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PSAT Prep: Help and Review18 chapters | 194 lessons

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Lesson Transcript

Instructor:
*Jennifer Beddoe*

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.

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.

There are four mathematical properties that deal with inverses.

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

7 + 0 = 7

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

13 * 1 = 13

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

2*x* + 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 2*x* = 14. The next step is to divide both sides by 2, since division is the opposite of multiplication. 2*x* / 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

All algebraic problems can be solved in this manner. Let's try another example.

Solve for *x*: *x* / 4 + 9 = 13

First, subtract 9 from both sides: *x* / 4 = 4. Then multiply each side by 4. This gives you *x* = 16.

There are other inverse operations that can help you solve more difficult problems:

- The inverse of an exponent is the root.

3^2 = 9

the square root of 9 = 3 - The inverse of a logarithm is 10^
*x*.

log7 = 0.845

10^0.845 = 7

Trigonometric functions such as sine, cosine, and tangent also have inverses. They are called the arcsine, arccosine, and arctangent. Inverse trigonometric functions can be used to solve for the angle of a right triangle when you only know the lengths of at least two sides.

For example, say you need to reach the second story window of your house, which is 10 feet off the ground. Your ladder is 18 feet tall. At what angle (*x*) do you have to place your ladder to reach the window?

Since you know the hypotenuse and the side opposite the angle, you need to use the sin equation, which is sin(*x*) = opposite / hypotenuse.

sin(*x*) = 10 / 18

To solve for *x*, we need to use the arcsine (sin^-1).

*x* = sin^(-1) * (10 / 18)

*x* = 34 degrees.

Mathematically, **inverse operations** are opposite operations. Addition is the opposite of subtraction; division is the opposite of multiplication, and so on. Inverse operations are used to solve simple algebraic equations to more difficult equations that involve exponents, logarithms, and trigonometry.

Once you are done, you should be able to:

- Identify the inverse operations
- State the properties of inverse operations
- Solve an algebraic problem using inverse operations

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PSAT Prep: Help and Review18 chapters | 194 lessons

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