✓
Check 56 × 8 = 448: 448 ÷ 8 = 56 ✓
Key Takeaways
- Addition and subtraction are inverse operations.
- Multiplication and division are inverse operations.
- Inverses are used to solve equations (do the opposite to isolate the unknown).
- Inverses are the most powerful checking tool available.
Practice Questions
- Use the inverse to check: 247 + 153 = 400.
- Solve: x + 14 = 31.
- Solve: 7x = 63.
- Use the inverse to check: 36 × 25 = 900.
- Solve: x ÷ 8 = 12.