1Cademy - Subtracting $$\frac{3x-1}{x^2-5x-6}

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Adding and Subtracting Rational Expressions Whose Denominators Are Opposites

Example

Subtracting $\frac{3x-1}{x^2-5x-6} - \frac{2}{6-x}$

Subtract two rational expressions where one denominator is the opposite of a binomial factor of the other: $\frac{3x - 1}{x^2 - 5x - 6} - \frac{2}{6 - x}$

Step 1 — Factor the denominator and resolve the opposites. Factor the first denominator: $x^2 - 5x - 6 = (x - 6)(x + 1)$ . The second denominator is $6 - x$ . Because $x - 6$ and $6 - x$ are opposites, multiply the second fraction by $\frac{-1}{-1}$ : $\frac{2}{6 - x} \cdot \frac{-1}{-1} = \frac{-2}{x - 6}$ Rewrite the subtraction: $\frac{3x - 1}{(x - 6)(x + 1)} - \frac{-2}{x - 6}$

Step 2 — Simplify the signs. Subtracting a negative is the same as adding a positive: $\frac{3x - 1}{(x - 6)(x + 1)} + \frac{2}{x - 6}$

Step 3 — Find the LCD and rewrite. The LCD is $(x - 6)(x + 1)$ . The first fraction already has the LCD. Multiply the numerator and denominator of the second fraction by $(x + 1)$ : $\frac{3x - 1}{(x - 6)(x + 1)} + \frac{2(x + 1)}{(x - 6)(x + 1)}$ Distribute the $2$ in the second numerator: $2(x + 1) = 2x + 2$ . $\frac{3x - 1}{(x - 6)(x + 1)} + \frac{2x + 2}{(x - 6)(x + 1)}$

Step 4 — Add the numerators. Combine the numerators over the common denominator: $\frac{3x - 1 + 2x + 2}{(x - 6)(x + 1)}$ Combine like terms to get the simplified result: $\frac{5x + 1}{(x - 6)(x + 1)}$

Updated 2026-06-17

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OpenStax Intermediate Algebra

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