If x≠0 and x/|x|<x, which of the following must be true?
A. x>1
B. x>−1
C. |x|<1
D. |x|>1
E. −1<x<0
Source: GMATClub Tests
OA: B
Tricky absloute question
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Prove that four of the five answer choices do NOT have to be true.Mo2men wrote:If x≠0 and x/|x|<x, which of the following must be true?
A. x>1
B. x>−1
C. |x|<1
D. |x|>1
E. −1<x<0
Constraint: x/|x| < x
It's possible that x=2, since 2/|2| < 2, with the result that C and E do not have to be true.
Eliminate C and E.
It's possible that x=-1/2, since (-1/2)/|-1/2| < -1/2, with the result that A and D do not have to be true.
Eliminate A and D.
The correct answer is B.
Algebra:
x/|x| < x
x < x|x|
0 < x|x| - x
0 < x (|x| - 1)
The CRITICAL POINTS are -1, 0 and 1.
These are the only values where x(|x|-1) = 0.
To determine the ranges where x(|x|-1) > 0, test one value to the left and right of each critical point.
Plug x = -2 into x/|x| < x:
-2/ |-2| < -2
-1 < -2.
Doesn't work.
x < -1 is not a valid range.
Plug x = -1/2 into x/|x| < x:
-1/2/ |-1/2| < -1/2
-1 < -1/2.
This works.
-1<x<0 is a valid range.
Plug x = 1/2 into x/|x| < x:
(1/2)/ |1/2| < 1/2
1 < 1/2
Doesn't work.
0<x<1 is not a valid range.
Plug x = 2 into x/|x| < x:
2/ |2| < 2
1 < 2.
This works
x > 1 is a valid range.
Thus, the valid ranges are -1<x<0 and x>1.
Case 1: x=2
Eliminate C, since it doesn't have to be true that |x|<1.
Eliminate E, since it doesn't have to be true that -1<x<0.
Case 2: x=-1/2
Eliminate A, since it doesn't have to be true that x>1.
Eliminate D, since it doesn't have to be true that |x|>1.
The correct answer is B.
Since both -1<x<0 and x>1 are to the right of -1, it must be true that x > -1.
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Excellent solution by Mitch.Mo2men wrote:If x≠0 and x/|x|<x, which of the following must be true?
A. x>1
B. x>−1
C. |x|<1
D. |x|>1
E. −1<x<0
Source: GMATClub Tests
OA: B
Just to add an important aspect:
He concluded that the valid ranges are -1 < x < 0 and x > 1, and the answer is [spoiler]B: x > -1[/spoiler]. Though the invalid range 0 < x < 1 falls within x > -1, this does not mean that [spoiler]OA: B[/spoiler] is also an incorrect option or none is correct.
This is a must be true kind of question, and for the solution range -1 < x < 0 and x > 1, the inequality x > -1 makes merry both!
Another take on it: If you are unsure of how many goals Messi score but these would be 2, [not 3], 4, 5, or more, you can say a MUST BE TRUE statement: Messi scored more than one goal.
Hope this helps!
-Jay
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GMAT/MBA Expert
- Jay@ManhattanReview
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Excellent solution by Mitch.Mo2men wrote:If x≠0 and x/|x|<x, which of the following must be true?
A. x>1
B. x>−1
C. |x|<1
D. |x|>1
E. −1<x<0
Source: GMATClub Tests
OA: B
Just to add an important aspect:
He concluded that the valid ranges are -1 < x < 0 and x > 1, and the answer is [spoiler]B: x > -1[/spoiler]. Though the invalid range 0 < x < 1 falls within x > -1, this does not mean that [spoiler]OA: B[/spoiler] is also an incorrect option or none is correct.
This is a must be true kind of question, and for the solution range -1 < x < 0 and x > 1, the inequality x > -1 makes merry both!
Another take on it: If you are unsure of how many goals Messi score but these would be 2, [not 3], 4, 5, or more, you can say a MUST BE TRUE statement: Messi scored more than one goal.
Hope this helps!
-Jay
_________________
Manhattan Review GMAT Prep
Locations: New York | Tokyo | Manchester | Geneva | and many more...
Schedule your free consultation with an experienced GMAT Prep Advisor! Click here.