Inverse Operations Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Inverse Operations.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Operations that undo each other: addition undoes subtraction, multiplication undoes division, and vice versa. Applying an operation followed by its inverse returns you to the starting value.

Adding 5 then subtracting 5 brings you back to where you started.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Inverse operations reverse each other so applying one then its inverse returns the start value.

Common stuck point: The procedure for inverse operations is the easy part; the trap is undoing multiplication with subtraction. Asking "Am I applying an operation to cancel a previous one and return to the start?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I applying an operation to cancel a previous one and return to the start?

Worked Examples

Example 1

easy

Show that adding and subtracting the same number leaves you where you started: start with 15, add 8, then subtract 8. What do you get?

Answer

15 (back to the start)

First step

Start: 15.

Full solution

2
Add 8: $15 + 8 = 23$ .
3
Subtract 8: $23 - 8 = 15$ .
4
You are back to 15. This shows $a + b - b = a$ .

Addition and subtraction are inverse operations. Adding then subtracting the same number always returns you to the original value.

Example 2

medium

You start with a mystery number. You multiply it by 6 and get 54. Use the inverse operation to find the mystery number.

Example 3

easy

A magician thinks of a number, adds

9

, and tells you the result is

23

. Use the inverse operation to find the magician's number.

Example 4

medium

To find the side length of a square with area

81

sq cm, what inverse operation undoes squaring?

Example 5

medium

A baker triples a recipe, then adds

4

extra cookies. The final batch has

40

cookies. How big was the original recipe?

Example 6

hard

A function machine doubles its input, then adds

3

. If the output is

19

, find the input and explain the inverse machine.

Example 7

hard

A student claims the inverse of 'multiply by

4

then add

6

' is 'add

6

then divide by

4

.' Find the mistake and give the correct inverse.

Example 8

challenge

A number is first squared, then

5

is added, giving

30

. List every starting number using inverse operations.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Start with 20. Subtract 7, then add 7. What is the result? What property does this show?

Example 2

medium

A number divided by 4 equals 8. What is the number? Use inverse operations.

Example 3

easy

What operation undoes adding 8?

Example 4

easy

What operation undoes multiplying by 5?

Example 5

easy

Start with 12, add 7, then subtract 7. What is the result?

Example 6

easy

Solve for

x

x - 4 = 10

Example 7

easy

Solve for

x

3x = 21

Example 8

easy

What is the inverse operation of dividing by 6?

Example 9

easy

Start with 9, multiply by 4, then divide by 4. What is the result?

Example 10

easy

What is the inverse operation of squaring a positive number?

Example 11

medium

Solve for

x

2x + 5 = 17

Example 12

medium

A number is multiplied by 3 and then 7 is added, giving 28. Find the number.

Example 13

medium

Solve for

x

\frac{x}{4} - 3 = 5

Example 14

medium

Solve for

x

5x - 2 = 3x + 8

Example 15

medium

Undo the process: take a number, divide by 2, then add 6, result is 10. Find it.

Example 16

medium

Solve for

x

7 - x = 2

Example 17

medium

A bank account had some money, then

40

was withdrawn and later

25

deposited, leaving

\$110

. What was the start?

Example 18

medium

Solve for

x

\frac{2x+1}{3} = 5

Example 19

medium

f(x)=x+9

, what input gives output 4?

Example 20

challenge

Solve for

x

\sqrt{x-3}=4

Example 21

challenge

A machine doubles a number then subtracts 3; running its inverse on 11 gives what?

Example 22

challenge

For which value of

x

does

x^2-5

return to

x

after taking

\sqrt{x^2-5+5}

? Show the inverse pair undoes the square for

x\ge 0

Example 23

easy

What operation undoes subtracting

12

Example 24

easy

Start with

25

, divide by

5

, then multiply by

5

. What is the result?

Example 25

easy

Solve for

x

x + 17 = 25

Example 26

easy

Solve for

y

y \div 7 = 6

Example 27

medium

Solve for

x

4x - 9 = 15

Example 28

medium

I think of a number, multiply by

5

, then subtract

4

, and get

36

. What was my number?

Example 29

medium

Solve for

x

\dfrac{x+5}{2} = 9

Example 30

medium

Maria has some marbles. She gives away

7

, then her friend gives her

15

more. She now has

32

. How many did she start with?

Example 31

medium

Solve for

n

\dfrac{n}{3} + 4 = 10

Example 32

medium

Solve for

x

6(x - 2) = 30

Example 33

medium

What is the inverse of subtracting

3

then dividing by

4

? Give the steps in order.

Example 34

hard

Solve for

x

\dfrac{2x - 1}{5} = 3

Example 35

hard

Solve for

x

\sqrt{x + 7} = 5

Example 36

hard

Solve for

x

3x + 8 = 5x - 4

Example 37

hard

Solve for

x

2(3x - 1) + 4 = 20

Example 38

challenge

A machine

f

does:

x \to (x+1)^2

. Find every input

x

that the machine sends to

9

, using inverse operations.

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Related Concepts

Addition Subtraction Multiplication Division Solving Linear Equations

Background Knowledge

These ideas may be useful before you work through the harder examples.

additionsubtractionmultiplicationdivision