Solving 2/x + 3/(x + 1) = 5

Algebra 1 · Axiom Academy

EXAMPLE Solving a Rational Equation Clear the fractions with the LCD, solve the resulting equation, then check for extraneous solutions. Solve the rational equation for x , and check whether either solution is extraneous. Nice work — you solved a rational equation end to end. Here's the playbook: Find the LCD: multiply the distinct factors from every denominator — here x and x+1 share no factor, so the LCD is x(x+1) . Clear the denominators: multiply both sides by the LCD to turn the fraction equation into a polynomial one. Solve what's left: after canceling, this collapsed to 5x^2 - 2 = 0 , giving . Always check for extraneous roots: reject any solution that makes a denominator zero. A root is extraneous when it satisfies the cleared polynomial but makes an original denominator zero. Here both keep x and x+1 nonzero, so both survive — but the check is never optional.

This is the written version of the interactive lesson above. See the full Algebra 1 course.