Source: MGMAT question
If x is not equal to 0, is |x| less than 1?
(1) x / |x| < x
(2) |x| > x
Statement 1)
Cross multiply with Mod (x) --it's positive so we can multiply isn't it?
So that tells that x < x* mod(x)
Essentially it's telling us that X is >0
Example: x =2
2<2*2 =4
X= -2
2<-2*2---not possible
Statement 2) mod(x) > x
we can infer that x<0
So both are statements are insufficient.
But the explanation given is:
The question "Is |x| less than 1?" can be rephrased in the following way.
Case 1: If x > 0, then |x| = x. For instance, |5| = 5. So, if x > 0, then the question becomes "Is x less than 1?"
Case 2: If x < 0, then |x| = -x. For instance, |-5| = -(-5) = 5. So, if x < 0, then the question becomes "Is -x less than 1?" This can be written as follows:
-x < 1?
or, by multiplying both sides by -1, we get
x > -1?
Putting these two cases together, we get the fully rephrased question:
"Is -1 < x < 1 (and x not equal to 0)"?
Another way to achieve this rephrasing is to interpret absolute value as distance from zero on the number line. Asking "Is |x| less than 1?" can then be reinterpreted as "Is x less than 1 unit away from zero on the number line?" or "Is -1 < x < 1?" (The fact that x does not equal zero is given in the question stem.)
(1) INSUFFICIENT: If x > 0, this statement tells us that x > x/x or x > 1. If x < 0, this
statement tells us that x > x/-x or x > -1. This is not enough to tell us if -1 < x < 1.
(2) INSUFFICIENT: When x > 0, x > x which is not true (so x < 0). When x < 0, -x > x or
x < 0. Statement (2) simply tells us that x is negative. This is not enough to tell us if -1 < x < 1.
(1) AND (2) SUFFICIENT: If we know x < 0 (statement 2), we know that x > -1 (statement 1). This means that -1 < x < 0. This means that x is definitely between -1 and 1.
The correct answer is C.
GMAT Forum
6 postsPage 1 of 1
- Saurabh Malpani
- GMAT 5/18
Saurabh,
I didn't go through your entire post, but I thought I would shed some light on something I noticed earlier on in your reasoning. This is what you wrote early in your post:
Statement 1)
Cross multiply with Mod (x) --it's positive so we can multiply isn't it?
So that tells that x < x* mod(x)
Essentially it's telling us that X is >0
Example: x =2
2<2*2 =4
X= -2
2<-2*2---not possible
I don't agree that Statement I tells us X>0. I think it tells us that -1<X<0 and X>1. For example, you have already proved that x=2 works, but if we use x=1, it doesn't as 1<1*1 is not true. Also, if you try x=-0.25, it works as -1/4<-1/16.
Also, it's a good idea to test the Statements by putting the test numbers back into the original form of the numbers - that way you will know for sure whether the numbers work or don't work (I have caught myself incorrectly rephrasing from time to time).
Like I said, I didn't read anymore of your post, but hopefully this helps in you finding the answer!
I didn't go through your entire post, but I thought I would shed some light on something I noticed earlier on in your reasoning. This is what you wrote early in your post:
Statement 1)
Cross multiply with Mod (x) --it's positive so we can multiply isn't it?
So that tells that x < x* mod(x)
Essentially it's telling us that X is >0
Example: x =2
2<2*2 =4
X= -2
2<-2*2---not possible
I don't agree that Statement I tells us X>0. I think it tells us that -1<X<0 and X>1. For example, you have already proved that x=2 works, but if we use x=1, it doesn't as 1<1*1 is not true. Also, if you try x=-0.25, it works as -1/4<-1/16.
Also, it's a good idea to test the Statements by putting the test numbers back into the original form of the numbers - that way you will know for sure whether the numbers work or don't work (I have caught myself incorrectly rephrasing from time to time).
Like I said, I didn't read anymore of your post, but hopefully this helps in you finding the answer!
- Saurabh Malpani
Aha!!! ....thanks GMAT 5/18 I think I for the answer!! Great!!!
Thanks fore your help!!!
Saurabh Malpani
Thanks fore your help!!!
Saurabh Malpani
GMAT 5/18 Wrote:Saurabh,
I didn't go through your entire post, but I thought I would shed some light on something I noticed earlier on in your reasoning. This is what you wrote early in your post:
Statement 1)
Cross multiply with Mod (x) --it's positive so we can multiply isn't it?
So that tells that x < x* mod(x)
Essentially it's telling us that X is >0
Example: x =2
2<2*2 =4
X= -2
2<-2*2---not possible
I don't agree that Statement I tells us X>0. I think it tells us that -1<X<0 and X>1. For example, you have already proved that x=2 works, but if we use x=1, it doesn't as 1<1*1 is not true. Also, if you try x=-0.25, it works as -1/4<-1/16.
Also, it's a good idea to test the Statements by putting the test numbers back into the original form of the numbers - that way you will know for sure whether the numbers work or don't work (I have caught myself incorrectly rephrasing from time to time).
