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Manhattan CAT DS Number Line Positive

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Manhattan CAT DS Number Line Positive

by marshelle.slayton » Mon Nov 24, 2014 3:34 pm
Is the number x positive?

(1) On the number line, 0 is closer to x - 1 than to x.
(2) On the number line, 0 is closer to x than to x + 1.

The answer I find a bit confusing. Please answer and explain. I am confused by the description they gave.

OA A
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Source: — Data Sufficiency |
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by MartyMurray » Mon Nov 24, 2014 4:33 pm
marshelle.slayton wrote:Is the number x positive?

(1) On the number line, 0 is closer to x - 1 than to x.
(2) On the number line, 0 is closer to x than to x + 1.

The answer I find a bit confusing. Please answer and explain. I am confused by the description they gave.

OA A
Statement 1 says that 0 is closer to x - 1 than it is to x.

We can start by ruling out that x is negative. If x were negative, then x - 1 would be further from 0 than x is.

Next, from Statement 1 we can tell that x is not 0. If x were 0, 0 would be closer to x than it is to x - 1.

Finally, from Statement 1 x can be positive. If subtracting 1 from a number makes it closer to 0, that number is postive. So Statemment 1 is sufficient.

Now look at Statement 2.

x could be 0, and then 0 would be closer to x than to x + 1. Also x could be positive, in which case 0 would still be closer to x than it is to x + 1.

So Statement 2 is insufficient.

Choose A.
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by GMATGuruNY » Mon Nov 24, 2014 4:57 pm
Is the number x positive?

(1) On the number line, 0 is closer to x - 1 than to x.
(2) On the number line, 0 is closer to x than to x + 1.
|x| = the distance between 0 and x.
|x-y| = the distance between 0 and x-y.
|x+y| = the distance between 0 and x+y.

Statement 1: On the number line, 0 is closer to x - 1 than to x.
|x-1| < |x|.
Since there is absolute value notation on each side, we can square the inequality.
(x-1)² < x²
x² - 2x + 1 < x²
-2x < -1
x > 1/2.
Thus, x must be positive.
SUFFICIENT.

Statement 2: On the number line, 0 is closer to x than to x + 1.
|x| < |x+1|.
Since there is absolute value notation on each side, we can square the inequality.
x² < (x+1)²
x² < x² + 2x + 1
-2x < 1
x > -1/2.
Thus, x could be negative or positive.
INSUFFICIENT.

The correct answer is A.
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by [email protected] » Mon Nov 24, 2014 5:24 pm
Hi marshelle.slayton,

For this DS question, you'll find it pretty easy to TEST values:

We're asked if x is positive? This is a YES/NO question. We don't know anything about x to start.

For this question you'll find that drawing a picture might be helpful; Number Properties will also make this question a bit easier to handle. Remember what the question's asking for though: Is x positive?

If x > 0 then the answer is YES
If x = 0 then the answer is NO
If x < 0 then the answer is NO

Fact 1 tells us that 0 is closer to (x - 1) than to x.

Here, consider the possibilities...
Could x = 0??? Is 0 closer to -1 than to 0?
No, it's not, so x CANNOT = 0

Could x = a negative?
Try x = -1
Is 0 closer to -2 than to -1
No, it's not, so x CANNOT = a negative

If x CANNOT = 0 and x CANNOT = a negative, then all that's left are positives!!!!
So x MUST be positive and the answer to the question is ALWAYS YES.
Fact 1 is SUFFICIENT

Fact 2 tells us that 0 is closer to x than to (x+1)

Let's consider the possibilities here:

Could x = 0???
0 is closer to 0 than to 1, so 0 IS POSSIBLE and the answer to the question is NO.

Could x = 1???
0 is closer to 1 than to 2, so 1 IS POSSIBLE and the answer to the question is YES.
These results are inconsistent.
Fact 2 is INSUFFICIENT

Final Answer: A

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by marshelle.slayton » Tue Nov 25, 2014 5:52 am
All thanks for the help. During my practice test I did plug answers in and reach the same conclusion but when I plugged fractions in, it didn't work out so well.

If you try x=1/3 the first statement is no longer sufficient. 0 < |1/3| < |(1/3)-1| C would be closer to 1/3 than -2/3.

No where in the prompt did it describe if X had to be an integer. Can someone please explain why the OA is still A?
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by GMATGuruNY » Tue Nov 25, 2014 8:05 am
marshelle.slayton wrote:All thanks for the help. During my practice test I did plug answers in and reach the same conclusion but when I plugged fractions in, it didn't work out so well.

If you try x=1/3 the first statement is no longer sufficient. 0 < |1/3| < |(1/3)-1| C would be closer to 1/3 than -2/3.

No where in the prompt did it describe if X had to be an integer. Can someone please explain why the OA is still A?
When we evaluate statement 1, we may consider only values that SATISFY statement 1.

Statement 1: On the number line, 0 is closer to x - 1 than to x.
x=1/3 does not satisfy this constraint, since 0 is NOT closer to x-1 = -2/3 than to x = 1/3.
Thus, we may not consider x=1/3 when we evaluate statement 1.

As I noted in my solution above, only values greater than 1/2 will satisfy statement 1.
x=3/4 works, since 0 is closer to x-1 = -1/4 than to x=3/4.
x=1 works, since 0 is closer to x-1 = 0 than to x=1.
x=10 works, since 0 is closer to x-1 = 9 than to x=10.
Since statement 1 is satisfied only by values greater than 1/2, we know that x must be POSITIVE.
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by MartyMurray » Tue Nov 25, 2014 4:30 pm
There you have it Marshelle. Mitch laid it out well.

To get that high score, be clear about what you did to get that wrong answer.

I was tempted to do the same thing you did on this one.

Much of maximizing a GMAT score is about getting problems right.

This one for instance, used very simple concepts, and yet is still a little tricky. You certainly don't need to dredge up some arcane math concept to answer the question. You just need to get it right.

So to maximize your score it would probably help to see exactly what you did here and seek not to do that in the future, and if you do do something similar on another problem, to be aware of that and seek to change whatever it is that underlies that pattern.
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by marshelle.slayton » Wed Nov 26, 2014 11:36 am
Oh that makes more sense! I was making it to complicated and forgot that the statement defined the set!

Thanks for your help guys!
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