If x and y are negative and x^4-y^4

Question

If x and y are negative and  x^4-y^4<0  which of the following must be true?

I.  x<y

II.  xy<y^2

III.  (x+y)^2<(x-y)^2

A. I only 
B. II only 
C. III only 
D. I & II 
E. II & III

Bunuel · Answer

Give:  x^4-y^4<0 :

Re-arrange:  x^4<y^4 ;

Take the fourth root:  |x| < |y| ;

Since both are negative, then  -x < -y ;

Re-arrange:  y < x .

Evaluate options:

I:  x<y . Not true.

II:  xy<y^2 : reduce by y and flip the sign because y is negative:  x > y . TRUE.

III:  (x+y)^2<(x-y)^2 : expand  x^2 + 2xy + y^2 < x^2 - 2xy + y^2 . Simplify:  xy < 0 . Since both x and y are negative, then  xy = negative*negative = positive , thus this options is also not true.

Answer: B.

Hope it's clear.

paddy41 · Answer

Step 1: Rephrasing x^4-y^4<0 to (x^2-y^2)(x^2+y^2)<0 . From this we know that x^2-y^2 has to be negative. Step 2: Rephrasing x^2-y^2<0 to (x-y)(x+y)<0 . Since x and y are both negative we know that (x-y)>0 which means x>y Evaluating the answers: I. not true II. If we divide both sides by y we end up at x>y - true III. If we factor out we end up at 4xy<0 . Since x and y are both negative this can´t be true Answer B.

paddy41 · Answer

Hi Bunuel,

can you shortly elaborate on the underlying rule of the above steps?

Thank you!

Bunuel · Answer

Absolute value properties:

When  x\leq{0}  then  |x|=-x , or more generally when  some \ expression\leq{0}  then  |some \ expression|={-(some \ expression)} . For example:  |-5|=5=-(-5) ;

When  x\geq{0}  then  |x|=x , or more generally when  some \ expression\geq{0}  then  |some \ expression|={some \ expression} . For example:  |5|=5 .

10. Absolute Value 
     Theory   
 Math: Absolute value (Modulus) 
 Absolute Value: Tips and hints 
 Properties of Absolute Values on the GMAT 
 Absolute modulus : A better understanding 
 GMAT Question Patterns: Abolute Value 
 The Reason Behind Absolute Value Questions on the GMAT 
     Questions   
 Inequality and absolute value questions from my collection 
 The E-GMAT Question Series on ABSOLUTE VALUE  
 DS Questions 
 PS Questions

For more check  Ultimate GMAT Quantitative Megathread

Hope it helps.

Mo2men · Answer

From question stem, We can organize exmaples

Let x = -1 & y = -2

Let x =  \frac{-1}{4}  & y =  \frac{-1}{2}

Let x =  \frac{-1}{2}  & y = -1

I:  x<y ........From examples above..........  NOT True

II:  xy<y^2 .... xy-y^2<0 .......y<0 & x>y........From examples above........ True

III:  (x+y)^2<(x-y)^2 ..........Take first example to DISPROVE........  NOT True

Answer: B

TheGuyDangerous · Answer

If Y^4 is greater than X^4 and both X & Y are negative, then Y has to be less than X (for example -8 & -2. (-8)^4 is greater than (-2)^4, but -8 is less than -2)

Substitute and check for x= -2 & y= -3

Smash B

bhanu29 · Answer

Given x^4 < y^4 => |x| < |y| Stmt I: xy Eliminate A,D Stmt 2: xy (|x| + |y|)^2 > (|x| - |y|)^2 because x and y are of the same sign the above expressions will hold true for (x+y)^2 and (x-y)^2 Therefore (x+y)^2 < (x-y)^2 must be false. Eliminate E Correct Answer: B

RC3 · Answer

I did it by assumption method,
 
where x=-2 and y=-3

Am i wrong?

RC3 · Answer

I did it by assumption method,
 
where x=-2 and y=-3

Am i wrong?

Bunuel · Answer

These values also lead to the correct result that only II must be true. However, generally, for must-be-true questions, relying on specific values is not a good approach. A statement might hold for some chosen values but fail for others. That’s why, for must-be-true questions, number plugging is useful mainly to disprove an option, not to establish that it is always true.

If x and y are negative and x^4-y^4<0 which of the following...

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