If x is an integer, is x|x|<2^x ?

If x is an integer, is x*|x|<2^x

This is YES/NO data sufficiency question: In a Yes/No Data Sufficiency question, each statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no".

Now, you should notice that the RHS (right hand side) of the expression is always positive (\(2^x>0\)), but the LHS is positive when \(x>0\) (\(x>0\) --> \(x*|x|=x^2\)), negative when \(x<0\) (\(x<0\) --> \(x*|x|=-x^2\)) and equals to zero when \(x={0}\).

(1) x<0 --> according to the above \(x*|x|<0<2^x\), so the answer to the question "is x*|x|<2^x" is YES. Sufficient.

(2) x=-10, the same thing here \(x*|x|=-100<0<\frac{1}{2^{10}}\), so the answer to the question "is x*|x|<2^x" is YES. Sufficient.

Answer: D.



When \(x=-10\) then \(|x|=|-10|=10\) and \(x*|x|=-10*10=-100\).

Hope it's clear.

1) If x<0, then x*|x| is always negative. 2^x is positive. So, this statement is sufficient.

2) Just plug it in. Sufficient.

Answer is D.

If x is an integer, is x|x| < 2^x?

(1) x < 0
(2) x = -10

I can understand the second part:
-10|-10| < 2^-10 --> -10 * 10 < 1/2 ^ 10
|-10| --> reduced to 10 as its numeric.. is my reasoning correct?
B is sufficient

For (1) .. however i am not able to decipher anything..
-x|-x| < 2^-x --> -x * -x < 1/2 ^x
|-x| --> reduced to -x as x < 0 .. is my reasoning correct?
But what should be the next steps .. Please help

Question : Is x|x|<2^x ?

Statement 1: x < 0

For x to be Negative LHS i.e. x|x| will always be NEGATIVE
and 2^x will be positive for any value of x
i.e. x|x|<2^x will always be true
SUFFICIENT

Statement 1: x = -10
For x =-10 LHS i.e. x|x| will always be NEGATIVE (-100)
and 2^x will be positive for given x (1/2^10)
i.e. x|x|<2^x will always be true
SUFFICIENT

Answer: option D

Target question: Is x|x|< 2^x ?

Given: x is an integer

Statement 1: x < 0
In other words, x is NEGATIVE
So, x|x| = (NEGATIVE)(|NEGATIVE|) = (NEGATIVE)(POSITIVE) = NEGATIVE

IMPORTANT: 2^x will be POSITIVE for all values of x.

Since x|x| must be NEGATIVE, and since 2^x must be POSITIVE, we can be certain that x|x|< 2^x
Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: x = -10
So, x|x| = (-10)(|-10|) = (-10)(10) = -100 = a NEGATIVE
On the other hand, 2^x = 2^(-10) = 1/(2^10) = some POSITIVE number
Since x|x| is NEGATIVE, and since 2^x must be POSITIVE, we can be certain that x|x|< 2^x
Since we can answer the target question with certainty, statement 2 is SUFFICIENT

Answer:

No. The right hand side is \(2^{(-2)}\), which is the same as \(\frac{1}{2^2}\) but the left hand side (-2*|-2|) stays the same.

Negative powers:
\(a^{-n}=\frac{1}{a^n}\)
Important: you cannot rise 0 to a negative power because you get division by 0, which is NOT allowed. For example, \(0^{-1} = \frac{1}{0}=undefined\).

8. Exponents and Roots of Numbers

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Hope it helps.

what is wrong in my approach :

x |x| < 2^x
x *sqrt(x^2) < 2^x
square on both sides,
x^2 * x^2 < 2^2x
x^4 < 2^2x

given 1 stmt, x as -ve, always x^4 > 2^2x, whereas I know i am making some mistake.
are we not allowed to take square on both sides?

Hi Farina,

Most of what you wrote is correct. This DS question is based on a couple of Number Properties:

when X is NEGATIVE....
X|X| will ALWAYS be NEGATIVE
2^(X) will ALWAYS be POSTIIVE

This means that the answer to the question is ALWAYS YES and Fact 1 is SUFFICIENT.

However, what you noted about POSITIVE values is not true. For example....

when X = 1...
1|1| = 1
2^1 = 2
so the answer to the question "is X|X| < 2^X?".... is YES.

when X = 2...
2|2| = 4
2^2 = 4
so the answer to the question "is X|X| < 2^X?".... is NO.

This means that when X is POSITIVE, the answer to the question changes and this situation would be INSUFFICIENT.

GMAT assassins aren't born, they're made,
Rich

If X is an integer is X |x| < 2^X

1. X<0
2. X=-10

I solved it - using two options
-10 |10| < 1/2^10 AND -10 -|10| < 1/2^10. This method gives two solutions and therefore not sufficient. However my logic is wrong. Please explain why there are not two options. I have come across questions where one is required to use the two options. why not in this case? thanks

I don't quite understand your question.

Original question asks whether \(x*|x|<2^x\) is true, it's YES/NO DS question, it doesn't ask for specific value of \(x\). AGAIN: In a Yes/No Data Sufficiency question, each statement is sufficient if the answer is “always yes” or “always no” while a statement is insufficient if the answer is "sometimes yes" and "sometimes no". As EACH statement ALONE gives the definite answer YES x*|x| is less than 2^x then EACH statement ALONE is sufficient to answer the question which means than answer is D .

