Simplify, and state the restrictions on the variable : (x^2-x-6/x^2-4)/(x^2-2x+1/x^2-1)

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a.) Essentially, you have one rational expression divided by another. So let's simplify each one individually:

(x^2-x-6) = (x-3)(x+2) = (x-3)

(x^2-4) (x-2)(x+2) (x+2)

In the second step, in its factored form, we find the restrictions for the denominator are x cannot equal positive or negative 2.

Repeat for the second expression:

(x^2-2x+1) = (x-2)(x+1) = (x-2)

(x^2-1) (x-1)(x+1) (x-1)

Our restrictions here: x cannot be positive or negative 1.

Division is the same as multiplying by the reciprocal. Now take the first simplified expression and multiply it by the reciprocal of the second.

(x-3) (x-1)

(x+2) (x-2) with the given restrictions x is not 1, -1, 2, -2

You are allowed to ask only one question at a time. I have therefore edited your question.

For the expression (x^2-x-6/x^2-4)/(x^2-2x+1/x^2-1) the simplified form is:

(x^2-x-6/x^2-4)/(x^2-2x+1/x^2-1)

=> ((x^2 - 3x + 2x - 6)/(x - 2)(x + 2))/((x - 1)^2/(x - 1)(x + 1))

=> ((x - 3)(x + 2)/(x - 2)(x + 2))/((x - 1)^2/(x - 1)(x + 1))

=> ((x - 3)/(x - 2))/((x - 1)/(x + 1))

=> (x - 3)(x + 1)/(x - 2)((x - 1)

By restrictions I assume you mean what values of the variable do not give a meaningful result. Here if x = -2, x = 2, x = 1 and x = -1, we get a form where one number is being divided by 0.

**The simplified form is: (x - 3)(x + 1)/(x - 2)((x - 1). Also, x cannot take on the values {-2, -1, 1, 2}**

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