Solve rational equations without restoring excluded inputs
Record denominator restrictions before clearing fractions, and filter candidates afterward.
Loading your saved responses…
Try
A fictional 12 km journey takes 12/v hours at constant speed v in km/hour. Solve 12/v = 3. Before multiplying, decide whether v = 0 belongs to the original domain.
Predict
The journey should take three hours at 4 km/hour. A zero speed cannot be entered in 12/v. A negative speed might be algebraically allowed in other signed-velocity models, but this travel-time story requires positive speed.
Experiment
For 2/(x − 1) = 1, record x ≠ 1 first. Multiplying both sides by x − 1 gives 2 = x − 1, so x = 3. Substitute into the original quotient: 2/2 = 1. Change the right side to zero. Clearing the denominator now gives 2 = 0, so the solution set is empty.
Understand
A rational equation contains quotients of polynomials. Its original denominator restrictions remain in force throughout the solution. Clearing denominators is reversible only on that restricted domain, where the multiplier is nonzero.
Consider 1/x + 1/(x + 2) = 3/4. Exclude 0 and −2. Multiply every term by 4x(x + 2):
The first term becomes 4(x + 2).
The second becomes 4x.
The right side becomes 3x(x + 2).
Collecting gives 3x² − 2x − 8 = 0.
Factor (3x + 4)(x − 2) = 0, giving −4/3 and 2.
Both candidates are allowed. At 2 the sum is 1/2 + 1/4 = 3/4. At −4/3 it is −3/4 + 3/2 = 3/4.
A canceled factor still matters. In (x² − 1)/(x − 1) = 2, cancellation gives x + 1 = 2 only for x ≠ 1. Its only candidate is 1, so the original equation has no solution. “Simplified” does not mean “defined at more inputs.”
Apply
Use three written columns: original exclusions, algebraic candidates, accepted solutions. The checker below evaluates finite rational solution sets; an identity on a restricted domain requires a written answer instead. Check the original, not merely the cleared equation.
Continue
Squaring a radical equation can also produce candidates that fail the original. Unlike multiplication by a nonzero denominator on a fixed domain, squaring generally loses sign information. The next lesson makes that distinction explicit.
Apply · explain · revise
Practice and reasoning
Drafts save locally as you type. Reveal worked solutions when useful; the site remembers that exposure. Automatic matches check the requested answer, while written arguments require review against the rubric.
Solve 3/(x − 1) = 2 over the real numbers.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
Solve (x² − 4)/(x − 2) = 4.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
Solve 1/x + 1/(x + 1) = 3/2.
Integers, fractions and finite decimals are accepted. Separate values with commas; type none for no solutions. Expressions such as sqrt(2) are not parsed by this checker.
What is the original domain of (x − 5)/[(x − 5)(x + 2)]?
Solve (x + 1)/(x − 3) = 4/(x − 3) + 1. Explain why saying every real number is inaccurate.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Reflect and continue
Before marking this lesson complete, explain one key idea without the worked solution and identify a condition that matters. Marking complete records participation, not independent mastery.
