Absolute Value (lost in a)

1. Is 0<x<1?

a) 1/x > [x]

b) [x]<1

Stmt 1:
1/x > |x|
ie 1/x - |x| > 0

The LHS (left hand side) can be >0 ONLY if x is a +ve fraction
U can try the following smart numbers to prove the above.
-2, -1/2, 0, 1/2, 1, 2

So sufficient.


stmt 2:
|x| < 1

|x| is always >= 0.

So here |x| < 1.

But -1 < x < 1 so it is insufficient.

Hence A.

Ht Helps

Ian,
can you please help it solve algebrically?

1. Is 0<x<1?

a) 1/x > [x]

-1/x<x<1/x

by looking at this you should be able to determine two things
1) x is positive becuse if x were negative -1/-#<-# would be violated because the two negatives make a positive.
2) x must be <1 because an integer such as 2 would yield 2<1/2 which wouldn't make sense but 1/2<2 does make sense.

These two things make A sufficient


b) [x]<1

-1<x<1

This alone is not sufficent

now 1/x > |x|

Part 1 :- since |x| is always +ve and 1/x is greater than tht +ve value, 1/x shud definitely be +ve.

Part 2 :- From statement 1 we also get 1>x*|x|.

So x>0 since x is +ve and less than 1 since x*|x|<1

Hence A

First of All, let me make it clear:

[x] and |x| are different

[x] means greatest integer function, this function rounds off the number to nearest smallest integer value.

eg : [8.3] = 8
and [-9.2] =-10


|x| = gives a positive value,no matter x is negative or positive

so |-4| = 4
and |-4.5| = 4.5

a) 1/x > [x]

In the original question
we use greatest integer function

Everyone has misinterpreted the [x] as |x|

(I) this statement means
0< x < 1

because if we take x = 0.5

1/0.5 = 2

and [x] = [0.5] =0

so 1/x > [x]

And if we take -1<x< 0

because x has to be negative fraction lets say: -0.5

1/-0.5 = -2

and [-0.5] = -1

so 1/x not greater than [x]

rest other options where x>1 or x<-1 also cannot satisfy this inequation

So A is Suff

I looks big but it is very easy.


Although one can get an answer correct but I believe it is much more important to learn the correct method, that is what learning is.

https://www.beatthegmat.com/is-x-y-t10389.html#141440

Hope this Helps

Karan

I never saw any q before testing []. Are we sure the original post did not have a typo?

Oops
I have never seen that notation [] before. I suppose the question should have a explanation of []. Since this was absent, I assumed it to be | |.

gmat740 wrote:First of All, let me make it clear:

[x] and |x| are different

[x] means greatest integer function, this function rounds off the number to nearest smallest integer value.

eg : [8.3] = 8
and [-9.2] =-10


|x| = gives a positive value,no matter x is negative or positive

so |-4| = 4
and |-4.5| = 4.5

a) 1/x > [x]

In the original question
we use greatest integer function

Everyone has misinterpreted the [x] as |x|

(I) this statement means
0< x < 1

because if we take x = 0.5

1/0.5 = 2

and [x] = [0.5] =0

so 1/x > [x]

And if we take -1<x< 0

because x has to be negative fraction lets say: -0.5

1/-0.5 = -2

and [-0.5] = -1

so 1/x not greater than [x]

rest other options where x>1 or x<-1 also cannot satisfy this inequation

So A is Suff

I looks big but it is very easy.


Although one can get an answer correct but I believe it is much more important to learn the correct method, that is what learning is.

https://www.beatthegmat.com/is-x-y-t10389.html#141440

Hope this Helps

Karan

IMO C

Karan , what if the value of x = 1?

Karan , what if the value of x = 1?

well In that case

[x] = [1] = 1

so 1/x = [x]

inequality changes to equality

Hope this helps

Karan