Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:
The fastest way to solve this problem is first to recognize that an algebraic approach will take a little time. Essentially, we will have to multiply through by the product (x – 2)(x + 2)(x – 1), then simplify.
If, instead, we glance at the answer choices, we see that 3 of them make one of the denominators zero, a result that is not allowed (we cannot divide by zero). Specifically, x cannot be –2 because one denominator is x + 2; likewise, x cannot be 1 or 2, since we have x – 1 and x – 2 as denominators as well.
Thus, the only two possible answers are –1 and 0. We try each in turn.
If x = –1, then we have the following:
1/(–3) = 1/(1) + 1/(–2)?
–1/3 = 1 – 1/2?
This is not true.
However, if x = 0, then we have the following:
1/(–2) = 1/(2) + 1/(–1)?
–1/2 = 1/2 – 1?
–1/2 = –1/2?
This is true, so x can be equal to 0.
Alternatively, we could take the algebraic approach.
First, we multiply through by the product (x – 2)(x + 2)(x – 1) to eliminate denominators.
(x – 1)(x + 2) = (x – 2)(x – 1) + (x – 2)(x + 2)
x2 + x – 2 = x2 – 3x + 2 + x2 – 4
0 = x2 – 4x
0 = x(x – 4)
x = 0 or x = 4
The correct answer is (C).