Like I said, I didn't read anymore of your post, but hopefully this helps in you finding the answer!
- Saurabh Malpani
I forgot to mention in my previous Post that the explanation given by MGMAT for statement 1 concludes that:
1) INSUFFICIENT: If x > 0, this statement tells us that x > x/x or x > 1. If x < 0, this
statement tells us that x > x/-x or x > -1
The bold part is confusing!!! shouldn't it be what GMAT 5/18 suggested!! -1<X<0
Please confirm!
1) INSUFFICIENT: If x > 0, this statement tells us that x > x/x or x > 1. If x < 0, this
statement tells us that x > x/-x or x > -1
The bold part is confusing!!! shouldn't it be what GMAT 5/18 suggested!! -1<X<0
Please confirm!
Saurabh Malpani Wrote:Aha!!! ....thanks GMAT 5/18 I think I for the answer!! Great!!!
Thanks fore your help!!!
Saurabh MalpaniGMAT 5/18 Wrote:Saurabh,
I didn't go through your entire post, but I thought I would shed some light on something I noticed earlier on in your reasoning. This is what you wrote early in your post:
Statement 1)
Cross multiply with Mod (x) --it's positive so we can multiply isn't it?
So that tells that x < x* mod(x)
Essentially it's telling us that X is >0
Example: x =2
2<2*2 =4
X= -2
2<-2*2---not possible
I don't agree that Statement I tells us X>0. I think it tells us that -1<X<0 and X>1. For example, you have already proved that x=2 works, but if we use x=1, it doesn't as 1<1*1 is not true. Also, if you try x=-0.25, it works as -1/4<-1/16.
Also, it's a good idea to test the Statements by putting the test numbers back into the original form of the numbers - that way you will know for sure whether the numbers work or don't work (I have caught myself incorrectly rephrasing from time to time).
Like I said, I didn't read anymore of your post, but hopefully this helps in you finding the answer!
- Guest
I rephrased the question as: is x a fraction (the abs val of a fraction will be < 1)
I think that starting with statement 2 is a litte easier.
#2 says: |x| > x .... so X has to be a negative number, either an integer or fraction, so insufficient alone
#1 says: x < x * |x| therefore, x could be a positive integer (such as 2) or a negative fraction (such as -1/2), so insufficient alone
together it says that x has to be negative, so it could only ne a negative fraction, and therefore the answer is C
I think that starting with statement 2 is a litte easier.
#2 says: |x| > x .... so X has to be a negative number, either an integer or fraction, so insufficient alone
#1 says: x < x * |x| therefore, x could be a positive integer (such as 2) or a negative fraction (such as -1/2), so insufficient alone
together it says that x has to be negative, so it could only ne a negative fraction, and therefore the answer is C
- dbernst
- ManhattanGMAT Staff
- Posts: 300
- Joined: Mon May 09, 2005 9:03 am
A great debate! Just the type of discourse we like to see on our forums.
First, I would definitely rephrase the question to Is -1< x <1 (with x not equal to zero)?
Then, start with statement (2) and a BD/ACE grid. Statement 2 simply tells us that x < 0, so clearly this is not sufficient. Eliminate BD.
Statement (1) says x / |x| < x. This is the tricky one. If x > 0, we can simply manipulate by multiplying both sides by x. Thus, we have x^2 > x or x > 1. If x < 0, we can rewrite as x/-x < x; then, we can multiply both sides by -x (the opposite of x, which is a positive value because x <0) to get x < x(-x). Divide both sides by x (don't forget to flip sign since x is a negative value) to get 1 > -x; finally divide both sides by -1 to get x > -1.
To summarize statement (1) says the following: If x is positive, x > 1; If x is negative x > -1. This is insufficient since when x is negative it gives us a "yes" to our original rephrased question, but when x is positive it gives us a "no."
Together, we are only concerned with x as a negative (info from statement (2)). Combine this with statement (1) and we get -1 < x < 0. This is sufficient to definitively answer our question.
The correct answer is C.
First, I would definitely rephrase the question to Is -1< x <1 (with x not equal to zero)?
Then, start with statement (2) and a BD/ACE grid. Statement 2 simply tells us that x < 0, so clearly this is not sufficient. Eliminate BD.
Statement (1) says x / |x| < x. This is the tricky one. If x > 0, we can simply manipulate by multiplying both sides by x. Thus, we have x^2 > x or x > 1. If x < 0, we can rewrite as x/-x < x; then, we can multiply both sides by -x (the opposite of x, which is a positive value because x <0) to get x < x(-x). Divide both sides by x (don't forget to flip sign since x is a negative value) to get 1 > -x; finally divide both sides by -1 to get x > -1.
To summarize statement (1) says the following: If x is positive, x > 1; If x is negative x > -1. This is insufficient since when x is negative it gives us a "yes" to our original rephrased question, but when x is positive it gives us a "no."
Together, we are only concerned with x as a negative (info from statement (2)). Combine this with statement (1) and we get -1 < x < 0. This is sufficient to definitively answer our question.
The correct answer is C.
6 posts Page 1 of 1