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Master simplifying algebraic fractions, adding with common denominators, and solving fraction equations using clear steps, worked examples, and practice.
Algebraic fractions combine polynomial expressions in numerators and denominators. This lesson builds fluency in three linked skills: reducing fractions by factoring, combining fractions through a least common denominator, and solving equations that contain fractions. Each section uses checkable arithmetic and a consistent convention for handwritten work so that valid alternative forms can be recognized.
Practice materials: Printable downloads are not available for this lesson yet. Use the instructions and four exercises below.
A fraction is simplified when numerator and denominator share no common factor other than 1. For algebraic fractions, factor both parts completely, then cancel identical factors. For example, (x^2 - 9)/(x + 3) factors as (x - 3)(x + 3)/(x + 3); canceling (x + 3) gives x - 3, provided x is not -3. State excluded values when they arise from the original denominator.
Use the convention that handwritten cancellation marks are a single diagonal stroke through the common factor, with the reduced factor written nearby. This convention is a communication aid, not a universal answer format; equivalent correct simplifications are accepted.
To add or subtract algebraic fractions, rewrite each fraction with a common denominator, preferably the least common denominator (LCD). The LCD is the product of each distinct factor raised to its highest power. Multiply numerator and denominator of each fraction by the missing factors, then combine numerators over the LCD.
After combining, simplify the resulting fraction if possible. Keep the denominator factored in the final answer unless expansion is requested; factored form makes common factors visible and supports checking.
Multiply every term by the LCD to clear denominators, then solve the resulting polynomial equation. Because multiplying by an expression can introduce extraneous solutions, substitute each candidate back into the original equation or check that it does not make any original denominator zero.
For simple linear cases, isolate the fraction first, then cross-multiply. Always record the check step explicitly; a solution that satisfies the cleared equation but zeroes a denominator is rejected.
Simplify (x^2 - 4)/(x^2 + x - 6) and state excluded values.
(x + 2)/(x + 3), with x not equal to 2 and x not equal to -3.
Simplify (x^2 - 9)/(x^2 - x - 6).
(x + 3)/(x + 2), with x not equal to 3 and x not equal to -2.
Why: Factor numerator as (x - 3)(x + 3) and denominator as (x - 3)(x + 2). Cancel (x - 3). Excluded values come from the original denominator: x = 3 and x = -2.
Check: Substitute x = 0 into original and simplified forms: original gives (-9)/(-6) = 1.5; simplified gives 3/2 = 1.5. Both match.
Add 2/(x + 1) + 3/(x - 1).
(5x + 1)/((x + 1)(x - 1)), with x not equal to 1 and x not equal to -1.
Why: The LCD is (x + 1)(x - 1). Rewrite as 2(x - 1)/LCD + 3(x + 1)/LCD. Combine numerators: 2x - 2 + 3x + 3 = 5x + 1.
Check: Let x = 2. Original: 2/3 + 3/1 = 2/3 + 3 = 11/3. Simplified: (10 + 1)/((3)(1)) = 11/3. Match.
Solve 3/(x + 2) = 1/2.
x = 4.
Why: Cross-multiply: 3 times 2 equals 1 times (x + 2), so 6 = x + 2, giving x = 4. The original denominator x + 2 is 6 at x = 4, not zero.
Check: Substitute x = 4: left side 3/6 = 1/2, right side 1/2. Match.
Solve 1/(x - 1) + 1/(x + 1) = 2/(x^2 - 1).
No solution; x = 1 and x = -1 are excluded.
Why: The LCD is (x - 1)(x + 1). Multiply every term: (x + 1) + (x - 1) = 2, so 2x = 2 and x = 1. But x = 1 makes the original denominator zero, so it is extraneous. x = -1 is also excluded. No valid solution remains.
Check: Substitute x = 1 into the original equation: 1/0 is undefined, confirming rejection. The cleared equation alone would falsely suggest x = 1.
No. Cancellation requires identical factors multiplying the entire numerator and entire denominator. Factor the whole expression first.
No. Factored form is usually preferred because it reveals common factors and excluded values. Follow any explicit instruction to expand.
Substitute it into the original equation. If any original denominator becomes zero, the candidate is extraneous and must be rejected.
No. The diagonal-stroke convention is defined for clarity, but any consistent, legible marking that identifies the canceled factor is acceptable under the rubric.