I need some help on three math problems?

Question

How do I rewrite these expressions into a rational expression in the lowest terms?

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

(2x)/(x - x/4)

(x-5) / (2x- 50/x) <----- also what would the restrictions be

Answer 1

a) common denominator is (x-1)(x+1)

3x (x+1) - 2x (x-1) / common

3x^2 + 3x - 2x^2 + 2x / common

x^2 + 5x / common

x (x+5) / ((x-1)(x+1)) or (x^2+5x)/(x^2-1)

(x not equal ±1)

b)

2x / ((4x - x)/4)

8x / 3x

8/3   (x not equal 0)

c)

(x - 5) / (2x - 50/x)

(x - 5) / ((2x^2 - 50)/x)

x (x-5) / (2 (x^2 - 25))

x (x-5) / (2 (x-5) (x+5))

x / (2x + 10)

(x not equal -5)

The general restriction is: denominator not equal 0.

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