What this lesson covers
- The four-step method: exclude, clear the fractions, solve, then CHECK (x^2/(x-2) = 4/(x-2) gives x = 2 or x = -2, but x = 2 is extraneous, so the answer is just x = -2)
- The twist where the only candidate is banned, so the equation has NO solution (x/(x-4) = 4/(x-4))
- A practice equation where both answers survive (x + 4/x = 5 gives x = 1 and x = 4)
- The three traps that quietly cost points
Questions worked in the video
- 0:59Solve x^2/(x-2) = 4/(x-2). Which candidate is extraneous?
- 1:39Solve x/(x-4) = 4/(x-4). How many solutions are there?
- 2:15Solve x + 4/x = 5.
Chapters
- 0:00The answer that's secretly illegal
- 0:24The method: exclude, clear, solve, check
- 0:59Solve it: one root is extraneous
- 1:39When there's no solution at all
- 2:15Practice: x + 4/x = 5
- 2:55Three traps to avoid
- 3:30Recap
Lesson transcript
The narration of the video, word for word, under its chapter headings.
Welcome to SAT Climb. Rational equations put a variable in the denominator, and they hide one of the SAT's favorite traps. You do the algebra, you get an answer, and one of those answers is secretly illegal. Here's how to catch it every time.
Start with the one rule that drives everything. A denominator can never be zero. So before you solve, look at the bottoms and note what x can't be. Then multiply every term by the common denominator to clear the fractions, solve the equation you get, and finally check each answer against those forbidden values. Take this one. x squared over x minus two equals four over x minus two. First question: what can x not be?
Since x minus two is on the bottom, x can't be two. Now multiply both sides by x minus two, and the denominators cancel, leaving x squared equals four. So x is two, or x is negative two. Two candidates. But watch, x equals two is exactly the value we banned, it would make the denominator zero, so it's extraneous and we throw it out. x equals negative two checks out. The only real solution is negative two.
Now the twist the SAT loves. Sometimes the only answer you get is the banned one. Look at x over x minus four equals four over x minus four. Here x can't be four. Multiply through and you get x equals four, and that's it, that's the only candidate. But four is exactly the forbidden value, so it's extraneous too. Nothing survives. This equation has no solution, and you'd only know that if you checked.
Your turn. x plus four over x equals five. Only x over zero is banned, so x can't be zero. Multiply every term by x, and don't forget the plain x and the five, that gives x squared plus four equals five x. Move it over, x squared minus five x plus four equals zero, which factors into x minus one times x minus four. So x is one or four. Check both, neither is zero, and both actually work. This time you keep both answers.
Three traps. One, skipping the check. Any answer that makes a denominator zero is extraneous, and it has to go. Two, multiplying only the fractions by the common denominator. Every term gets multiplied, including the ones that aren't fractions. Three, canceling across a plus or minus sign. You can only cancel factors that multiply the whole top and bottom, never single terms. Note the excluded values, clear the fractions, solve, then check.
Rational equations, solved. Note what x can't be, clear the fractions, solve, and always check your answers against those forbidden values. Start practicing free at satclimb.com. Your SAT is closer than you think.
Read the written version: the Nonlinear equations · one variable strategy guide, then try 25 hard Nonlinear equations · one variable questions with full explanations. Both are free, no account needed.