Question

Simplify the complex rational expression. When typing your answer for the numerator and denominator be sure to type the term with the variable first with no spaces between characters. If an exponent is needed use the carrot key (^) by pressing shift and 6 at the same time.\frac{\left(\frac{3}{x+1}+\frac{2}{x-1}\right)}{\frac{\left(x-1\right)}{x+1}}The numerator is AnswerThe denominator is Answer

Answer 1

Answer

Numerator:

$5x-1$

Denominator:

$(x-1)^2$

EXPLANATION

Problem Statement

The question asks us to simplify the complex rational expression given below:

$\frac{\mleft(\frac{3}{x+1}+\frac{2}{x-1}\mright)}{\frac{(x-1)}{x+1}}$

Solution

To solve the question, we will proceed to simplify the Numerator and Denominator separately.

Numerator:

$\begin{gathered} \frac{3}{x+1}+\frac{2}{x-1} \\ \text{Multiply the numerator and denominator by (x+1)(x-1)} \\ \mleft(\frac{3}{x+1}+\frac{2}{x-1}\mright)\times\frac{(x+1)(x-1)}{(x+1)(x-1)} \\ \text{Expand the bracket} \\ \frac{3}{(x+1)}\times\frac{(x+1)(x-1)}{(x+1)(x-1)}+\frac{2}{(x-1)}\times\frac{(x+1)(x-1)}{(x+1)(x-1)} \\ \\ \frac{3(x-1)}{(x+1)(x-1)}+\frac{2(x+1)}{(x+1)(x-1)} \\ \\ =\frac{3(x-1)+2(x+1)}{(x+1)(x-1)} \end{gathered}$

Denominator:

$\begin{gathered} \frac{1}{\frac{(x-1)}{x+1}} \\ we\text{ can re-write this expression as:} \\ \frac{x+1}{x-1} \end{gathered}$

Now, let us combine the Numerator and Denominator as follows:

$\begin{gathered} \frac{(\frac{3}{x+1}+\frac{2}{x-1})}{\frac{(x-1)}{x+1}}=\mleft(\frac{3}{x+1}+\frac{2}{x-1}\mright)\times\frac{1}{\frac{(x-1)}{x+1}} \\ \\ =\mleft(\frac{3}{x+1}+\frac{2}{x-1}\mright)\times\frac{x+1}{x-1} \\ \\ =\frac{3(x-1)+2(x+1)}{(x+1)(x-1)}\times\frac{(x+1)}{(x-1)} \\ \\ (x+1)\text{ crosses out.} \\ \\ =\frac{3(x-1)+2(x+1)}{(x-1)}\times\frac{1}{(x-1)} \\ \\ =\frac{3(x-1)+2(x+1)}{(x-1)^2}=\frac{3x-3+2x+2}{(x-1)^2} \\ \\ =\frac{5x-1}{(x-1)^2} \end{gathered}$

Final Answer

Numerator:

$5x-1$

Denominator:

$(x-1)^2$