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BetterGrades Algebra · Unit A10 · Unit Hub

Unit A10: Rational Expressions, Equations, and Variation

Develop algebra with variable denominators, explicit restrictions, and inverse variation. What changes when a variable expression appears in a denominator?

Act X - Tracking where division is allowed

The 10-lesson path

A denominator creates excluded inputs. Factoring enables cancellation but cannot restore missing domain points. Common denominators permit addition and equation solving; sign analysis introduces rational inequalities and discontinuities.

Governing question

What changes when a variable expression appears in a denominator?

Develop algebra with variable denominators, explicit restrictions, and inverse variation.

Mastery design

Know when to continue

Every answer must include restrictions and rational-equation candidates must be checked in the original equation.

Unit outcomes
  1. Identify excluded values and preserve original restrictions.
  2. Simplify and perform arithmetic with rational expressions.
  3. Solve rational equations and check candidate solutions.
  4. Simplify complex rational expressions.
  5. Model variation and interpret holes and asymptotes informally.
Core sequence

Lessons

Move in order when this material is new; use prerequisites and repair links when entering mid-unit.

  1. 01Rational expressions and restrictionsDetermine where a denominator is zero and state the valid input set.
  2. 02Simplifying rational expressionsFactor and cancel common factors without cancelling terms or erasing restrictions.
  3. 03Multiplication of rational expressionsFactor before multiplying and cancel across products.
  4. 04Division of rational expressionsMultiply by the reciprocal while adding restrictions from the divisor.
  5. 05Least common denominatorsAssemble an LCD using every factor at its greatest required power.
  6. 06Addition and subtractionCreate equivalent rational expressions with a common denominator before combining numerators.
  7. 07Complex rational expressionsSimplify a fraction containing fractions by multiplying numerator and denominator by a common LCD.
  8. 08Rational equationsClear denominators to produce candidates and verify every candidate in the original equation.
  9. 09Direct, inverse, and joint variationTranslate proportional structures into formulas with a constant parameter.
  10. 10Rational inequalities and introductory graphsUse zeros and restrictions as critical values and distinguish holes from vertical asymptotes.

Practice around the path

Review, mixed practice, mastery, and investigation

Every route contains a fixed set of concrete questions, with detailed solutions available after an attempt.

ReviewCumulative reviewRevisit this unit with prior-unit retrieval.PracticeMixed practiceClassify the structure before choosing a method.MasteryMastery checkUse the supplied grading and cumulative-share contract.Response guideMastery response guideAttempt-gated rubric support; no invented canonical answers.

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