Meaning Preservation Math Example 4

Follow the full solution, then compare it with the other examples linked below.

Example 4

medium

A student divides both sides of

x(x-2) = 0

x

, getting

x - 2 = 0

, so

x = 2

. Identify the meaning-preservation error and give the full solution.

Solution

1
Error: dividing both sides by $x$ is only valid when $x \ne 0$ . If $x = 0$ , dividing by $x$ is undefined — this step discards the solution $x = 0$ .
2
Correct approach: $x(x-2) = 0$ means $x = 0$ or $x - 2 = 0$ (zero-product property).
3
Full solution: $x = 0$ or $x = 2$ .

Answer

x = 0 \text{ or } x = 2

Dividing by a variable expression is not meaning-preserving unless you can guarantee it is non-zero. The zero-product property is the correct approach for factored equations: if a product is zero, at least one factor must be zero.

About Meaning Preservation

Meaning preservation is the principle that valid mathematical transformations must maintain the truth and relationships of the original expression — changing form without changing content.

Learn more about Meaning Preservation →

More Meaning Preservation Examples

Example 1 easy

When solving [formula], list each algebraic step and explain why it preserves the meaning (solution

Example 2 medium

Squaring both sides of [formula] can introduce extraneous solutions. Solve the equation and check wh

Example 3 easy

Which operation preserves the solution set of an equation: (a) multiply both sides by 3, (b) multipl

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Related Concepts

Equivalence Transformation