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by Brent@GMATPrepNow » Thu May 03, 2012 6:29 am
rahulvsd wrote:Is x positive?

1) (1/(x+1)) < 1.
2) x - 1 is a perfect square.

[spoiler]OA: B [/spoiler]
Statement 1: (1/(x+1)) < 1
case a: x=2, in which case x is positive
case b: x=-2, in which case x is negative
Since we can't answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x - 1 is a perfect square
In other words, x = (a perfect square) + 1
Since perfect squares are greater than or equal to zero, a perfect square + 1 (i.e., x) must be positive.
So, statement 2 is SUFFICIENT and the answer is B

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by aneesh.kg » Fri May 04, 2012 12:48 am
Statement(1):
1/(x + 1) < 1
This inequality holds true many negative values (-2, -3,..) and many positive values (1, 2,..)
INSUFFICIENT.

Statement(2):
If (x - 1) is a perfect square
(x - 1) is positive. x - 1 > 0, or x > 1
That's enough to say that x is positive.
The answer is YES, and this statement is
SUFFICIENT

[spoiler](B)[/spoiler] is the answer
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by rahulvsd » Fri May 04, 2012 9:50 am
I do realize that we get option A as insufficient alone. I simplified it this way:

(1/(x+1))<1
Multiplying denominator:

x+1>1
Hence x>0

What am I missing here?
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by Brent@GMATPrepNow » Fri May 04, 2012 12:01 pm
rahulvsd wrote:I do realize that we get option A as insufficient alone. I simplified it this way:

(1/(x+1))<1
Multiplying denominator:

x+1>1
Hence x>0

What am I missing here?

Excellent question, in fact this exemplifies the trap that the GMAT has created for this question.

We cannot take 1/(x+1)<1 and multiply both sides by x+1 to get 1<x+1

Here's why:
Notice that, if we take the inequality 2 < 3 and multiply both sides by 5 we get 10 < 15. Great, the resulting inequality holds true.
Conversely, if we take the inequality 2 < 3 and multiply both sides by -5 we get -10 < -15. The resulting inequality does not hold true.

So, we can't just multiply both sides of an equality by any number we choose. If we multiply both sides by a positive number, the inequality holds true. If we multiply both sides by a negative number, the inequality does not hold true.

Now take the original inequality: 1/(x+1)<1
If we multiply both sides by x+1, does the inequality hold true? Well, it depends on whether or not x+1 is positive or negative.

Since we cannot be 100% certain x+1 is positive (or negative for that matter), we cannot multiply both sides of the inequality by it.

Takeaway: Before multiplying both sides of an inequality by some variable expression, we must be 100% certain that the variable expression is always positive. Otherwise, we must resort to different kinds of algebraic manipulation (if possible)

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Brent
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by aneesh.kg » Fri May 04, 2012 7:30 pm
Woa, Woa, Woa.

Hold on.

You have to be really careful while multiplying with anything on both sides of an inequality.
Inequalities behave a little differently compared to an equation. Let's see how.

For e.g.
5 > 3
Lets multiply with a positive number on both sides (Let's say 2).
5*2 ? 3*2
What happens to the sign of the inequality?
15 is greater than 6. So, the sign of the inequality does not change.
5*2 > 3*2

Lets multiply with a negative number on both sides (Let's say -2).
5*(-2) ? 3*(-2)
What happens to the sign of the inequality?
-15 is smaller than -6. So, the sign of the inequality REVERSES.
5*(-2) < 3*(-2)

As we saw: (two simple guidelines)
(i) When we multiply both sides of an inequality with a positive quantity, the sign of the inequality DOES NOT CHANGE. Yes, you're right. NO EFFECT.
(ii) When we multiply both sides of an inequality with a negative quantity, the sign of an inequality REVERSES. It exactly reverses.

Do you see your mistake now?
You multiplied both sides by a quantity without changing the sign of the inequality.
That is, you assumed the quantity (x + 1) to be positive. And, we don't know if it is positive or negative.
Had (x + 1) been known to be positive, what you did was correct. But, since we don't, what you did is almost criminal.

This is an important concept and a very common mistake. Good that you made mistake, asked it and got it clarified in time.

Wait, This is getting interesting. Let's probe further?
We will solve Statement (1) by your method but we will consider both the possibilities.

Given: 1/(x + 1) < 1

If (x + 1) > 0, i.e. if x > -1,
(multiplying both sides by (x + 1))
1 < x + 1
or x > 0
What is the common solution of x > -1 and x > 0?
x > 0

However, if (x + 1) < 0, i.e. if x < -1,
(multiplying both sides by (x + 1) and reversing the sign)
1 > x + 1
or x < 0
What is the common solution of x < -1 and x < 0?
x < -1

So, what is the overall solution?
x > 0 OR x < -1

Is this the same solution that we got before also?
Yes. 'x' can be positive as well as negative.

Wow.
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