Bunuel wrote:
Hussain15 wrote:
If x is not equal to 0, is |x| less than 1?
(1) x/|x|< x
(2) |x| > x
Will really appreciate if answer is supported by explanation.
\(x\neq{0}\), is \(|x|<1\)? Which means is \(-1<x<1\)? (\(x\neq{0}\))
(1) \(\frac{x}{|x|}< x\)
Two cases:
A. \(x<0\) --> \(\frac{x}{-x}<x\) --> \(-1<x\). But remember that \(x<0\), so \(-1<x<0\)
B. \(x>0\) --> \(\frac{x}{x}<x\) --> \(1<x\).
Two ranges \(-1<x<0\) or \(x>1\). Which says that \(x\) either in the first range or in the second. Not sufficient to answer whether \(-1<x<1\). (For instance \(x\) can be \(-0.5\) or \(3\))
Second approach: look at the fraction \(\frac{x}{|x|}\) it can take only two values:
1 for \(x>0\) --> so we would have: \(1<x\);
Or -1 for \(x<0\) --> so we would have: \(-1<x\) and as we considering the range for which \(x<0\) then completer range would be: \(-1<x<0\).
The same two ranges: \(-1<x<0\) or \(x>1\).
(2) \(|x| > x\). Well this basically tells that \(x\) is negative, as if x were positive or zero then \(|x|\) would be equal to \(x\). Only one range: \(x<0\), but still insufficient to say whether \(-1<x<1\). (For instance \(x\) can be \(-0.5\) or \(-10\))
Or two cases again:
\(x<0\)--> \(-x>x\)--> \(x<0\).
\(x>0\) --> \(x>x\): never correct.
(1)+(2) Intersection of the ranges from (1) and (2) is the range \(-1<x<0\) (\(x<0\) (from 2) and \(-1<x<0\) or \(x>1\) (from 1), hence \(-1<x<0\)). Every \(x\) from this range is definitely in the range \(-1<x<1\). Sufficient.
Answer: C.
I did mine a little differently.
If x ≠ 0, is |x| < 1?
Means: Is -1 < x < 1?
(1) \(\frac{x}{|x|}< x\)
x < |x|*x
If x is positive, then: \(\frac{x}{x} < |x|\) which is the same as \(1 < |x|\)
This means that \(-1 > x > 1\)
If x is negative, then: \(\frac{x}{x} > |x|\) which is the same as \(1 > |x|\)
This means that \(-1 < x < 1\) (we switch the direction of < to > because we divided by -1 to put |x| by itself).
These two answers are inconsistent: x is both less than 1 and greater than 1. So, insufficient.
(2) \(|x| > x\)
This means that x is negative since the negative value of something is always less than the absolute value of something; whereas, a positive or 0 value of something is equal to its absolute value.
If x is equal to a negative value less than -1, then it's within the range -1 < x < 1. But, if x is equal to a large negative value, |x| can be out of range, for example, x = -1,000. Therefore, (2) is insufficient.
(1) + (2)
|x| > x and |x| > 1
x is negative, and we know that x < -1. So, pick an arbitrary value less than -1, say, -5.
Plug it into equation (1): \(\frac{-5}{|-5|} < -5\). Is \(-1< -5\)? False. x does not refer to values < -1.
|x| > x and |x| < 1
x is negative, and we know that x > -1. Therefore, \(-1 < x < 0\).
Pick an arbitrary value between -1 and 0, non-inclusive, say, -0.5.
Plug it into equation (1): \(\frac{-0.5}{|-0.5|} < -0.5\). Is \(-1 < -0.5\)? Yes, this works. Sufficient.
Answer C.