Is 3^x 1 (2) 0

Question

Is  3^x < x^3 ?

(1)  3^x > 1

(2)  0 < x < 1

pushpitkc · Answer

(1)  3^x > 1 
If x=1 or 3, the equation  3^x < x^3  is not true
But, if x =  \frac{5}{2} , the equation  3^x < x^3  is true.(Insufficient)

(2)  0 < x < 1 
If value of x is between 0 and 1,
x could be  \frac{1}{4}  or  \frac{1}{2} 
When we plug x= \frac{1}{2} ,  3^x < x^3  is false
Similarly when we plug x= \frac{1}{4}  also,  3^x < x^3  is false.
Whatever the value of x in this range, the equation is always false(Sufficient)  (Option B)

niks18 · Answer

Statement 1 : implies that x>0 but at x=3 ; 3^x=x^3 so the relation does not always hold true. Insufficient Statement 2 : this implies x is positive. Any number which is greater than 1 raised to any positive power will yield a number greater than 1. this can be derived as - 3>1 now raise both sides of the inequality to power x so 3^x>1^x =>3^x>1 again as x<1 , raise both sides of the inequality to power 3 so x^3<1^3 => x^3<1 So we have 3^x>x^3 . Hence a NO for our question stem. Sufficient Option B You can also use the plug-in method to arrive at Option B but I guess it will involve multiple testing and calculations.

dhruv solanki · Answer

what should be ideal approach for this question plugin or concept??

niks18 · Answer

Hi dhruv solanki Any approach that helps you solve the question within 2 minutes and strike you first should be the ideal approach. The so called short-cut methods you might have come across will rarely give you confidence under GMAT exam pressure and chances are you might miss a hidden concept required to solve the question. Anyways so far I have not come across a real GMAT question that cannot be solved within 2 minutes using proper mathematical concept

chetan2u · Answer

Hi..

you will agree any question done on concept is surely correct and increases your confidence level..
as here if you know - 
x is negative :- 3^x will be <1... it in all probability will be a positive fraction AND x^3 will be negative
x is 0 :- 3^x will be 1 AND x^3 will be 0
and so x>0 will give you a value more than 3^0 or 1 AND x^3 is less than 1 if 0<x <1

so when you know these, you will be sure of answer..

OR you will have to plug three values - highest possible, lowest possible and something in middle and still can be in some doubt.
ofcourse if you do not know concept, go for substitution

substitution on the other hand is very effective in few cases 
when you are asked for value of x in some equation and choices are given

dhruv solanki · Answer

in this question, suppose x=3 then it will be equal so is the information insufficient?

chetan2u · Answer

Hi..

If one of the statement said that x=3..
so our answer for "Is  3^x<x^3  ?" will be NO and the statement will be sufficient as it is giving us a clear NO

vatsal323 · Answer

1) How did you think of 5/2 in this case?
2) Even if you did, calculating the values is so tough without a calculator. The first term is 15.58, while the second one is 15.62. Little too tough don't you think?

MBAB123 · Answer

Hi  chetan2u  and other members, can anybody help me in breaking the 1st statement? How do you solve 3^x>x^3?

chetan2u · Answer

Hi Brian123 The question requires to be looked at from perspective of what the GMAT can test you at max. You won’t require to solve the inequality completely, but analyse it from the perspective of GMAT. Let us analyse 3^x0 x^3 <0. So answer will be NO 2) When x is a positive fraction <1 3^x will be greater than 1 even if x is 0.0000001, because even 3^0=1 x^31 Now there can be certain situations where we will get yes and somewhere no. Both 3^x and x^3 are different curves which surely intersect at x=3. If they intersect at 3, surely somewhere x^3>3^x Just for information, 2.51 We know this is true for all positive values of x, that is x>0. Case (3) gives both yes and no. Insufficient 2. \ \ 0

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Is 3^x < x^3? (1) 3^x > 1 (2) 0 < x < 1

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