Must be true problem

[sam2304]

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

[pemdas]

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

when x>0 we have 1/x < 1 and 1 < x
when x<0 we have 1/(-x) > 1 and 1/x < -1

answer deleted
i missed sign above

[Anurag@Gurome]

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

For x > 0, |x| = x ---> x/|x| = x/x = 1

• Now if |x|/x < x, then 1 < x

For x < 0, |x| = -x ---> x/|x| = x/(-x) = -1

• Now if |x|/x < x, then -1 < x. Hence, -1 < x < 0

Therefore, either -1 < x < 0 or x > 1.

In both cases, x > -1

The correct answer is B.

[sam2304]

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

For x > 0, |x| = x ---> x/|x| = x/x = 1

• Now if |x|/x < x, then 1 < x

For x < 0, |x| = -x ---> x/|x| = x/(-x) = -1

• Now if |x|/x < x, then -1 < x. Hence, -1 < x < 0

Therefore, either -1 < x < 0 or x > 1.

In both cases, x > -1

The correct answer is B.

I am confused with the bold part. I tried to plot in graph for better understanding, yet I wasn't clear. How is it, x > -1 from both the statements ? Where am I going wrong ?

[GMATGuruNY]

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

Try to DISPROVE the answer choices.

A states that x>1.
B states that x>-1.
Try a value BETWEEN -1 and 1.
Plug x=-.5 into x/|x| < x:
-.5/|-.5| < -.5
-1 < -.5.
This works.
Eliminate A (since it doesn't have to be true that x>1) and D (since it doesn't have to be true that |x|>1.)

B states that x>-1.
E states that -1<x<0.
Try a value GREATER than 0.
Plug x=2 into x/|x| < x:
2/|2| < 2.
1 < 2.
This works.
Eliminate E (since it doesn't have to be true that -1<x<0) and C (since it doesn't have to be true that |x|<1).

The correct answer is B.

[Anurag@Gurome]

How is it, x > -1 from both the statements ?

Because x > 1 means x is greater than -1. Also -1 < x < 0 means x greater than -1.

The question asked which of the following must be true for x. None of the options other than C, i.e. x > -1 is true about x. To make it more clear,

• A. x > 1 ---> If -1 < x < 0, then this is not true about x.
  B. x > -1 ---> This is true about x for both the ranges.
  C. |x| < 1 ---> If x > 1, then this is not true about x.
  D. |x| > 1 ---> If -1 < x < 0, then this is not true about x.
  E. -1 < x < 0 ---> If x > 1, then this is not true about x.

Keep in mind that what the question have asked is not same as "For what range of values of x, x/|x| < x?"

Hope that helps.

[sam2304]

B. x > -1 ---> This is true about x for both the ranges.

Keep in mind that what the question have asked is not same as "For what range of values of x, x/|x| < x?"

These two points cleared my doubts. I was interpreting it the way you mentioned. Thanks a lot Anurag.

Thanks a lot Mitch for a different approach

[bubbliiiiiiii]

How is it, x > -1 from both the statements ?

Because x > 1 means x is greater than -1. Also -1 < x < 0 means x greater than -1.

The question asked which of the following must be true for x. None of the options other than C, i.e. x > -1 is true about x. To make it more clear,

• A. x > 1 ---> If -1 < x < 0, then this is not true about x.
  B. x > -1 ---> This is true about x for both the ranges.
  C. |x| < 1 ---> If x > 1, then this is not true about x.
  D. |x| > 1 ---> If -1 < x < 0, then this is not true about x.
  E. -1 < x < 0 ---> If x > 1, then this is not true about x.

Keep in mind that what the question have asked is not same as "For what range of values of x, x/|x| < x?"

Hope that helps.

What if x = 1? This does not fit into the statement! Could you please help me what am I missing?

[sam2304]

What if x = 1? This does not fit into the statement! Could you please help me what am I missing?

You are thinking the same way I did. First we have to understand the question properly. Given x/|x| < x is true. Break down the inequality and find the range which comes for the given condition. Check the diagram which I have posted above. x > 1 and -1 < x < 0 are two intervals which are satisfied by the inequality x/|x| < x. What the question asks is to find the range which fits both these intervals or shaded region in the picture. Now look at the answer choices and you will find the answer easily.