Inverse Property of Addition — Definition, Formula & Examples

The Inverse Property of Addition states that every number has an opposite (called its additive inverse), and when you add the two together, the result is always zero.

For every real number $a$ , there exists a unique real number $-a$ such that $a + (-a) = 0$ and $(-a) + a = 0$ , where $0$ is the additive identity.

Key Formula

a + (-a) = 0

Where:

$a$ = Any real number
$-a$ = The additive inverse (opposite) of a

How It Works

To use this property, identify the additive inverse of any number — just flip its sign. The additive inverse of

7

-7

, and the additive inverse of

-3

3

. This property is essential when solving equations because adding the inverse of a term to both sides cancels that term out, isolating the variable.

Worked Example

Problem: Solve for x: x + 9 = 14

Identify the inverse: The additive inverse of 9 is −9.

Add the inverse to both sides: Add −9 to both sides of the equation.

x + 9 + (-9) = 14 + (-9)

Apply the Inverse Property: On the left side, 9 + (−9) = 0 by the Inverse Property of Addition, leaving x alone.

x + 0 = 5

Answer:

x = 5

Why It Matters

This property is the reason you can "move" terms across the equals sign when solving equations. Every time you subtract a number from both sides, you are really adding its inverse. You will rely on it constantly in algebra, calculus, and any field that uses equations.

Common Mistakes

Mistake: Confusing the additive inverse with the multiplicative inverse (reciprocal). For example, thinking the inverse of 5 is 1/5.

Correction: The additive inverse of 5 is −5 (because 5 + (−5) = 0). The reciprocal 1/5 is the multiplicative inverse (because 5 × 1/5 = 1). These are two different properties.