Simplify and solve rational expressions and equations; respect domain restrictions. | Algebra | ACT Mathematics

Factor first, then simplify or solve while keeping denominator restrictions.

Core Idea

Factor first, then either cancel common factors or clear denominators to solve. Keep the original denominator restrictions the whole time, even if a factor cancels later.

Understanding

Rule: Rational expressions behave like fractions, so structure matters before any algebra. Start by factoring every numerator and denominator and marking any value that makes an original denominator zero.

If the task is to simplify, cancel only common factors. If the task is to solve an equation, use the factored form to identify restrictions first, then multiply by the least common denominator to clear fractions. After solving, check that any candidate does not violate an original denominator restriction.

ACT answer choices often include a restricted value or a result from canceling terms instead of factors. Clean factoring and a final restriction check usually remove those traps.

Step by Step

Factor every numerator and denominator completely.
Record any value that makes an original denominator zero.
If you are simplifying, cancel only matching factors, not matching terms.
If you are solving an equation, clear denominators after stating the restrictions, then check each candidate against the original denominator.

Misconceptions

Canceling terms instead of factors.
Forgetting that any original denominator restriction still applies after simplification.
Clearing denominators before identifying the values that make them zero.

Question

Worked Example

For $𝑥 \neq 0$ and $𝑥 \neq 3$ , which expression is equivalent to $𝑥 2 - 9 𝑥 2 - 3 𝑥$ ?

Select an answer to see the explanation