Strategy to Solve Equations with Rational Expressions
To solve an equation that contains rational expressions, apply a five-step strategy that extends the clearing-fractions technique used for linear equations with numerical fraction coefficients to equations whose denominators may contain variables:
Step 1. Identify every variable value that would make any denominator in the equation equal zero. Record these restricted values before doing any algebra. Note them by writing next to the equation for each restricted value .
Step 2. Find the least common denominator (LCD) of all denominators that appear in the equation.
Step 3. Clear the fractions by multiplying both sides of the equation by the LCD. Because the LCD is divisible by each denominator, every rational expression simplifies to a polynomial expression, producing a fraction-free equation.
Step 4. Solve the resulting equation using standard techniques (combining like terms, factoring, applying properties of equality, etc.).
Step 5. Check the solutions:
- If any algebraic solution matches one of the restricted values recorded in Step 1, discard it — it is an extraneous solution that would make a denominator zero in the original equation.
- Substitute each remaining solution back into the original equation to verify that it produces a true statement.
The key difference between this strategy and the simpler fraction-clearing method for linear equations is that rational equations can have variable expressions in the denominators, so certain variable values are off-limits. Steps 1 and 5 specifically address this risk by screening for extraneous solutions before and after solving.
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