Simplify: $\dfrac{x-2}{x^2-1} - \dfrac{x+1}{x^2-2x+1}$
This is a class 10 Question From the Simplification of Rational Expressions chapter of Unit Algebra (Mathematics). All the steps for the solutions are mentioned in the description below. If that's hard for you to navigate, you can always visit the Facebook link given at the end of every posts. 

Solution:

Given,

= $\dfrac{x-2}{x^2-1} - \dfrac{x+1}{x^2-2x+1}$

= $\dfrac{x-2}{x^2-1} -\dfrac{x+1}{(x-1)^2}$

= $\dfrac{x-2}{x^2-1}$ x $\dfrac{x-1}{x-1} -\dfrac{x+1}{(x-1)^2}$ x $\dfrac{x+1}{x+1}$

= $\dfrac{(x-2)(x-1)}{(x^2-1)(x-1)} - \dfrac{(x+1)^2}{(x^2-1)(x-1)}$

= $\dfrac{(x-2)(x-1) -(x+1)^2}{(x-1)(x^2-1)}$

= $\dfrac{x^2 -3x+2 -(x^2 +2x+1)}{(x-1)(x^2-1)}$

= $\dfrac{x^2-3x+2-x^2-2x-1}{(x-1)(x^2-1)}$

= $\dfrac{1-5x}{(x-1)(x^2-1)}$

= Answer


Explanation to the above answer.


Step 1: Copy the same question given.

Step 2: The denominator of the second term of the expression had a formula of (a-b)² = (a²-2ab+b²). So, we write (a-b)² instead. 

Step 3: We need to make the denominators same to perform the subtraction. Therefore, we take the LCM of the two terms and do mathematics accordingly.

Step 4: Now, in the numerator, we write the expressions in factor form. On the denominator, we write the expressions as (a²-b²)(a-b). Because, when we factor every terms in the denominator, we get (a-b)(a-b)(a+b). This can be written as (a²-b²)(a-b) or (a-b)(a²-b²).

Step 5: We subtract the two terms as they have like denominators. While subtracting, we write only one denominator. 

Step 6: Now, we write the expanded form of these factors in the numerator.

Step 7: As we had a subtraction sign, we need to multiply the second term's expression by the '-' sign before we can perform further operations. So, we do this.

Step 8: Similar terms with opposite signs in the numerator gets added and results 0. The numbers are subtracted. And, we get the expression {(1-5x)/(x-1)(x²-1)} as our answer. We write a positive sign in the numerator before a negative sign. 


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Question: Simplify: (x-2)/(x²-1) - (x+1)/(x²-2x+1). | Simplification of Rational Expressions | SciPiPupil

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