## Manhattan GMAT Challenge Problem of the Week – 14 Jan 10

Welcome back to this week’s Challenge Problem! As always, the problem and solution below were written by one of our fantastic instructors. Each challenge problem represents a 700+ level question. If you are up for the challenge, however, set your timer for 2 minutes and go!

## Question

If 1/(

x– 2) = 1/(x+ 2) + 1/(x– 1), which of the following is a possible value ofx?

(A) –2

(B) –1

(C) 0

(D) 1

(E) 2

## Solution

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)

+ *x* – 2 = – 3*x* + 2 + – 4

0 = – 4*x*

0 = *x*(*x* – 4)

*x* = 0 or *x* = 4

**The correct answer is (C) 0.**

To view the current Challenge Problem, simply visit the Challenge Problem page on Manhattan GMAT’s website.

## 2 comments

aryan on January 15th, 2010 at 10:29 am

well easy one

Srikanth on January 15th, 2010 at 4:27 pm

1/(x-2) = 1/(x+2) + 1/(x-1)

=> 1/(x-2) = (2x+1)/(x+2)(x-1)

=> (x+2)/(x-2) = (2x+1)/(x-1)

By componendo-Dividendo, we have

2x/4 = 3x/(x+2)

x^2 + 2x = 6x

x^2 - 4x = 0 x=0 or 4.

hence, C is the answer.

Yep, that was an easy one.