Solve after Recording a Nonzero Variable Restriction
Clearing a variable denominator requires excluding its zero before multiplying.
About 25 minutes, plus any extended practice
Supplied materials
- Paper or a browser scratch area; a calculator is optional unless rounding is requested.
- Supplied teaching case: Solve 2/x=1/2, retaining the restriction x≠0. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8.
- Independent check inputs: Solve 3/x=1/2, retaining the restriction x≠0. Candidate answers: (A) 12 (B) 5 (C) 6 (D) 7.
Learn the idea
Rational expressions divide polynomials, so denominator zeros are excluded from their domains. Factoring can reveal cancellation or help solve equations, but it does not erase the original restrictions or turn inverse variation into linear change. Clearing a variable denominator requires excluding its zero before multiplying. Solving the transformed equation gives a candidate, and substitution verifies that it remains allowed and satisfies the original fraction.
Here is the method on the supplied teaching case. Multiply by 2x, permitted only for x≠0: 2(2)=x. The solution 4 is nonzero and substitutes to 2/4=1/2.
A solution that makes an original denominator zero must be rejected even if a later transformed equation accepts it. Record restrictions first so simplification does not accidentally hide them.
- Write or sketch the supplied teaching case, labeling its quantities and restrictions.
- Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
- Solve the independent check with its changed inputs before revealing feedback.
- Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
See the method in action
Teaching case: Solve 2/x=1/2, retaining the restriction x≠0. Candidate answers: (A) 3 (B) 4 (C) 5 (D) 8. Solution: Multiply by 2x, permitted only for x≠0: 2(2)=x. The solution 4 is nonzero and substitutes to 2/4=1/2. The final answer is 4. A solution that makes an original denominator zero must be rejected even if a later transformed equation accepts it. Record restrictions first so simplification does not accidentally hide them.
Check your reasoning
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Helpful tips
- Clearing a variable denominator requires excluding its zero before multiplying.
- A solution that makes an original denominator zero must be rejected even if a later transformed equation accepts it.
See the answer and explanation
6
Multiply by 2x, permitted only for x≠0: 2(3)=x. The solution 6 is nonzero and substitutes to 3/6=1/2.
Take it further
Changed-input task: Solve 9/x=1/2, retaining the restriction x≠0. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. Show the mathematical reason, not only the selected answer. Expected result: 18. Multiply by 2x, permitted only for x≠0: 2(9)=x. The solution 18 is nonzero and substitutes to 9/18=1/2.
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