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

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To simplify the given expression, we need to find a common denominator for the two fractions in the numerator and the two fractions in the denominator.

Given expression:

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

First, let's factor the denominators in all four fractions:

(x^2 - 4) = (x + 2)(x - 2)
(2x - 2) = 2(x - 1)

Now, rewrite the expression with factored denominators:

2x/[(x + 2)(x - 2)] - 2/[(x + 2)(x - 2)] : [(x + 1)/[2(x - 1)] - 1/(x - 1)]

Next, find a common denominator for all four fractions:

Common denominator = [(x + 2)(x - 2)(2)(x - 1)]

Rewrite the fractions with the common denominator:

[2x - 2(2)] / [(x + 2)(x - 2)] - [2(x + 2)] / [(x + 2)(x - 2)] : [2(x + 1)(x - 2) - (x - 2)] / [(2)(x + 2)(x - 2)]

Now, simplify the expression:

[(2x - 4) - 2(x + 2)] / [(x + 2)(x - 2)] : [(2x + 2 - 2)(x - 2)] / [(2)(x + 2)(x - 2)]

[2x - 4 - 2x - 4] / [(x + 2)(x - 2)] : [(2x + 2 - 2)(x - 2)] / [(2)(x + 2)(x - 2)]

-8 / [(x + 2)(x - 2)] : [2x(x - 2)] / [(2)(x + 2)(x - 2)]

-8 / [(x + 2)(x - 2)] : [2x^2 - 4x] / [2(x + 2)(x - 2)]

Since both denominators are the same, we can now divide the numerators:

-8 / [2x^2 - 4x]

Therefore, the simplified expression is -8 / [2x^2 - 4x].