confused with this DS

[gmatwarroom]

Here is DS question:

If x is an integer and 4 power x < 100, what is x?

(1) 4 power (x + 1) - 4 power (x - 1) > 100
(2) 4 power (x + 1) + 4 power x > 100

My answer was E) but actual answer says each one is sufficient.

My approach:
from given that 4 power x < 100 ==> x < or = 3

1) derives finally that x > or = 3
2) derives finally that x > or = 3

Now, dont understand how each would suffice?? I know, it's given that x < or =3 but combined with 1), how can I infer that x =3 ???????

any help to explain it

[anuprajan5]

Hi,

Question states that 4 power x < 100 which means that x < = 3

Statement 1- 4 power (x + 1) - 4 power (x - 1) > 100

Simplifying,
4 power x * 15 > 400
4 power x > 80/3
or 4 power x > 26,6

and the only integer value of x which satisfies that and the condition in the question is 3.

Statement 2 - 4 power (x + 1) + 4 power x > 100

Simplifying
4 power x > 20

and the only integer value of x which satisfies that and the condition in the question is 3.

hence answer D

Regards
Anup

[neelgandham]

If x is an Integer and if (4^x)<100, what is x?

4^x < 100 < 4^4
x < 4.

The question can be rephrased to "If Integer x < 4, then what is the value of x ?"

(1) 4^(x + 1) - 4^(x - 1) > 100

4^(x + 1) - 4^(x - 1) > 100
[4^x * 4^1] - [4^x - 4^-1] > 100 > 64
4^x *[4-(1/4)] > 4^3
4^x * 15/4 > 4^3
4^x * 15 > 4^4
15 > 4^(4-x)
16 > 15 > 4^(4-x)
4^2 > 4^(4-x)
2 > 4-x
x > 2. We already know that x < 4 and x is an integer. So 3 is the only value of x between 2 and 4.
So statement I is sufficient to answer the question.

(2) 4^(x + 1) + 4^ x > 100

4^(x + 1) + 4^ x > 100 > 64
4^x [4+1] > 100
4^x > 20
4^x > 20 > 16
4^x > 16
4^x > 4^2
x > 2. We already know that x < 4 and x is an integer. So 3 is the only value of x between 2 and 4.
So statement II is sufficient to answer the question.

p.s: I tried to explain each step in detail for the benefit of the members who requested me to solve in detail.

[LalaB]

1)
4^(x+1)-4^(x-1)>100

4^(x+1)-4^x/4>100

(4^x)*4*4-4^x>400
4^x(16-1)>400
(4^x)*15> 400
26<4^x<100

x=3

2) 4^(x + 1) + 4^x > 100

4^x(4+1)>100
4^x>20

20<4^x<100
x=3
ans is D

[gmatwarroom]

thanks everyone for solving the equations.

However, my issue was not really "solving" each of the equations but solving the problem i.e. deriving the conclusion.

As I mentioned, I was able to derive
Given: x < or = 3 (i.e. 3,2,1....)
statement 1) derives finally that x > or = 3 (i.e. 3,4,5...)

what I was confused with (and unfortunately still is) that, how it would mean that I should take the common value (of "Given" and "Statement 1"), which is 3, as the answer -- ??
Seems I am missing something basic about DS problem

[anuprajan5]

Hi,

Well the given condition and the statement give you the parameters that you have to work with. The statement is only sufficient if it along with the given information can conclusively answer the question.

Taking your statements:

Given: x < or = 3 (i.e. 3,2,1....) - a
statement 1) derives finally that x > or = 3 (i.e. 3,4,5...) - b

The only way both a and b are possible is if x = 3. (the commonality). Since we can conclude that we get only 1 value for x, this means that the condition in the question and the statement are sufficient to answer the question.

Regards
Anup

[gmatwarroom]

The statement is only sufficient if it along with the given information can conclusively answer the question.

I think, I would rephrase it to make it more understandable --
The statement is only sufficient if it along with the given information can get UNIQUE value/answer to the question.

Also, I think I kind of understood my confusion -- I was confused with the fact that both "Given" and "Statement" separately was having multiple values of "x".

The point that I should look for is that, if "Given" and "Statement" together gives "UNIQUE" answer/value or not -- and that made me understand it.