Inverse operations are operations that undo each other. Addition and subtraction are inverses: 9 + 6 − 6 = 9. Multiplication and division are inverses: 7 × 4 ÷ 4 = 7. You use them to check answers and to solve equations, undoing each step in reverse order until the variable is alone.
Inverse operations make round trips: add then subtract, or multiply then divide, and you are back where you started.
Formal definition and notation
Two operations are inverse operations when applying one and then the other with the same number returns the starting value. For every number a and b: (a + b) − b = a and (a − b) + b = a; for b ≠ 0: (a × b) ÷ b = a and (a ÷ b) × b = a. Squaring and taking the square root are inverses for a ≥ 0: √a2 = a.
To solve an equation, undo the operations in the reverse of the order of operations: undo addition and subtraction first, then multiplication and division.
Whatever you do to one side of an equation, do to the other side so it stays balanced.
Multiplying by 0 cannot be undone: 5 × 0 = 0 and 9 × 0 = 0, so dividing by 0 cannot tell them apart.
Squares and square roots are inverses for nonnegative numbers: √49 = 7 because 72 = 49.
Solving with inverse operations
Solving with inverse operations
If ax + b = c, then x = c − ba
b
added last when building the expression, so undone first (subtract it)
a
the multiplier, undone second (divide by it, a ≠ 0)
For 3x + 8 = 29: subtract 8 to get 3x = 21, then divide by 3 to get x = 7.
Try it: solve with inverse operations
Interactive
Solve 5x − 4 = 31 by doing the same inverse operation to both sides: first add 4, then divide by 5.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Inverse Operations examples, step by step
Example 1: Undo a subtraction
Warm-up
Solve n − 13 = 29.
13 is subtracted from n, so add 13 to both sides.
n = 29 + 13 = 42.
Check: 42 − 13 = 29.
Answer:n = 42
Example 2: Two steps, reverse order
Core
Solve 6x + 11 = 71.
Building 6x + 11, you multiply by 6 first and add 11 last. Undo the last step first.
Subtract 11: 6x = 60.
Divide by 6: x = 10.
Check: 6 × 10 + 11 = 71.
Answer:x = 10
Example 3: Work backward
Test-style
Jaden thinks of a number, divides it by 4, then subtracts 9. The result is 6. What was his number?
An additive inverse is a number, the opposite: the additive inverse of 8 is −8. Inverse operations are pairs of operations, such as adding 8 and subtracting 8.
A fact family is four facts from three numbers, like 4 × 5 = 20 and 20 ÷ 5 = 4. It is a way to practice inverse operations.
Quick check: 3 questions
Question 1 of 3
Which operation undoes multiplying by 8?
Show answer and explanation
Answer: C. Division by 8 reverses multiplication by 8: 5 × 8 ÷ 8 = 5. Adding or subtracting 8 undoes an addition, and multiplying again doubles the effect.
Question 2 of 3
Solve x3 + 2 = 9.
Show answer and explanation
Answer: A. Subtract 2 to get x3 = 7, then multiply by 3: x = 21. 33 multiplies before subtracting; 5 and 7/3 divide instead of multiplying.
Question 3 of 3
Which equation checks that 156 ÷ 12 = 13?
Show answer and explanation
Answer: D. Multiplication is the inverse of division, so multiply the quotient by the divisor: 13 × 12 = 156. The other choices use addition or subtraction, or multiply the wrong pair.
Inverse Operations: frequently asked questions
What are the four inverse operations?
There are two pairs. Addition and subtraction undo each other, and multiplication and division undo each other. In later courses you meet more pairs: squaring and square roots, cubing and cube roots, and exponents and logarithms, such as 25 = 32 and log2 32 = 5. Each pair works the same way: do one, then the other, and you get back the start.
Why do you undo operations in reverse order?
