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# If x ≠ 0 and x/|x| < x, which of the following must be true?

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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]

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20 Jun 2017, 07:32
Marcab wrote:
If $$x\neq{0}$$ and $$\frac{x}{|x|}<x$$, which of the following must be true?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|>1$$

(E) $$-1<x<0$$

We can simplify the given inequality:

x/|x| < x

x < (x)|x|

(x)|x| > x

If x is positive, we can divide both sides by x and obtain |x| > 1.

If x is negative, we can also divide both sides by x, but we have to switch the inequality sign, so we have |x| < 1.

We see that if x is positive, |x| > 1, which is choice C, and if x is negative, |x| < 1, which is choice D. However, since we don’t know whether x is positive or negative, both choice C and choice D “can be true,” not “must be true.”

Let’s analyze further. If x is positive, |x| = x. So, |x| > 1 means x > 1, which is choice A. If x is negative, |x| = -x. So, |x| < 1 means -x < 1 or x > -1. However, because x is negative, we have -1 < x < 0, which is choice E. Again, since we don’t know whether x is positive or negative, both choice A and choice E “can be true,” not “must be true.”

This leaves choice B as the correct answer. In fact, it’s the correct choice because the inequality x > -1 includes both x > 1 and -1 < x < 0. So, regardless of whether x is positive or negative, we can say x > -1.

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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]

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10 Aug 2017, 22:25
Bunuel wrote:
Marcab wrote:
If $$x\neq{0}$$ and $$\frac{x}{|x|}<x$$, which of the following must be true?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|>1$$

(E) $$-1<x<0$$

Explanations required for this one.
Not convinced at all with the OA.

My range is -1<x<0 and x>1.

Notice that we are asked to find which of the options MUST be true, not COULD be true.

Let's see what ranges does $$\frac{x}{|x|}< x$$ give for $$x$$. Two cases:

If $$x<0$$ then $$|x|=-x$$, hence in this case we would have: $$\frac{x}{-x}<x$$ --> $$-1<x$$. But remember that we consider the range $$x<0$$, so $$-1<x<0$$;

If $$x>0$$ then $$|x|=x$$, hence in this case we would have: $$\frac{x}{x}<x$$ --> $$1<x$$.

So, $$\frac{x}{|x|}< x$$ means that $$-1<x<0$$ or $$x>1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the range $$-1<x<0$$ or $$x>1$$ will definitely be more the $$-1$$.

As for other options:

A. $$x>1$$. Not necessarily true since $$x$$ could be -0.5;
C. $$|x|<1$$ --> $$-1<x<1$$. Not necessarily true since $$x$$ could be 2;
D. $$|x|>1$$ --> $$x<-1$$ or $$x>1$$. Not necessarily true since $$x$$ could be -0.5;
E. $$-1<x<0$$. Not necessarily true since $$x$$ could be 2.

P.S. Please read carefully and follow: http://gmatclub.com/forum/rules-for-pos ... 33935.html Please pay attention to the rules #3 and 6. Thank you.

If in case I multiply |x| both sides then inequality will not change and then if i approach like this then how the inequality should be solved :-

X<X|X|
x|x|-x>0
x(|x|-1)>0
now either x>0 or |x|>1,x<0 |x|<1

how to proceed further to solve it to get the range as per the question.

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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]

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10 Aug 2017, 23:43
himanshukamra2711 wrote:
Bunuel wrote:
Marcab wrote:
If $$x\neq{0}$$ and $$\frac{x}{|x|}<x$$, which of the following must be true?

(A) $$x>1$$

(B) $$x>-1$$

(C) $$|x|<1$$

(D) $$|x|>1$$

(E) $$-1<x<0$$

Explanations required for this one.
Not convinced at all with the OA.

My range is -1<x<0 and x>1.

Notice that we are asked to find which of the options MUST be true, not COULD be true.

Let's see what ranges does $$\frac{x}{|x|}< x$$ give for $$x$$. Two cases:

If $$x<0$$ then $$|x|=-x$$, hence in this case we would have: $$\frac{x}{-x}<x$$ --> $$-1<x$$. But remember that we consider the range $$x<0$$, so $$-1<x<0$$;

If $$x>0$$ then $$|x|=x$$, hence in this case we would have: $$\frac{x}{x}<x$$ --> $$1<x$$.

So, $$\frac{x}{|x|}< x$$ means that $$-1<x<0$$ or $$x>1$$.

Only option which is ALWAYS true is B. ANY $$x$$ from the range $$-1<x<0$$ or $$x>1$$ will definitely be more the $$-1$$.

As for other options:

A. $$x>1$$. Not necessarily true since $$x$$ could be -0.5;
C. $$|x|<1$$ --> $$-1<x<1$$. Not necessarily true since $$x$$ could be 2;
D. $$|x|>1$$ --> $$x<-1$$ or $$x>1$$. Not necessarily true since $$x$$ could be -0.5;
E. $$-1<x<0$$. Not necessarily true since $$x$$ could be 2.

P.S. Please read carefully and follow: http://gmatclub.com/forum/rules-for-pos ... 33935.html Please pay attention to the rules #3 and 6. Thank you.

If in case I multiply |x| both sides then inequality will not change and then if i approach like this then how the inequality should be solved :-

X<X|X|
x|x|-x>0
x(|x|-1)>0
now either x>0 or |x|>1,x<0 |x|<1

how to proceed further to solve it to get the range as per the question.

$$x(|x|-1)>0$$

Case 1: $$x > 0$$ and $$|x| > 1$$ ($$x < -1$$ or $$x > 1$$) --> $$x > 1$$.

Case 2: $$x < 0$$ and $$|x| < 1$$ ($$-1 < x < 1$$) --> $$-1 < x < 0$$.

Finally, $$-1 < x < 0$$ or $$x > 1$$.
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Re: If x ≠ 0 and x/|x| < x, which of the following must be true? [#permalink]

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30 Aug 2017, 10:56
according to explanation from BB, such kinds of math problem will often appear in gmat, and the answer is based on the assumption of the values of x. In other words, what can be inferred from the interval of values of x is the right answer.

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Re: If x ≠ 0 and x/|x| < x, which of the following must be true?   [#permalink] 30 Aug 2017, 10:56

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