A False Answer
Part of the Polynomials & Rational Expressions section of Coddy's Algebra 2 journey. Lesson 40 of 54.
Solve (x + 1)/(x - 1) = 2/(x - 1) + 4. Multiply both sides by x - 1 to get x + 1 = 2 + 4(x - 1). Expand: x + 1 = 4x - 2, so 3x is 3 and x is 1. Now put 1 back into the original: x - 1 is 0, and you cannot divide by 0. x = 1 is not a solution at all.
Where did it come from? At x = 1, multiplying by x - 1 meant multiplying both sides by 0, and that makes any two sides equal. So the new equation has a solution the original never had. Such an answer is called extraneous: a false answer that the working produced and the original equation refuses.
With x = 1 thrown out and nothing else left, this equation has no solution. That is a correct, finished answer, not a failure.
Challenge
Solve x/(x - 4) = 4/(x - 4) + 2. Check every answer in the original equation.
Try it yourself
Solve for x
Test yourselfThis lesson includes a short quiz. Start the lesson to answer it and track your progress.
All lessons in Polynomials & Rational Expressions
1Polynomial Long Division
How Long Division WorksOne Term at a TimeA Remainder Left OverA Missing TermDividing by a QuadraticRecap: Long Division7Adding and Subtracting
Same BottomDifferent BottomsSubtractingFactor the Bottoms FirstRecap: Adding Fractions2Remainders and Factors
The Remainder TheoremThe Factor TheoremFinding a Missing NumberUsing a Known FactorRecap: Remainders and Factors5Sign Charts
Between the RootsOutside the RootsA Negative Leading TermReading a Sign ChartRecap: Sign Charts8Rational Equations
Clearing Fractionsx in the BottomCross-MultiplyingA False AnswerOne True, One FalseRecap: Rational Equations