GMAT Question of the Day

1. The equation is true for all real $x$ except $x = \pm 2$ (correct answer)
2. The equation has exactly one solution: $x = 0$
3. The equation is an identity and holds for $x = \pm 2$
4. The equation has no solutions because it leads to $0 = 2x$

Explanation: Start with the left side: $\frac{1}{x-2} + \frac{1}{x+2} = \frac{(x+2) + (x-2)}{(x-2)(x+2)} = \frac{2x}{x^2-4}$. This matches the right side exactly, making it an identity. However, both sides are undefined when $x = \pm 2$ (since $x^2-4 = 0$), so the equation holds for all real $x$ except $x = \pm 2$. Choice B incorrectly thinks we need to solve $2x = 2x$, which would give $0 = 0$ (true for all $x$), not $x = 0$. Choice C incorrectly claims the equation holds at $x = \pm 2$ where it's undefined. Choice D incorrectly suggests we get $0 = 2x$, which would come from a calculation error.