Each week Manhattan GMAT posts a GMAT question on our blog and follows up with the answer the next day. Are you up for the challenge?
We can attack this problem by doing Direct Algebra. First, carry out the replacement. That is, literally replace every x in the expression with 1 – x, putting parentheses around the 1 – x in order to preserve proper order of operations:
Original: 1/x – 1/(1 – x)
Replacement:
1/(1 – x) – 1/(1 – (1 – x))
Now simplify the second denominator: (1 – (1 – x)) = (1 – 1 + x) = x
So the replacement expression becomes this:
1/(1 – x) – 1/x
This should make sense. If we replace x by 1 – x, then it turns out that we are also replacing 1 – x by x (since 1 – (1 – x) = x). Thus, the denominators of the original expression are simply swapped.
Now we can either combine these fractions first (by finding a common denominator) or go ahead & multiply by x2 – x, as we are instructed to. Let’s take the latter approach.
[1/(1 – x) – 1/x] (x2 – x)
Instead of FOILing this product right away, we should factor the expression x2 – x first. If we do so, we will be able to cancel denominators quickly.
x2 – x factors into (x – 1)x. We can now rewrite the product:
[1/(1 – x) – 1/x] (x – 1)x
= (x – 1)x/(1 – x) – (x – 1)x/x
The second term, (x – 1)x/x, becomes just x – 1 after we cancel the x’s.
Since (x – 1) = –(1 – x), we can rewrite the first term as –(1 – x)x/(1 – x) and then cancel the (1 – x)’s, leaving –x.
So, the final result is
–x – (x – 1) = –x – x + 1 = 1 – 2x
This is the answer.
Separately, since this is a Variables In Choices problem, we could instead pick a number and calculate a target. Since 0 and 1 are disallowed, let’s pick x = 2. We are told that x should be replaced by 1 – x, so we calculate 1 – x = –1 and put in –1 wherever x is in the original expression.
1/x – 1/(1 – x) = 1/(–1) – 1/(1 – (–1))
= –1 – ½
= –3/2
Now multiply this number by x2 – x = 22 – 2 = 2. We get –3 as our target number.
Finally, we plug x = 2 into the answer choices and look for –3:
(A) x + 1 = 2 + 1 = 3
(B) x – 1 = 2 – 1 = 1
(C) 1 – x2 = 1 – 22 = –3
(D) 2x – 1 = 2(2) – 1 = 3
(E) 1 – 2x = 1 – 2(2) = –3
We can eliminate choices A, B, and D, but to choose between C and E, we would need to pick another number. For instance, if we pick x = 3, we get a target of –5. Only E fits this target.
The correct answer is (E).
