1Cademy - Adding and Subtracting $$\frac{v}{v+1} + \frac{3}{v-1}

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How to Add or Subtract Rational Expressions

Example

Adding and Subtracting $\frac{v}{v+1} + \frac{3}{v-1} - \frac{6}{v^2-1}$

Simplify the expression $\frac{v}{v+1} + \frac{3}{v-1} - \frac{6}{v^2-1}$ . To add and subtract these rational expressions, first find the Least Common Denominator (LCD). Factor the third denominator, $v^2-1$ , into $(v-1)(v+1)$ . Thus, the LCD for all three terms is $(v-1)(v+1)$ . Rewrite each fraction with this common denominator: the first term becomes $\frac{v(v-1)}{(v+1)(v-1)}$ , the second becomes $\frac{3(v+1)}{(v-1)(v+1)}$ , and the third remains $\frac{6}{(v-1)(v+1)}$ . Combine the numerators over the common denominator to get $\frac{v(v-1) + 3(v+1) - 6}{(v-1)(v+1)}$ . Distribute to remove parentheses in the numerator, yielding $\frac{v^2 - v + 3v + 3 - 6}{(v-1)(v+1)}$ . Combine like terms to simplify the numerator to $v^2 + 2v - 3$ . Factor this trinomial as $(v+3)(v-1)$ . Write the factored expression as $\frac{(v+3)(v-1)}{(v-1)(v+1)}$ and divide out the common factor of $(v-1)$ . The final simplified expression is $\frac{v+3}{v+1}$ .

Updated 2026-05-25

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OpenStax Intermediate Algebra

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