1Cademy - Simplifying $$\frac{\frac{y}{y+1}}{1+\frac{1}{y-1}}$$ Using the LCD

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Simplifying a Complex Rational Expression by Using the LCD

Example

Simplifying $\frac{\frac{y}{y+1}}{1+\frac{1}{y-1}}$ Using the LCD

Simplify the complex rational expression using the LCD method:

$\frac{\frac{y}{y+1}}{1+\frac{1}{y-1}}$

Step 1 — Find the LCD of all inner fractions. The inner denominators are $(y+1)$ and $(y-1)$ . Since these are distinct linear factors, the LCD is $(y+1)(y-1)$ .

Step 2 — Multiply the numerator and denominator of the complex fraction by the LCD $(y+1)(y-1)$ . Distribute to each term:

$\frac{(y+1)(y-1) \cdot \frac{y}{y+1}}{(y+1)(y-1) \cdot 1 + (y+1)(y-1) \cdot \frac{1}{y-1}}$

Step 3 — Simplify by canceling matching factors. In the numerator, $(y+1)$ cancels, leaving $y(y-1)$ . In the denominator, the first term becomes $(y+1)(y-1)$ and in the second term $(y-1)$ cancels, leaving $(y+1)$ :

$\frac{y(y-1)}{(y+1)(y-1) + (y+1)}$

Step 4 — Simplify the denominator. Expand and combine like terms: $(y+1)(y-1) = y^2-1$ , so the denominator becomes $y^2-1+y+1 = y^2+y$ .

$\frac{y(y-1)}{y^2+y}$

Step 5 — Factor the denominator and cancel common factors. Factor $y^2+y = y(y+1)$ . The numerator and denominator share the factor $y$ :

$\frac{y(y-1)}{y(y+1)} = \frac{y-1}{y+1}$

The simplified result is $\frac{y-1}{y+1}$ . This example demonstrates the LCD method when the overall numerator is a single fraction but the overall denominator is a whole number plus a fraction. The LCD $(y+1)(y-1)$ clears all inner fractions in one step, and factoring the resulting denominator reveals a common factor with the numerator that allows further simplification.

Updated 2026-04-21

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OpenStax Elementary Algebra

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