Yesterday, Manhattan GMAT posted a GMAT question on our blog. Today, they have followed up with the answer:

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 *x*^{2} – *x*, as we are instructed to. Let’s take the latter approach.

[1/(1 – *x*) – 1/*x*] (*x*^{2} – *x*)

Instead of FOILing this product right away, we should factor the expression *x*^{2} – *x* first. If we do so, we will be able to cancel denominators quickly.

*x*^{2} – *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 – 2*x*

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 *x*^{2} – *x* = 2^{2} – 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 – *x*^{2} = 1 – 2^{2} = –3

(D) 2*x* – 1 = 2(2) – 1 = 3

(E) 1 – 2*x* = 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).**

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