Around the World in 80 Questions (Day 5): Is x

Question

Is  x < 0  ?

(1)  5^x < x^6

(2)  6^x < x^5

Muskaan09 · Answer

The answer would be B.
1. For any negative value, integer or fraction, 
x=-1
5^(-1)<(-1)^6
1/5<1
Yes
x=(-1/2)
5^(-1/2)<(-1/2)^6
1/sqrt(5)<1/(2^6) No

Therefore not sufficient.

2. 6^x<x^5
For -1
1/6<(-1)^5
1/6<-1 no

For(-1/2)
6^(-1/2)<(-1/2)^5
1/sqrt(6)<-1/(2^5) no.
Sufficient to say x is not less than 0.

Posted from my mobile device

Archit3110 · Answer

target is x -ve
#1
 5^x < x^6 
this will be true
when x = -2 and x = 5
insufficient
#2
 6^x < x^5  
this will be true only for x +ve values
x = 3
x >0 ; sufficient
  option B is correct

kratos0906 · Answer

My Answer: B, Statement(2) alone is sufficient.

Statement 1: Condition holds true for both x=-1 and x=2. So, we can't conclude that x<0.

Statement 2: If x<0, then 6^x will always be positive and x^5 will always be negative. So, 6^x<x^5 is only possible when x is non-negative. Hence, sufficient.

missionmba2025 · Answer

(1)  5^x < x^6

A) Assume x is a negative number very very close to -1

5^{-0.999999999} < (-0.999999999) ^6

\frac{1}{5^{0.999999999}} < (-0.999999999) ^6

\frac{1}{5^{0.999999999}}   approximately is equal to  \frac{1}{5}

(-0.999999999) ^6  approximately is equal to  1

Hence, we can have a negative value of x.

B) Assume x = 2

5^{2} < (2) ^6

Hence, we can be a positive value as well.

Not sufficient.

(2)  6^x < x^5

If x is negative, x^5 is always negative.

6^x is always positive regardless the value of x.

Hence, a positive value cannot be less than a negative value.

Therefore x cannot be negative.

Sufficient.

IMO B

Kinshook · Answer

Asked: Is x < 0 ? (1) 5^x < x^6 If x=0; 5^x = 5^0 = 1; x^6 = 0 ; 5^x =1 > x^6=0 If x=1; 5^x = 5; x^6 = 1^6 = 1; 5^x = 5 > x^6 = 1 If x=2; 5^2 = 25; x^6 = 2^6 = 64; 5^x = 25 < x^6 = 64 For all x>1; 5^x < x^6 If x=-1; 5^x = 1/5 = 0.2; (-1)^6 = 1; 5^x = .2 < x^6 = 1 If x=-2; 5^x = 1/25 = .04; (-2)^6 = 64; 5^x = .04 < x^6 = 64 For all x<0; 5^x < x^6 5^x < x^6 if x<0 or x>1 NOT SUFFICIENT (2) 6^x < x^5 If x=0; 6^x = 6^0 = 1; x^5 = 0 ; 6^x =1 > x^5=0 If x=1; 6^x = 6; x^5 = 1^5 = 1; 6^x = 6 > x^5 = 1 If x=2; 6^x = 6^2 = 36; x^5 = 2^5 = 32; 6^x = 36 > x^5 = 32 If x=3; 6^x = 6^3 = 216; x^5 = 3^5 = 243; 6^x = 216 < x^5 = 243 For all x>2; 6^x < x^5 If x=-1; 6^x = 1/6 ; (-1)^5 = -1; 6^x =1/6 > x^5 = -1 If x=-2; 6^x = 1/36 ; (-2)^5 = -32; 6^x =1/36 < x^5 = -32 For all x<0; 6^x > x^5 If 6^x < x^5 then x is NOT <0 SUFFICIENT IMO B

Around the World in 80 Questions (Day 5): Is x < 0 ? (1) 5^x

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