SAT Math · Passport to Advanced Math
Rational expressions and equations
Simplifying ratios of polynomials and solving rational equations.
What's tested in this subtopic
A rational expression is a ratio of two polynomials. SAT items ask you to simplify, to identify restricted values where the denominator equals zero, to add or subtract two rational expressions, or to solve a rational equation by clearing denominators.
Tactics that actually move your score
Always factor numerator and denominator first; common factors usually cancel. When solving, multiply both sides by the LCD to clear denominators, solve the resulting polynomial equation, and check each candidate against the restricted values. Extraneous solutions are common. The single biggest leverage point on most subtopics isn't learning more math — it's recognizing the test's preferred surface forms quickly enough that you don't burn 30 seconds re-reading the question. The first time you see a particular phrasing it might take you a full minute. The tenth time you see it, you should be reaching for your method before you've finished the sentence. Repetition is what builds that recognition. Fifteen problems in a row of the same shape is more useful than fifty mixed.
Practice questions (14)
- Easy For what value of x is the expression (x + 6) / (x − 8) undefined?
- Easy For what value of x is the expression (x + 2) / (x − 1) undefined?
- Medium For what value of x is the expression (x + 7) / (x − 8) undefined?
- Medium For what value of x is the expression (x + 2) / (x − 2) undefined?
- Medium For what value of x is the expression (x + 4) / (x − 8) undefined?
- Hard For what value of x is the expression (x + 2) / (x − 2) undefined?
- Easy For what value of x is the expression (x + 3) / (x − 2) undefined?
- Easy For what value of x is the expression (x + 7) / (x − 6) undefined?
- Medium For what value of x is the expression (x + 7) / (x − 8) undefined?
- Medium For what value of x is the expression (x + 5) / (x − 3) undefined?
- Medium For what value of x is the expression (x + 5) / (x − 1) undefined?
- Hard For what value of x is the expression (x + 4) / (x − 1) undefined?
- Easy For what value of x is the expression (x + 6) / (x − 8) undefined?
- Easy For what value of x is the expression (x + 2) / (x − 2) undefined?
How to drill
Work through the questions above untimed. After each one, read the worked solution from start to finish — even when you got it right. Note which solution method you used, and which method we used; if they differ, ask yourself which would have been faster on test day. Speed in SAT math comes from shortening your method-selection step, not from doing arithmetic faster. Most fast students are doing the same arithmetic everyone else is — they're just spending less time deciding what to do.
Once you can clear the easy and medium items in this subtopic at 90% accuracy, attempt a timed mini-set of ten hard items at 75 seconds each. If you finish in time and score 7+ correct, you've effectively mastered the subtopic for test purposes and can move on.