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# help

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Simplify $$\dfrac{\frac{x}{x - 1} - \frac{x}{x + 1}}{\frac{x}{x^2 - 1}}$$

Jun 14, 2020

#1
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Let's first combine the top two fractions in the numerator.

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

This is what our fraction looks like now:
$$\frac{\frac{2x}{x^2-1}}{\frac{x}{x^2-1}}$$

We can multiply top and bottom by $$x^2-1$$, which cancels to $$\frac{2x}{x}=2$$

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Jun 14, 2020
edited by thelizzybeth  Jun 15, 2020
#2
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Multiplying both the numerator and the denominator by $$(x^2 - 1)$$, we have

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

This holds for $$x \neq 0$$

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Jun 14, 2020