Question

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

Short answer

The simplification of the expression is$\frac{5x+1}{\left(x+1\right)\left(x-6\right)}$.

Step-by-step solution

Given information 

The given rational expression is$\frac{3x-1}{x^2-5x-6}-\frac{2}{6-x}$.

The simplification of the rational expression 

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

Factor the denominator

$$\begin{gathered} =\frac{3x-1}{x^2-6x+x-6}-\frac{2}{6-x} \\ =\frac{3x-1}{x\left(x-6\right)+\left(x-6\right)}-\frac{2}{6-x} \\ =\frac{3x-1}{\left(x+1\right)\left(x-6\right)}-\frac{2}{6-x} \end{gathered}$$

Since $\left(n-6\right)and\left(6-n\right)$are opposites ,we will multiply the second rational expression by $\frac{-1}{-1}$.

$$\begin{gathered} =\frac{3x-1}{\left(x+1\right)\left(x-6\right)}-\frac{\left(-2\right)}{-\left(6-x\right)} \\ =\frac{3x-1}{\left(x+1\right)\left(x-6\right)}-\frac{-2}{\left(x-6\right)} \end{gathered}$$

The LCD of the rational expression is $\left(x+1\right)\left(x-6\right)$.

Rewrite each rational expression as an equivalent rational expression with the LCD.

localid="1657069728617" $$\begin{gathered} =\frac{3x-1}{\left(x+1\right)\left(x-6\right)}-\frac{-2\left(x+1\right)}{\left(x-6\right)\left(x+1\right)} \\ Simplify \\ =\frac{3x-1+2x+2}{\left(x+1\right)\left(x-6\right)} \\ =\frac{5x+1}{\left(x+1\right)\left(x-6\right)} \end{gathered}$$