Simplify and solve rational expressions and equations; respect domain restrictions.
Factor first, then simplify or solve while keeping denominator restrictions.
核心知识
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.
深入理解
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.
分步讲解
- 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.
常见误解
- 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.
示例解析
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