Good one

[Deepthi Subbu]

S = x^3+3^x

Is S>0?

1. x<0

2. |x|>1

[bblast]

IMO A


we can plug in -2,-3 etc for X and check statment one always gives -ve S.


statement 2 says x can be -ve or +ve so S will vary accordingly

[Deepthi Subbu]

IMO A


we can plug in -2,-3 etc for X and check statment one always gives -ve S.


statement 2 says x can be -ve or +ve so S will vary accordingly

Even I chose A , but the answer is C .

And I have no idea as to how and why the OA is C

[Anurag@Gurome]

S = x^3+3^x, Is S>0?

1. x<0
2. |x|>1

While answering this kind of problems, always remember that there are some traps:

• For |x| < 1, x^3 < x
  For |x| > 1, x^3 > x

This also holds for any other integer powers of x too.
So always check for integer values of x as well as for fractional values of x, i.e. |x| < 1.

Now for this problem,

Statement 1: x < 0

• 1. x = -1: (-1)^3 + 3^(-1) = -1 + (1/3) = -(2/3) < 0
  2. x = -0.5: (-0.5)^3 + 3^(-0.5) = -0.125 + (1/sqrt(3)) = 0.57 - 0.125 > 0

Not sufficient

Statement 1: |x| > 1

• 1. x = -2: (-2)^3 + 3^(-2) = -8 + (1/9) < 0
  2. x = 2: (2)^3 + 3^(2) = 8 + 9 > 0

Not sufficient

1 & 2 Together: Now we have x < -1
Hence, S is always less than zero.

Sufficient

The correct answer is C.

[Night reader]

S = x^3+3^x

Is S>0?

1. x<0

2. |x|>1

1st method, traditional

st(1) x<0 clearly Not sufficient as x=-1/4 and -1/64 + 3^(-1/4)= -1/64 + 1/root^4(3) >0 whereas x=-1 and -1 + 3^(-1)=-1+1/3 < 0
st(2) |x|>1 means x can not be within the interval (-1;1) OR x<-1 ; x>1 Not sufficient as x=-3 and -27 + 3^(-3)= -27+1/81 < 0 whereas x=2 and 8 + 3^2 >0 Not Sufficient

Combining st(1&2) we get only selection in common area x<0 and x<-1 so x<-1 is the solution area. Sufficient

2nd method, non-traditional ... use differential function and seek inequality conditions within the statements given
f(x)=x^3 + 3^x; f`(x)=[x^3]` + [3^x]`=3x^2 + (3x)^(x-3) =3x^2 + 3x^x +1/(3x)^3

[Deepthi Subbu]

S = x^3+3^x, Is S>0?

1. x<0
2. |x|>1

While answering this kind of problems, always remember that there are some traps:

• For |x| < 1, x^3 < x
  For |x| > 1, x^3 > x

This also holds for any other integer powers of x too.
So always check for integer values of x as well as for fractional values of x, i.e. |x| < 1.

Now for this problem,

Statement 1: x < 0

• 1. x = -1: (-1)^3 + 3^(-1) = -1 + (1/3) = -(2/3) < 0
  2. x = -0.5: (-0.5)^3 + 3^(-0.5) = -0.125 + (1/sqrt(3)) = 0.57 - 0.125 > 0

Not sufficient

Statement 1: |x| > 1

• 1. x = -2: (-2)^3 + 3^(-2) = -8 + (1/9) < 0
  2. x = 2: (2)^3 + 3^(2) = 8 + 9 > 0

Not sufficient

1 & 2 Together: Now we have x < -1
Hence, S is always less than zero.

Sufficient

The correct answer is C.

Great explanation Anurag,thank you

[aleph777]

I'm always really cautious with DS problems that don't specify a limitation on a value. When we aren't told "x is a positive integer" or "y is a non-zero integer," that means we need to think about positive and negative integers, positive and negative fractions, and, of course, zero.

Statement 1: x < 0. From this we instantly know that x^3 is negative and 3^x is a fraction. And if x is an integer, then S < 0. But since the questions doesn't restrict us to think about specific numbers only, we can't solve with this statement alone, because what if x itself is a fraction? For instance, (-.5)^3 + 3^-.5 = -.125 + 1.72 = .47, which means S > 0.

