Practice Meaning Preservation in Math

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

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Quick Recap

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

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

Showing a random 20 of 50 problems.

Example 1

medium
To solve โˆฃxโˆฃ=3|x| = 3, a student writes x=3x = 3 only. Does dropping the absolute value preserve the solution set?

Example 2

hard
Why is the chain x2=xโ‡’x=1x^2=x\Rightarrow x=1 wrong, and what is the correct solution set?

Example 3

hard
State precisely when squaring both sides of f(x)=g(x)f(x)=g(x) produces an equivalent equation.

Example 4

hard
Solve x+7โˆ’x=1\sqrt{x+7}-\sqrt{x}=1 and verify the solution.

Example 5

easy
From x=3x = 3 you square both sides to get x2=9x^2 = 9. Does squaring preserve the solution set here?

Example 6

medium
From โˆฃ2xโˆ’3โˆฃ=5|2x-3|=5, conclude all values of xx.

Example 7

challenge
Solve sinโก2x+sinโกx=0\sin^2 x+\sin x=0 for xโˆˆ[0,2ฯ€]x\in[0,2\pi]. Use factoring, NOT division by sinโกx\sin x.

Example 8

challenge
To solve x+5+x=5\sqrt{x+5} + \sqrt{x} = 5, why must isolating one radical BEFORE squaring be done, and what root must be checked?

Example 9

medium
Solve 2x+1=xโˆ’1\sqrt{2x+1}=x-1. Check all candidates.

Example 10

hard
Solve 1x+1xโˆ’1=32\frac{1}{x}+\frac{1}{x-1}=\frac{3}{2} and check domain restrictions.

Example 11

medium
Is the rewrite x2=x\sqrt{x^2}=x correct for all real xx?

Example 12

medium
A student divides the inequality โˆ’2x<6-2x < 6 by โˆ’2-2 to get x<โˆ’3x < -3. Did this preserve the solution set?

Example 13

easy
You cancel to get xโˆ’2xโˆ’2=1\frac{x-2}{x-2} = 1. For which xx is this valid?

Example 14

easy
True or false: Adding the same number to both sides of an equation always preserves the solution set.

Example 15

medium
From 1xโˆ’1=1xโˆ’1+0\frac{1}{x-1} = \frac{1}{x-1} + 0, a student 'simplifies' to 1=11 = 1 and claims all xx work. What is wrong?

Example 16

medium
From 2(x+3)=2x+62(x+3)=2x+6, identify whether this is an identity, equation, or contradiction.

Example 17

medium
Is squaring x=โˆ’3x=-3 to get x2=9x^2=9 meaning-preserving?

Example 18

easy
Is replacing logโก(ab)\log(ab) with logโกa+logโกb\log a + \log b meaning-preserving for a,b>0a,b > 0?

Example 19

medium
You cross-multiply x+1xโˆ’2=31\frac{x+1}{x-2} = \frac{3}{1} to get x+1=3(xโˆ’2)x+1 = 3(x-2). Under what condition is this safe?

Example 20

medium
A student writes (xโˆ’2)(x+1)(xโˆ’2)=x+1\frac{(x-2)(x+1)}{(x-2)} = x+1. State this equality precisely.
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