Math101 · Independent Ontario learning library
Math101learn.math101.caRational Equations
Rational equations contain variable denominators and are solved by recording restrictions, clearing denominators, and checking candidates.
Clearing denominators simplifies the algebra, but it never makes forbidden inputs legal.
Identify restrictions first
A rational equation contains one or more rational expressions. Before manipulating it, find values that make any original denominator zero. These are excluded even if a factor later cancels.
For
the restriction is $x\ne1$.
Clear denominators with the LCD
Find the least common denominator and multiply every term on both sides by it. This creates an equivalent polynomial equation for allowed inputs.
Write parentheses around numerators before distributing. Skipping a term or part of a numerator changes the equation.
Worked example: a linear result
Worked example: a quadratic result
Solve
Multiply by $x(x-2)$:
Rearrange:
The quadratic formula gives
Neither value is excluded, so both are valid.
Extraneous solutions
Multiplying by a variable expression can make an excluded value appear as a solution of the cleared equation. Such a value is extraneous because the original equation was undefined there.
Checking restrictions is mandatory; direct substitution also catches algebra errors.
Equations with proportional structure
Some rational equations are proportions:
Cross multiplication gives $ad=bc$ only when $b$ and $d$ are nonzero. Cross multiplication is simply multiplication by the common denominator—it does not remove the need for restrictions.
Equations with no solution
After clearing denominators, an equation may reduce to a false statement such as $3=7$, indicating no solution. A candidate set may also consist only of excluded values, again leaving no solution.
If the result is an identity such as $0=0$, every allowed domain value solves the equation, not the excluded ones.
Applications and units
Rational equations model combined work rates, travel time, lens formulas, electrical resistance, and inverse variation. Define variables and units before solving.
Context may add restrictions beyond denominators: time, speed, length, and count are often positive. Algebraic candidates must satisfy both mathematical and contextual domains.
Graphical interpretation
An equation $f(x)=g(x)$ asks where two graphs intersect. Vertical asymptotes and holes can explain excluded candidates or a lack of solutions.
Graphing is a useful check, but exact solutions require algebra when the question asks for exact form.
A structured solution routine
- Factor denominators if needed.
- Record all restrictions.
- Identify the LCD.
- Multiply every term by the LCD.
- Solve the resulting equation.
- reject forbidden or contextually invalid values.
- verify in the original equation.
Common mistakes
Writing restrictions after cancelling. Use original denominators.
Multiplying only the fractions by the LCD. Every term must be multiplied.
Cross multiplying an equation with more than one fraction per side without combining first. Use the full LCD method.
Keeping an excluded candidate. A zero denominator is never allowed.
Checking only the cleared equation. Verify in the original.
Quick self-check
- Which inputs make an original denominator zero?
- Is the LCD fully factored and complete?
- Did every term receive the multiplier?
- Are all resulting candidates in the domain?
- Do they satisfy the original equation and context?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
Solve 2/(x − 1) + 1 = 5/(x − 1).
- 2 + (x − 1) = 5
- x + 1 = 5
- x = 4, which is allowed.
End of lesson
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