Complicated absolute value

[El Cucu]

Is [x] <1?

a) x / [x] < x

b) x < [x]


OA is C

I know the problem asks whether x is between -1 and 1. But I don't know how to solve or think the choices.Tksvm!

[kobel51]

I'll be looking forward to this explanation as well. This is a tough question. what is the source?

[mjjking]

At first I was confused too, but it simply requires you to pick numbers.

try for instance:

stmt 1.

4/4 =1 holds
-2/2 = -1 not right
1/2 / 1/2 =1 not right
-1/2 / 1/2 = -1 holds

So we find that to satisfy stmt 1 x could either be >1 or -1<x<0.

Stmt 2 tells us what we need to add to stmt 1 to get C as answer.

Hope this helps!

[vittalgmat]

@El Cucu,
R u sure that u posted the entire question?
I think u missed "X is not equal to 0" Otherwise stmt 1 will fail and GMAC avoids infinity/indeterminate values.
Next time pls copy the problem entirely.

Let me assume that x is != 0 and solve the problem. (here ! implies "not")

stmt 1.
x/|x| < x

since x != 0, we can cross multiply.
x < x* |x|-----------------------1

case 1. x is +ve.

we can divide by x and < sign will remain the same.
so we get 1 < |x|

ie 1 < x (coz x is +ve)

case 2: x is -ve.
< will become ">"

Dividing by x,
1 > |x|

Now x is -ve so we get 1 > -x or -1 < x < 0 ( x < 0 coz we assumed that x is negative.. in case 2)

combining case 1 and 2. we get.
x > 1 OR -1 < x < 0 ====> ambiguous
Hence stmt 1 is insufficient.


stmt 2.

x < |x|

again .. intuitively one can say that x has to be -ve. for this to be true.

eg. let x = -2
so -2 < |-2| ==> -2 < 2 OK.
But it just says that x is -ve and nothing else. hence insufficent

Combining 1 and 2.

Apply x is -ve from stmt 2 to stmt 1.
we get -1 < x < 0.

hence x < 1

Answer is C.

HT Helps

[El Cucu]

I see now that the part x is different from 0 was blurred in the sheet, that's why I couldn't post it in the statement. Sorry for that. All the best