Solving Rational Equations

Pre-Calculus · Axiom Academy

LESSON Solving Rational Equations Clear the denominators with the LCD, solve the polynomial that remains, then check every candidate and throw out the extraneous ones. Take . Both fractions sit over x , so the LCD is just x . Multiply every term by x and each denominator cancels — the fractions vanish and a plain linear equation is left. Denominators cleared — solve what's left Every rational equation yields to the same routine. The first three steps get you a candidate answer; the last two are what separate a right answer from a wrong one. The least common denominator of every fraction in the equation. Multiply each term by the LCD so every denominator cancels. Simplify and solve the polynomial equation that remains. Test each candidate in the original; reject any that make a denominator 0 . Solve . Forbidden value: x = 0 (it would zero both denominators), so any answer must avoid it. 4. Example 2 — Catch the Extraneous Root Solve . The denominator is x-2 , so x = 2 is forbidden . Keep that in mind — it's about to come back. x = 2 makes the denominator 0 — division by zero. It is extraneous and must be rejected. Only x = -2 survives. 5. Example 3 — Several Fractions, One LCD Solve . Two different denominators appear, so the LCD is their product, x(x+1) . Forbidden values: x = 0 and x = -1 . You can clear any rational equation with its LCD, solve the polynomial that's left, and — most important — check every candidate so no extraneous root slips through.

This is the written version of the interactive lesson above. See the full Pre-Calculus course.