If r is not equal to 0, is r^2/|r| -1 (2)

Question

If r is not equal to 0, is r^2/|r| < 1?

(1) r > -1

(2) r < 1

AS far as i know the option B looks sufficient. Since, r<1, it can take values that are negative like -2 or fraction values like 
1/2. in either case the value of r^2/ |R| is <1. The OA suggests other wise.

Bunuel · Accepted Answer

Is \frac{r^2}{|r|}<1 ? --> reduce by |r| --> is |r|<1 ? or is -11 . Hope it's clear.

esledge · Answer

The first thing to note is that the question isn't testing sign. They tell us that r is not 0, and by definition, both r^2 and |r| are positive. So neither of these statements would be more useful than the other alone.

Since pos/pos = pos, we are ok doing a little creative manipulation of r^2/|r| = |(r*r)/r| = |r|. This move (putting the absolute value sign around the whole thing) isn't a rule to memorize or anything. I'm just ignoring sign temporarily, cancelling, then just assuring the positive result I need with the bars.

This question is really asking "Is r a fraction, or is it larger than 1 (in absolute value)?"

sunnyarora · Answer

C, value of 'r' shall fall in the range -1 and 1 for a single solution to exist.

kylexy · Answer

I guess i did make a mistake in the calc....my bad!!! thanks for the info bunuel!!!

hfbamafan · Answer

I just did this question on MGMAT, and learning to use number lines as a tool to answer these questions.

And I got it going well so far. Here is how I did it.

First I simplified the statement, but this is how I did it.

Instead of using long drawn out algebra I made it into 2 conditions.

I made \(\frac{r^2}{|r|} < 1\) into two conditions; first where, both r, and |r| is positive.

Creating the equation r<1, then I took the reverse and said that <-1 creating r>1

Then I took the absolute value into the picture and made it into and created r>1 and r>-1

Which makes it that the answer has to r is between -infinity and positive infinity.

And the only solution that satisfies those conditions is C.

I may have made an error in one of my rationales above but it works

sanjoo · Answer

How r^2/lrl reduce to lrl only ???

Bunuel · Answer

r^2=|r|*|r|  -->  \frac{r^2}{|r|}  -->  \frac{|r|*|r|}{|r|}  -->  |r| .

Hope it's clear.

Temurkhon · Answer

r^2/|r|<1 ---> r^2<|r| Logically, the only way when any number squared is less than the same number not squared is when the number is between -1 and 1 S1. r>-1 only one part of interval, so INSUFFICIENT S2. r<1 again, only one part of interval, INSUFFICIENT S1+S2 gives full interval, SUFFICIENT C

BillyZ · Answer

Since  |r|  is always positive, we can multiply both sides of the inequality by  |r|  and rephrase the question as: Is  r^{2} < |r |  ? The only way for this to be the case is if  r is a nonzero fraction between  -1  and  1 .
 
(1)  INSUFFICIENT : This does not tell us whether  r  is between  -1  and  1 . If  r = - \frac{1}{2}  ,  |r| = \frac{1}{2}  and  r^{2} = \frac{1}{4}  , and the answer to the rephrased question is  YES .  However, if  r = 4,  ,  |r| = 4 and  r^{2} = 16 , and the answer to the question is  NO .

(2)  INSUFFICIENT : This does not tell us whether  r  is between  -1  and  1 . If  r = \frac{1}{2}  ,  |r| = \frac{1}{2}  ans  r^{2} = \frac{1}{4}  , and the answer to the rephrased question is  YES .  However, if  r = -4 ,  |r| = 4  and  r^{2}=16 , and the answer to the question is  NO .
 
(1) AND (2)  SUFFICIENT : Together, the statements tell us that r is between  -1  and  1 . The square of a proper fraction (positive or negative) will always be smaller than the absolute value of that proper fraction.
 
The correct answer is  C .

CEdward · Answer

If r is not equal to 0, is r^2/|r| < 1?

(1) r > -1
r = 2 NO
r = 1/2 YES
Insufficient

(2) r < 1
r = -2 NO
r = -1/2 YES
Insufficient

C: Sufficient
-1 < r < 1 and r =/ 0 implies that r is a fraction

fraction / fraction is always < 1 if the x/y and x < y

sujoykrdatta · Answer

We know that: r^2 = |r|^2 Thus, the question essentially is: Is |r|^2/|r| < 1? i.e. Is |r| < 1? i.e. Is -1 < r < 1 From (1): r > -1 is not sufficient since we need to also ensure that r < 1 => Insufficient From (2): r < 1 is not sufficient since we need to also ensure that r > -1 => Insufficient Combining (1) and (2): We have -1 < r < 1 => Sufficient Answer C

KarishmaB · Answer

Note that r^2 is the same as |r|^2.

\frac{r^2}{|r|} = \frac{|r|^2}{|r|}

Since we are given that r cannot be 0, we can easily cancel off |r| to get: 
Is |r| < 1?

When will the distance of r from 0 be less than 1? 
When r lies between -1 to 1. So we need both statements to say that |r| will be less than 1.

Answer (C)

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If r is not equal to 0, is r^2/|r| < 1? (1) r > -1 (2)

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