# Subtract these two rational expressions: {2 x + 3} / {x - 1} - {x - 1} / {x + 1}.

## Question:

Subtract these two rational expressions: {eq}\dfrac {2 x + 3} {x - 1} - \dfrac {x - 1} {x + 1} {/eq}.

## Distributive Property

An algebraic expression is an expression which consists of constants, variables and algebraic operations like addition, subtraction, product and division.

If there are two linear algebraic expressions as

{eq}\displaystyle f(x)\ =\ ax\ +\ b\\ \displaystyle g(x)\ =\ cx\ +\ d {/eq}

then according to the distributive property, their product can be written as

{eq}\displaystyle f(x)\ \times\ g(x)\ =\ (ax\ +\ b)\ \times\ ( cx\ +\ d)\ =\ ax\ \times\ (cx\ +\ d)\ +\ b\ \times\ (cx\ +\ d) {/eq}

Here we have,

{eq}\displaystyle \dfrac {2 x + 3} {x - 1} - \dfrac {x - 1} {x + 1}\\ \displaystyle =\ \frac{(2x\ +\ 3)((x\ +\ 1)\ -\ (x\ -\ 1)(x\ -\ 1)}{(x\ -\ 1)(x\ +\ 1)}\\ \displaystyle =\ \frac{2x(x\ +\ 1)\ +\ 3(x\ +\ 1)\ -\ x(x\ -\ 1)\ +\ 1(x\ -\ 1)}{x(x\ -\ 1)\ +\ 1(x\ -\ 1)}\\ \displaystyle =\ \frac{2x^2\ +\ 2x\ +\ 3x\ +\ 3\ -\ x^2\ +\ x\ +\ x\ -\ 1}{x^2\ -\ x\ +\ x\ -\ 1}\\ \displaystyle =\ \frac{2x^2\ -\ x^2\ +\ 2x\ +\ 3x\ +\ x\ +\ x\ +\ 3\ -\ 1}{x^2\ -\ 1}\\ \displaystyle =\ \frac{x^2\ +\ 7x\ +\ 2}{x^2\ -\ 1} {/eq} 