SAT Math: Rational Equations and Excluded Values Practice Questions
Solve equations containing variable denominators, record excluded values, and check candidate solutions in the original equation. Try each original question before checking its worked explanation. This focused practice is free and requires no account.
Free original practice questions for study purposes. Open Exam Prep is an independent study resource and is not affiliated with, endorsed by, or sponsored by the makers of SAT.
Answer the questions below — you get instant feedback and a full explanation for each.
1. For x not equal to 1 or -2, what value of x satisfies 3/(x - 1) = 6/(x + 2)?
Explanation. Multiply by (x - 1)(x + 2): 3(x + 2) = 6(x - 1). Thus 3x + 6 = 6x - 6, so x = 4. It is allowed, and both original fractions then equal 1.
2. For x not equal to 2, solve (x + 3)/(x - 2) = 2. What is x?
Explanation. Multiply by the nonzero denominator: x + 3 = 2x - 4. Therefore x = 7. Substituting gives 10/5 = 2, so the solution is valid.
3. For x not equal to -1, what value of x satisfies x/(x + 1) = 3/4?
Explanation. Cross-multiply to obtain 4x = 3(x + 1). Hence x = 3. It is not excluded, and 3/(3 + 1) equals 3/4.
4. For nonzero x, solve 1/x + 1/6 = 1/3. What is x?
Explanation. Subtract 1/6 from both sides: 1/x = 1/6. Taking reciprocals gives x = 6, which satisfies the nonzero restriction.
5. How many real solutions does x/(x - 2) = 2/(x - 2) have?
Explanation. The denominator requires x not equal to 2. Multiplying by x - 2 gives x = 2, but that candidate makes both original fractions undefined. Therefore there are no solutions.
6. For x not equal to 3, what value of x satisfies (x^2 - 9)/(x - 3) = 8?
Explanation. Factor x^2 - 9 as (x - 3)(x + 3). On the allowed domain, cancel x - 3 to get x + 3 = 8, so x = 5. The excluded value 3 remains excluded even after cancellation.
📘 Looking for more study material? Browse SAT prep books and study resources. Some links are affiliate links; we may earn a commission at no cost to you.
FAQ
Why record excluded values before multiplying?
A denominator cannot equal zero. Multiplication can produce a candidate outside the original domain, so retain the restrictions and check each candidate.
Does cancellation remove a domain restriction?
No. A simplified expression agrees with the original only where the original expression is defined.