simplify the complex fraction x x+2 1/x+1/x+2

Question

GPT 3.5 · Answer

To simplify the complex fraction $$\frac{x}{x+2} \div \left(\frac{1}{x} + \frac{1}{x+2}\right)$$, first combine the individual fractions on the right side: $$\frac{1}{x} + \frac{1}{x+2} = \frac{x+2 + x}{x(x+2)} = \frac{2x+2}{x(x+2)} = \frac{2(x+1)}{x(x+2)}$$ Now we can simplify the complex fraction by dividing the numerator by the denominator. So, we get: $$\frac{x}{x+2} \div \left(\frac{1}{x} + \frac{1}{x+2}\right) = \frac{x}{x+2} \div \frac{2(x+1)}{x(x+2)} = \frac{x}{x+2} \times \frac{x(x+2)}{2(x+1)} = \frac{x^2}{2(x+1)}$$ Therefore, the simplified form of the given complex fraction is $$\frac{x^2}{2(x+1)}$$.