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.

Algebraic expressions follow distributive law.

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}

Answer and Explanation:

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}


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Distributive Property: Definition, Use & Examples

from High School Algebra II: Help and Review

Chapter 2 / Lesson 20
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