Inverse Operations in Math: Definition & Examples

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  • 0:05 What are Inverse Operations
  • 1:01 Properties of inverses
  • 1:54 How to Use Inverse Operations
  • 3:05 Other Inverse Operations
  • 4:29 Lesson Summary
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Lesson Transcript
Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

In mathematics, inverse operations are operations that 'undo' each other. Most operations have an inverse. This lesson describes the most common operations and their inverses, and it provides some examples and a quiz to cement your knowledge.

What are Inverse Operations?

The word 'inverse' means reverse in direction or position. It comes from the Latin word 'inversus,' which means to turn upside down or inside out. In mathematics, an inverse operation is an operation that undoes what was done by the previous operation.

The four main mathematical operations are addition, subtraction, multiplication, division. The inverse of addition is subtraction and vice versa. The inverse of multiplication is division and vice versa. Let's look at some examples to show how inversion works.

Take this simple addition problem: 4 + 3 = 7. If we want to reverse the addition, we just subtract 7 - 3 = 4 and we are back to where we started. The same is true for multiplication and division: 2 * 8 = 16 and 16 / 8 = 2. These are very simple examples, but the rule holds true for even complex addition, subtraction, multiplication, and division problems.

Properties of Inverses

There are four mathematical properties that deal with inverses.

The Additive Inverse Property

The additive inverse property states that when you add a number to its opposite, the result is always 0.

2 + (-2) = 0
369 + (-369) = 0

The Multiplicative Inverse Property

The multiplicative inverse property states that when you multiply any number by its opposite, the result is always 1.

6 * 1/6 = 1
213 * 1/213 = 1

The Additive Property

The additive property states that when you add any number to zero, the result is the same number.

7 + 0 = 7

The Multiplicative Property

The multiplicative property states that any time you multiply a number by 1, the number does not change.

13 * 1 = 13

How to Use Inverse Operations

Inverse operations can be used to solve algebraic problems. Let's solve for x:

2x + 3 = 17

In order to solve this problem, we must isolate the x on one side of the equation. The first step is to remember that the inverse operations of addition and multiplication are subtraction and division. The next step is to 'move' the 3 to the right side of the equation by subtracting it from both sides of the equation. This gives you 2x = 14. The next step is to divide both sides by 2, since division is the opposite of multiplication. 2x / 2 = 14 / 2. This gives you x = 7.

The answer to this problem is x = 7. If you are unsure, you can always go back and check your answer. To do this, substitute 7 for x in the original problem.

2(7) + 3 = 17
Then solve 14 + 3 = 17
17 = 17

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