Statement 2: |x| > 1. This is also insufficient. It tells us that the absolute value of x is greater than 1, but we don't know if it's positive or negative.

Combined: We still cannot answer because we still cannot determine whether whether x is an integer. If x = 2, then S = -2^3 + 3^-2 = -8 + 1/9 = -7 7/9. But if x = 1.5, then 1.5^3 + 3^-3/2 = -3.375 + SQRT27, which is greater than 0.

[anshumishra]

I'm always really cautious with DS problems that don't specify a limitation on a value. When we aren't told "x is a positive integer" or "y is a non-zero integer," that means we need to think about positive and negative integers, positive and negative fractions, and, of course, zero.

Statement 1: x < 0. From this we instantly know that x^3 is negative and 3^x is a fraction. And if x is an integer, then S < 0. But since the questions doesn't restrict us to think about specific numbers only, we can't solve with this statement alone, because what if x itself is a fraction? For instance, (-.5)^3 + 3^-.5 = -.125 + 1.72 = .47, which means S > 0.

Statement 2: |x| > 1. This is also insufficient. It tells us that the absolute value of x is greater than 1, but we don't know if it's positive or negative.

Combined: We still cannot answer because we still cannot determine whether whether x is an integer. If x = 2, then S = -2^3 + 3^-2 = -8 + 1/9 = -7 7/9. But if x = 1.5, then 1.5^3 + 3^-3/2 = -3.375 + SQRT27, which is greater than 0.

Combining (1) and (2), x < -1, so your test case should include only the values of x which satisfy this.

[anshumishra]

S = x^3+3^x

Is S>0?

1. x<0

2. |x|>1

1st method, traditional

st(1) x<0 clearly Not sufficient as x=-1/4 and -1/64 + 3^(-1/4)= -1/64 + 1/root^4(3) >0 whereas x=-1 and -1 + 3^(-1)=-1+1/3 < 0
st(2) |x|>1 means x can not be within the interval (-1;1) OR x<-1 ; x>1 Not sufficient as x=-3 and -27 + 3^(-3)= -27+1/81 < 0 whereas x=2 and 8 + 3^2 >0 Not Sufficient

Combining st(1&2) we get only selection in common area x<0 and x<-1 so x<-1 is the solution area. Sufficient

2nd method, non-traditional ... use differential function and seek inequality conditions within the statements given
f(x)=x^3 + 3x; f`(x)=[x^3]` + [3x]`=3x^2 + 3; the inequality can be found by plugging the +ve and -ve values into the statement [3x^2 + 3], if the statement increases with the plugging the +ve values then it's increasing function if not then it's decreasing function; plugging in the values within the range limits of st(1) suggests that the function does not increase within the interval (-1;1); st (2) suggests that the function increases below (- 1) and above 1 i.e. intermittently
combination of st(1&2) suggests only possible increase on x<-1

f(x) = x^3+3^x

[Night reader]

Anshu, thanks for pointing this out. I have deleted the solution introduced in the earlier post not to confuse the test-takers and GMAT aspirants. And yes with the function f(x) = x^3+3^x differentiation brings nothing rather than another complex function, which we can differentiate several times and still have the power of x:
the 1st differentiation of function f(x) = x^3+3^x brings this f`= 3x^2 + 3x^x +1/(3x)^3

I have posted the use of differentiation here as well you may assess its strength and weaknesses https://www.beatthegmat.com/value-of-y-t ... tml#331062

[anshumishra]

Anshu, thanks for pointing this out. I have deleted the solution introduced in the earlier post not to confuse the test-takers and GMAT aspirants. And yes with the function f(x) = x^3+3^x differentiation brings nothing rather than another complex function, which we can differentiate several times and still have the power of x:
the 1st differentiation of function f(x) = x^3+3^x brings this f`= 3x^2 + 3x^x +1/(3x)^3

I have posted the use of differentiation here as well you may assess its strength and weaknesses https://www.beatthegmat.com/value-of-y-t ... tml#331062

Thanks Night reader,

In fact it becomes even more complex :
To differentiate 3^x, you will have to use the formula shown below :
d/dx(a^x) = d/dx {e^(x ln a)}
= e^(x ln a)* d/dx(x ln a) = a^x*ln(a)