Think of putting on socks, then shoes. To undo it, take off the shoes first. In 2x + 7, the order of operations multiplies by 2 first and adds 7 last, so to solve 2x + 7 = 19 you subtract 7 first (2x = 12) and divide by 2 last (x = 6).
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 1.OA.B.4, CCSS 3.OA.B.6, CCSS 6.EE.B.7
Cite this entry
Effortless Math Team. “Inverse Operations.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/inverse-operations/
Flashcard
TermInverse Operations+ and −, × and ÷
DefinitionOperations that undo each other. Adding 6 then subtracting 6 gets you back: 9 + 6 − 6 = 9. Used to check answers and solve equations.
NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Square RootA square root of a number is a value that, multiplied by itself, gives that number. The square root of 36 is 6 because 6 × 6 = 36. Every positive number has two square roots, such as 6 and −6, but the radical sign √ means the principal square root, which is the non-negative one.Number System · Grade 8, Algebra 1SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4Order of OperationsThe order of operations is the agreed sequence for evaluating an expression: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. Without it the same expression would have several answers. Many students remember it as PEMDAS.Algebra · Grade 5, Grade 6, Grade 7SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2EquationAn equation is a statement that two expressions are equal. It always has an equal sign, such as x + 5 = 12 or 3a = 18. Solving an equation means finding the value of the variable that makes it true: here x = 7 and a = 6. An expression like 4x + 7 has no equal sign, so it is not an equation.Algebra · Grade 6, Grade 7, Grade 8SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3NonnegativeNonnegative means greater than or equal to zero: such a number is either 0 or positive, so x ≥ 0. The nonnegative integers are 0, 1, 2, 3 and so on. It is not the same as positive, because 0 is nonnegative but not positive. Distances, absolute values and principal square roots are always nonnegative.Problem Solving · Grade 6, Grade 7, Algebra 1Additive InverseThe additive inverse of a number is the value that adds to it to give zero. The additive inverse of 9 is −9, because 9 + (−9) = 0, so it is the same thing as its opposite. To find it, change the sign: the additive inverse of −14 is 14, and 0 is its own.Number System · Grade 6, Grade 7, Grade 8Fact FamilyA fact family is a group of related math facts that use the same three numbers. With 6, 9 and 15 you get 6 + 9 = 15, 9 + 6 = 15, 15 − 6 = 9 and 15 − 9 = 6. Multiplication and division make fact families too, such as 4, 7 and 28. They show that operations undo each other.Operations · Grade 1, Grade 2, Grade 3Cube RootThe cube root of a number is the value that, used as a factor three times, gives that number. It is written 3√x. Because 4 × 4 × 4 = 64, 3√64 = 4. Unlike square roots, cube roots of negative numbers are real numbers: 3√−27 = −3.Number System · Grade 8, Algebra 1, Algebra 2ExponentAn exponent tells how many times to use a base as a factor. It is the small raised number to the upper right of the base: in 53, the base is 5 and the exponent is 3, so 53 = 5 × 5 × 5 = 125. Any nonzero number with an exponent of 0 equals 1.Algebra · Grade 6, Grade 7, Grade 8SolutionA solution is a value that makes an equation or inequality true. To test a value, substitute it for the variable and check that the statement holds: x = 4 solves x + 2 = 6 because 4 + 2 = 6. A solution can also be an ordered pair, a point on a graph that satisfies the relationship.Algebra · Grade 6, Grade 7, Grade 8VariableA variable is a letter or symbol that represents a number or value, usually one that can change or is not yet known. In n + 5 = 12, the variable n is an unknown that equals 7. In A = lw, the variables l and w can take many values. Any letter can be a variable, not just x.Algebra · Grade 6, Grade 7, Grade 8Inverse FunctionAn inverse function undoes another function: if f sends a to b, its inverse f−1 sends b back to a. For f(x) = 2x + 3, the inverse is f−1(x) = x − 32. Only one-to-one functions have inverses, and the graph of f−1 is the mirror image of f across y = x.Functions · Algebra 2, Precalculus