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Inverse Properties of Addition and Multiplication

Solving mathematical equations often involves applying the opposite of a number or operation to cancel it out in order to get the variable isolated on one side of the equation. The opposite operation of addition is subtraction and the opposite of multiplication is division. But the identity properties and inverse properties present additional opposites that present not a method of elimination, but a method of solving for a desired answer.

Addition Identity and Inverse Properties
- The addition identity property states that x + 0 = x, meaning that any number plus 0 equals the number itself. The inverse property solves for the identity property, which is 0 in this case. Using algebra to get all instances of the variable on one side, the property becomes x + -x = 0. That means that a positive number plus its negative equivalent equals 0.
Example
- Use the equation 3x = 3x + 0 for an example. Subtract 3x from both sides to get the instances of the variable on the same side of the equation: 3x - 3x = 3x - 3x + 0 becomes 3x + -3x = 0 becomes 0 = 0.
Multiplication Identity and Inverse Properties
- The multiplication identity property states that x * 1 = x, or a number multiplied by 1 equals the number itself. The inverse property sets this formula equal to the identity property value, which is 1 in this case. Divide both sides of the equation by x: (x * 1) / x = x / x becomes x * 1/x = 1. This means that a number multiplied by an inverse fraction with 1 in the numerator and the number in the denominator equals 1.
Example
- Use the equation 5x = 5x as an example. Divide both sides by 5x, which is the same as multiplying them by (1/5x): 5x * (1/5x) = 5x * (1/5x) becomes 1 = 1.

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