Another one \(|x+1|=|3x-2|\) (I believe it's \(|x+1|=|3x-2|\) and not |x+1|= x*|3x-2| as you wrote, as solutions you provided 1/4 and 3/2 satisfy the first equation and not the second one), seems to be another type of DS question, the one which asks for a certain value of an unknown. For this type of questions statement is sufficient if it gives single numerical value of this unknown. So as \(|x+1|=|3x-2|\) has two solutions \(x=\frac{1}{4}\) and \(x=\frac{3}{2}\) then this statement (if this is the only thing we know for certain statement) is not sufficient, as it does not give single numerical value of \(x\).

For more absolute values and inequalities:

Check Walker's topic on ABSOLUTE VALUE: math-absolute-value-modulus-86462.html

For practice check collection of 13 tough inequalities and absolute values questions with detailed solutions at: inequality-and-absolute-value-questions-from-my-collection-86939.html

700+ PS and DS questions (also have some inequalities and absolute values questions with detailed solutions):
tough-problem-solving-questions-with-solutions-100858.html

700-gmat-data-sufficiency-questions-with-explanations-100617.html

Hope it helps.

We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality). Here x|x| is negative if x is negative, so we cannot square.


RAISING INEQUALITIES TO EVEN/ODD POWER

1. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
\(2<4\) --> we can square both sides and write: \(2^2<4^2\);
\(0\leq{x}<{y}\) --> we can square both sides and write: \(x^2<y^2\);

But if either of side is negative then raising to even power doesn't always work.
For example: \(1>-2\) if we square we'll get \(1>4\) which is not right. So if given that \(x>y\) then we cannot square both sides and write \(x^2>y^2\) if we are not certain that both \(x\) and \(y\) are non-negative.

2. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
For example:
\(-2<-1\) --> we can raise both sides to third power and write: \(-2^3=-8<-1=-1^3\) or \(-5<1\) --> \(-5^3=-125<1=1^3\);
\(x<y\) --> we can raise both sides to third power and write: \(x^3<y^3\).

Adding, subtracting, squaring etc.: Manipulating Inequalities.

9. Inequalities

• Theory
  Inequalities Made Easy!
  Inequalities: Tips and hints
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  Inequalities - Quadratic Inequalities
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  INEQUATIONS(Inequalities) Part 1
  INEQUATIONS(Inequalities) Part 2
  
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    Inequality and absolute value questions from my collection
    DS Questions
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For more check Ultimate GMAT Quantitative Megathread

Given: x is an integer

Asked: Is x|x| < \(2^x\) ?

(1) x < 0
|x| = -x
-x^2 < 2^x
-x^2 < 0
2^x >0
-x^2 < 2^x
\(x|x| < 2^x\)
SUFFICIENT

(2) x = -10
|x| = 10
x|x| = -10 * 10 = -100
\(2^x = 2^{-10} = \frac{1}{1024}\)
\(-100 < \frac{1}{1024}\)
\(x|x| < 2^x\)
SUFFICIENT

IMO D

Thanks Kp.

But if x<0 so we get |-x| => -x
Am i missing something?

Thank you for the explanation.

Please help me understand what the difference (in regards to having two solutions in terms of absolute and non absolute values) is between the two questions
is x*|x|<2^x and the question
|x+1|= x*|3x-2|what are the possible values for x from advanced equations of MGMAT Equations and inequalities - the answer is 1/4 and 3/2

Maybe i am overanalyzing the questions

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

There is 1 variable. Thus D is the answer most likely.

Condition 1) \(x < 0\)
Since \(|x| = -x\) if \(x < 0\), the question \(x|x| < 2^x\) is equivalent to \(-x^2 < 2^x\).
We have the left hand side \(-x^2 < 0\) and the right hand side \(2^x > 0\) all times.
Thus \(-x^2 < 0 < 2^x\).
This is sufficient.

Condition 2) \(x = -10\)
Since \(x = -10\) is negative, by the same logic of the condition 1), this condition is also sufficient.

Therefore, the answer is D as expected.


-> For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

Hi,

For statement 1 I tested values i.e. x = -1 or x = -2.This is more so a question regarding reciprocals and inequalities. If x = -2, then -2 (|-2|) = 2^-2. Then, this is equal to -2 (2) = 1/2^2. In the second step where I converted 2^-2 to 1/2^2 -- would I have to also flip the other side to become 1 / -2 (2) or is that wrong?

Thanks,

infinitemac

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

Since we have 1 variables and 0 equation, D could be the answer most likely.

Condition 1)
Since x < 0 and |x|≥0, x|x|≤0.
2^x > 0
Thus x|x| < 2^x.
This is sufficient.

Condition 2)
Since x = -10, x|x| = (-10)*10 = -100 < 0
And 2^(-10) = 1/(2^10) > 0
Thus x|x| < 2^x
This is also sufficient.

Therefore, D is the answer.

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both con 1) and con 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. D is most likely to be the answer using con 1) and con 2) separately according to DS definition. Obviously, there may be cases where the answer is A, B, C or E.