. If x≠0, is |x| <1?
(1) x2<1
(2) |x| < 1/x
OA is D
I thought (1) will be true for all fractions between -1 and 1. So |x| < 1. - 1 is sufficient
In (2) , we can express the equation as x < 1/x => x2 < 1 - This is same as 1. => so |x| < 1
But, we can express this equation also as -x < 1/x which makes it x2 > 1. -> This will make |x| > 1.
So, I thought it is A. But not sure why is it OA. Help on this would be great...
Is |x| < 1?
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You are right till this point:sairamGmat wrote:. If x≠0, is |x| <1?
(1) x2<1
(2) |x| < 1/x
OA is D
I thought (1) will be true for all fractions between -1 and 1. So |x| < 1. - 1 is sufficient
In (2) , we can express the equation as x < 1/x => x2 < 1 - This is same as 1. => so |x| < 1
But, we can express this equation also as -x < 1/x which makes it x2 > 1. -> This will make |x| > 1.
So, I thought it is A. But not sure why is it OA. Help on this would be great...
-x < 1/x when x<0
now multiply both sides by x
We know that x is negative. So when we multiply an inequality with negative number, the inequality reverses.
For example. -2 < 1/2
now multiply both sides by -3, it should become
6>-3/2
But not 6<-3/2 .
So,
-x^2>1
x^2<-1 [By multiplying both sides by -1]
This is absurd.
Hence x is always greater than 0
Hope this helps!!
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This is the point where error crept in. If -x < 1/x thensairamGmat wrote:.
But, we can express this equation also as -x < 1/x which makes it x2 > 1. -> This will make |x| > 1.
- x2 > 1 (multiplying both sides by the negative number x)
x2 < -1 (multiplying both sides by -1)
This would mean x2 is a negative number and hence x is an imaginary number. Now, in the GMAT all numbers are real unless specifically mentioned otherwise.
So, x<0 is invalid. Hence what we have is that x is a number greater than 0 yet less than it's reciprocal. A conclusion that can be drawn only for fractional numbers where numberator is less than the denominator. Hence x is fraction and is less than 1.
I didn't understand the statement 2 explanation.
|x| < 1/x
If x is +ve,
x < 1/x ==> x^2 < 1. This is okay. ---(a)
If x is -ve,
-x < 1/x ==> -x^2 < 1 {multiplied by x on both sides}.
OR x^2 > -1. This means, x^2 is > zero. --- (b) {squares cannot be negative}
Since, (a) says x^2 < 1 and (b) says x^2 > -1, we can say |x| < 1.
But how did you guys get
-x^2>1
x^2<-1
|x| < 1/x
If x is +ve,
x < 1/x ==> x^2 < 1. This is okay. ---(a)
If x is -ve,
-x < 1/x ==> -x^2 < 1 {multiplied by x on both sides}.
OR x^2 > -1. This means, x^2 is > zero. --- (b) {squares cannot be negative}
Since, (a) says x^2 < 1 and (b) says x^2 > -1, we can say |x| < 1.
But how did you guys get
-x^2>1
x^2<-1
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st 2: |x| < 1/xsairamGmat wrote:. If x≠0, is |x| <1?
(1) x2<1
(2) |x| < 1/x
OA is D
I thought (1) will be true for all fractions between -1 and 1. So |x| < 1. - 1 is sufficient
In (2) , we can express the equation as x < 1/x => x2 < 1 - This is same as 1. => so |x| < 1
But, we can express this equation also as -x < 1/x which makes it x2 > 1. -> This will make |x| > 1.
So, I thought it is A. But not sure why is it OA. Help on this would be great...
Actually here we need to apply the basic numeric theroy concepts...else plug in numbers
case 1 : X = 2
|X| = 2
1/x = 1/2
|x| < 1/x.. Does our picked numbers satisfy the core condition?? Not really.
So again try X= 1/2
|X| = |1/2|= 1/2
1/X = 1/(1/2)
= 2
|x| < 1/x : 1/2 is LESS THAN 2 ...Does our picked numbers satisfy the core condition?? YES
So from our picked numebrs we can deduce that this condition |x| < 1/x can hold true only for 0<X<1
Now coming back to the main query is |X| < 1;
yes ofcourse since 0<X<1, all values of X will be less than 1
Pick D
gmatmachoman, thanks for your reply. I do agree with you.
But my question is different. I was asking how did previous 2 respondents, before you, get
-x^2>1
x^2<-1
and then they deduce that statement 2 is also sufficient and hence the answer is D. Am I missing something here?
Thanks
But my question is different. I was asking how did previous 2 respondents, before you, get
-x^2>1
x^2<-1
and then they deduce that statement 2 is also sufficient and hence the answer is D. Am I missing something here?
Thanks
- kvcpk
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Hi,roh00kan wrote:gmatmachoman, thanks for your reply. I do agree with you.
But my question is different. I was asking how did previous 2 respondents, before you, get
-x^2>1
x^2<-1
and then they deduce that statement 2 is also sufficient and hence the answer is D. Am I missing something here?
Thanks
Let me come to your rescue
(2) |x| < 1/x
Two cases arise: 1)x>0, 2)x<0
Case1: x>0
When x>0, |x|=x
Hence x<1/x
x^2<1
x^2-1<0
This means that |x|<1
Case2: x<0
When x<0, |x|=-x
-x<1/x
Remember this thing. I have also stated the same on my above post.
Whenever an inequality is multiplied by a negative value, its sign reverses.
Example: 1<2 When multiplied by -1 on both sides,
it changes to -1>-2 and not -1<-2
With that in mind, I am trying to multiply both sides by x (which is negative)
Hence -x^2>1
Multiplying both sides by -1 again,
x^2<-1
this cannot be true. Because x^2 is positive.
hence this statement is sufficient.
Hope this helps!!
"Once you start working on something,
don't be afraid of failure and don't abandon it.
People who work sincerely are the happiest."
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don't be afraid of failure and don't abandon it.
People who work sincerely are the happiest."
Chanakya quotes (Indian politician, strategist and writer, 350 BC-275BC)
KVCPK, Thank you very much.
Well, I understood my mistake. I was considering the 'x' of 1/x as +ve and when I multiplied 'x' on both sides, I didn't reverse the sign as I was thinking that, 'x' is +ve.
Well, I understood my mistake. I was considering the 'x' of 1/x as +ve and when I multiplied 'x' on both sides, I didn't reverse the sign as I was thinking that, 'x' is +ve.
- kvcpk
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Glad it helpdedroh00kan wrote:KVCPK, Thank you very much.
Well, I understood my mistake. I was considering the 'x' of 1/x as +ve and when I multiplied 'x' on both sides, I didn't reverse the sign as I was thinking that, 'x' is +ve.
"Once you start working on something,
don't be afraid of failure and don't abandon it.
People who work sincerely are the happiest."
Chanakya quotes (Indian politician, strategist and writer, 350 BC-275BC)
don't be afraid of failure and don't abandon it.
People who work sincerely are the happiest."
Chanakya quotes (Indian politician, strategist and writer, 350 BC-275BC)