Questions: Simplify the complex fraction. ((x/y)-(y/x))/((x-y)/(xy)), x ≠ 0, y ≠ 0, x ≠ y

Solution Steps

To simplify the given complex fraction, we need to follow these steps:

1. Simplify the numerator: $\frac{x}{y} - \frac{y}{x}$.
2. Simplify the denominator: $\frac{x-y}{xy}$.
3. Divide the simplified numerator by the simplified denominator.

The numerator of the complex fraction is given by: $$\frac{x}{y} - \frac{y}{x}$$ To combine these fractions, we find a common denominator, which is $xy$: $$\frac{x^2 - y^2}{xy}$$

The denominator of the complex fraction is: $$\frac{x - y}{xy}$$

Now, we can express the complex fraction as: $$\frac{\frac{x^2 - y^2}{xy}}{\frac{x - y}{xy}}$$ This simplifies to: $$\frac{x^2 - y^2}{x - y}$$

The expression $x^2 - y^2$ can be factored using the difference of squares: $$x^2 - y^2 = (x - y)(x + y)$$ Thus, we have: $$\frac{(x - y)(x + y)}{x - y}$$ Since $x \neq y$, we can cancel $x - y$ from the numerator and denominator: $$x + y$$

Final Answer