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 Post subject: GMAT Algebra (Data Sufficiency)
PostPosted: Wed Aug 04, 2010 8:47 am 
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Joined: Sun May 30, 2010 3:15 am
Posts: 424
Is |x – 1| < 1 for all integers x?
(1) (x – 1)² > 1
(2) x < 0

A. Statement (1) BY ITSELF is sufficient to answer the question, but statement (2) by itself is not.
B. Statement (2) BY ITSELF is sufficient to answer the question, but statement (1) by itself is not.
C. Statements (1) and (2) TAKEN TOGETHER are sufficient to answer the question, even though NEITHER statement BY ITSELF is sufficient.
D. Either statement BY ITSELF is sufficient to answer the question.
E. Statements (1) and (2) TAKEN TOGETHER are NOT sufficient to answer the question, meaning that further information would be needed to answer the question.

(D) If we take the square root of both sides of the inequality in Statement (1), we get:
|x – 1| > 1.

This tells us that 1 is definitely not greater than |x – 1|, so Statement (1) is sufficient.

Remember: When you take the square root of a variable expression that has already been squared, you must take the absolute value of the squared expression in the process. A simple example is a² = 9. Let’s solve this:
a² = 9
|a| = 3
a = 3 or a = -3.

The same rule holds when dealing with inequalities, as we saw above.

The easiest way to evaluate the sufficiency of Statement (2) is probably to pick numbers. We are told that x is negative.

If x = -1, then the question becomes:
Is 1 > |-1 – 1|?
Is 1 > |-2|?
Is 1 > 2? The answer is NO.

If x = -2, then the question becomes:
Is 1 > |-2 – 1|?
Is 1 > |-3|?
Is 1 > 3? Again, the answer is NO.

The answer will always be NO if x is negative. The more negative the value of x, the greater the value of |x – 1|. So Statement (2) is also sufficient.

Since both statements are sufficient individually, the correct answer is choice (D).


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 Post subject: Re: math: algebra, inequalities
PostPosted: Wed Aug 04, 2010 9:19 am 
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Joined: Sun May 30, 2010 2:23 am
Posts: 498
We have three different inequalities in the question statement. Let us solve each one and draw the answer on the number line.

The first inequality is:
|x – 1| < 1
-1 < x – 1 < 1
0 < x < 2
Image

The second inequality is:
(x – 1)² > 1
|x – 1| > 1
x – 1 > 1 or x – 1 < -1
x > 2 or x < 0
Image


The third inequality is
x < 0
Image

We see that shaded area that represents possible values of x for the first inequality doesn't intersect with others. So both statements will give a definite "NO". (You may think of it as if we pick any number from the shaded area on second or third picture it will not be in the shaded area on the first one).

That means that both statements are sufficient on their own.

Furthermore if we had a situation when shaded region for the first inequality partly overlapped with another one then it would mean that corresponding statement is insufficient.

If we had a situation when shaded region for the first inequality completely enclosed another one then it would mean that corresponding statement sufficient (It will give a definite "YES").


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