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Rational Expressions and Equations

SAT MathAdvanced MathHard

Factor before canceling and track excluded values.

What you need to know

  • A rational expression is undefined where its denominator is zero.
  • Only factors cancel; terms separated by addition do not.
  • Multiplying by a common denominator can solve rational equations.
  • Every candidate must be checked against original restrictions.

The AYLOTI method

  1. List excluded values.
  2. Factor numerators and denominators.
  3. Cancel common factors or clear denominators.
  4. Solve and reject any excluded candidate.

AYLOTI • Sir Hunain Zia • SAT Score 1594/1600 • 154 Total A Grades (Personal), 11 World Records, 7 Distinctions

Worked example

Simplify (x² − 9)/(x² − 3x).

  1. Factor numerator: (x − 3)(x + 3).
  2. Factor denominator: x(x − 3).
  3. Cancel x − 3, while retaining x ≠ 3 and x ≠ 0.

Answer: (x + 3)/x, with x ≠ 0, 3.

Common SAT traps

  • Canceling x across x + 3.
  • Forgetting excluded values after cancellation.
  • Accepting a solution that makes an original denominator zero.
Quick check

Simplify (x² − 4)/(x − 2).

x + 2, with x ≠ 2.

Before you move on

  • Explain the core rule without looking back.
  • Redo the worked example using your own steps.
  • Write the trap you are most likely to fall for.
Hunain Zia

Hunain Zia has previously achieved considerable success in the high-school educational stream, gaining a massive 154 Total A Grades and 7 Major Distinctions, a process in which he broke/ set 11 different world records, including the most A grades ever achieved, most subjects ever appeared in, most distinctions, gaining distinctions across two separate boards (Pearson Edexcel and CAIE) within the same year/ ever, and even 19 hours of constant examination. All records stand intact today. Hunain pursued his Honors Accounting Degree from the prestigious Bentley University, and then completed an LLB (in Honors) from University of London. Currently, he manages multiple social and commercial projects, has founded digital streams and works tirelessly in the education sector.

Educate A Change focuses on truly bringing change in the educational sector worldwide. All our courses are designed to ensure success, and provide complete preparation without requiring any other external material. Therefore, it takes away the expense of academies, books, past paper compilations and other miscellaneous items. Instead, by just the courses we offer, the student can prepare fully for the best grades in their CAIEs, Edexcel and other high-school exams. The earnings from our course-sales are invested in social projects and growing the free educational activities of Educate A Change.