M02-21

Question

If  x  is a positive integer, is  y^2 (x^3 - x + 1) \gt 75?

(1)  y \gt 1 
 
(2)  (x-1)^2=16

Bunuel · Accepted Answer

Official Solution:

If  x  is a positive integer, is  y^2 (x^3 - x + 1) \gt 75?  
 
(1)  y>1 . No information about  x . Insufficient.
 
(2)  (x-1)^2=16 . 
 
Take the square root:  |x-1|=4 . This implies that  x=-3  or  x=5 . Since we are given that  x  is a positive integer, we discard the first root and have  x=5 . Thus,  the question becomes : is  121y^2>75 ? If  y=0 , then the answer is NO; but if  y=1 , then the answer is YES. Not sufficient..
 
(1)+(2) From (2), the question was rephrased to: is  121y^2>75 ? Statement (1) tells us that  y>1 . For any  y  greater than 1,  121y^2  will indeed be greater than 75. Sufficient.

Answer: C

disharupani · Answer

I solved statement 2 in following manner:

(x-1)^2=16
x^2-2x+1=16
x^2-2x=15
x(x-2)=15
So x=0 or x=17

Where did I go wrong?

Bunuel · Answer

You can equate multiples to 0, when the whole product equals 0. For example, if it were x(x - 2) = 0, then yes, x = 0 or x = 2. Notice that if x = 0, then x(x - 2) = 15, does not hold. Also, it's not clear how you got x = 17, there. Again, if x = 17, then x(x - 2) = 15, does not hold.

ttaiwo · Answer

Hi Bunuel,

I got the correct answer but I'd like clarity on statement 2:

(x-1)^2 = 16
I square rooted both sides to get:
x-1 = 4...then eventually x=5

The crux of my question is this...I got x=4 because the question stated that x is a positive integer.

If the question did not state that x was a positive integer, I would've gone with two solutions: x-1=4 and x-1=-4...giving x=5 and x=(-3) respectively.

Would that have been the correct approach if the question did not specify that x was a positive integer? I'm slightly confused because I think I remember you saying that in gmat land...they only take the positive of a root...so whether or not the question specified x= a positive integer...I still would've had to have taken the positive root?

Bunuel · Answer

(x-1)^2 = 16

x - 1 = 4 or x - 1 = -4

x = 5 or x = -3

When the GMAT provides the square root sign for an even root, such as a square root, fourth root, etc. then the only accepted answer is the positive root.  That is:

\sqrt{9} = 3 , NOT +3 or -3;
 \sqrt {16} = 2 , NOT +2 or -2;

Notice that in contrast, the equation  x^2 = 9  has TWO solutions, +3 and -3. Because  x^2 = 9  means that  x =-\sqrt{9}=-3  or  x=\sqrt{9}=3 .

sdacha · Answer

I think this is a  high-quality  question and I agree with explanation.

Basshead · Answer

Great question. Chose E initially because I didn't see that x had to be a positive integer.

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

ritwick2 · Answer

Hi  Bunuel  have 2 questions. 
1)With DS questions I try to breakdown the main stem first to the max extent and then move to each statement. Is this the right way to minimise time?
2) wrt this problem, i did not know how to use statement 2 given x^3-x+1. Could you help me understand this?

Bunuel · Answer

1. Yes, simplifying the question from the stem is generally a good thing to do. In this question, though, the question is already in its simplest form.
2. From (2), we can calculate x, as shown in the solution, and substitute it into x^3 - x + 1 to get its value.

M02-21

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