Practice Meaning Preservation in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

Quick Recap

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

Every algebraic step must be a valid equivalence — adding the same to both sides, multiplying by a non-zero quantity, or applying a one-to-one function preserves meaning.

Example 1

easy
When solving x + 5 = 12, list each algebraic step and explain why it preserves the meaning (solution set) of the equation.

Example 2

medium
Squaring both sides of \sqrt{x} = x - 2 can introduce extraneous solutions. Solve the equation and check which solutions are valid.

Example 3

easy
Which operation preserves the solution set of an equation: (a) multiply both sides by 3, (b) multiply both sides by 0, (c) add 7 to both sides?

Example 4

medium
A student divides both sides of x(x-2) = 0 by x, getting x - 2 = 0, so x = 2. Identify the meaning-preservation error and give the full solution.
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