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Concept

# Inverse Operations

Inverse operations are two operations that, all else being equal, undo one another. For instance, adding $6$ and subtracting $6$ are inverse operations because they cancel each other out. \begin{aligned} &x+6\\ &x+6{\color{#0000FF}{\ -\ 6}}\\ &x \end{aligned} In an expression like $x+6,$ the addition of $6$ to $x$ is eliminated by performing the inverse operation: a subtraction of $6.$ Using inverse operations on an equation, however, is a little different: in order to adhere to the Properties of Equality, any operation performed on one side of an equation must also be performed on the other side to maintain equality. \begin{aligned} x\div{2}&=1\\ x\div{2}{\color{#0000FF}{\ \times{\ 2}}}&=1{\color{#0000FF}{\ \times{\ 2}}}\\ x&=2 \end{aligned} In this case, the division by $2$ on one side of the equation could only be eliminated by a multiplication by $2$ on both sides of the equation. The result of applying the Properties of Equality on an equation is an equivalent equation.