Is $_(\frac{x}{2})^2 + (\frac{x}{3})^3 \geq 0$_?
$_x < 0$_
$_x > -\frac{27}{4}$_
How to approach this problem....???? What to assume ?
DS - Disguised eqn
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- karthikpandian19
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- eagleeye
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What language is this in? ahahaha. Sorry, I am cracking up at reading this. Could you please post an image or something. Seems like something went wonky here.karthikpandian19 wrote:Is $_(\frac{x}{2})^2 + (\frac{x}{3})^3 \geq 0$_?
$_x < 0$_
$_x > -\frac{27}{4}$_
How to approach this problem....???? What to assume ?
- karthikpandian19
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Even i was out of my mind when i saw this.......But here is the OE......Let me know if you understand the OE becoz its difficult for me to assume things..........
We are asked whether the expression $_(\frac{x}{2})^2 + (\frac{x}{3})^3$_ is greater than or equal to $_0$_.
Statement 1 tells us that $_x < 0$_. When $_x$_ is negative, the term $_(\frac{x}{2})^2$_ will always be positive because squaring any non-zero number will produce a positive result. However, the term $_(\frac{x}{3})^3$_ will always be negative because cubing any negative number will always produce a negative result.
Therefore, when $_x < 0$_, $_\left(\frac{x}{2}\right)^2 + \left(\frac{x}{3}\right)^3$_ is the sum of a positive number and a negative number. The value of the expression will be positive or negative depending on the magnitudes of the two terms. Testing cases will be helpful here. We choose simple, intuitive values to test.
When $_x = -1$_, the expression becomes $_\left(\frac{-1}{2}\right)^2 + \left(\frac{-1}{3}\right)^3 = \frac14 + \frac{-1}{27}$_. The magnitude of $_\frac14$_ is greater than the magnitude of $_-\frac{1}{27}$_, so the value of the expression is positive.
Let's see if we can find a value of $_x$_ that makes the expression negative-that is, a value of $_x$_ that makes the magnitude of $_\left(\frac{x}{3}\right)^3$_ greater than the magnitude of $_\left(\frac{x}{2}\right)^2$_. Since the second term, $_(\frac{x}{3})^3$_, is being cubed, it will increase in magnitude much faster than $_(\frac{x}{2})^2$_ as $_x$_ increases in magnitude. So, we should test a value where $_x$_ has a large magnitude; $_x = -10$_ is a good candidate to test since it is easy to work with.
When $_x = -10$_, the expression becomes $_(\frac{-10}{2})^2 + (\frac{-10}{3})^3 = \frac{100}{4} + \frac{-1000}{27}$_. The magnitude of the term $_\frac{-1000}{27}$_ is greater than that of $_\frac{100}{4}$_, so the sum will be negative. Thus, we have found a case where the value of the expression is positive and a case where it is negative. Statement 1 alone is insufficient. Eliminate answer choices A and D. The correct answer choice is either B, C, or E.
Statement 2 tells us that $_x > -\frac{27}{4}$_. It would be too difficult to plug that value into the expression, so let's see what we can do to the expression itself.
We can solve the inequality $_(\frac{x}{2})^2 + (\frac{x}{3})^3 \geq 0$_ to see where the expression is positive. First, subtract $_(\frac{x}{3})^3$_ from both sides:
$_(\frac{x}{2})^2 \geq -(\frac{x}{3})^3$_. Apply the exponent to the numerator and denominator of the fraction:
$_\frac{x^2}{4} \geq -\frac{x^3}{27}$_. Cross multiply:
$_27x^2 \geq -4x^3$_. Cancel $_x^2$_ from both sides. Note that because x^2 must be positive, we don't have to worry about flipping the inequality; we can also ignore the case in which $_x=0$_, since $_x=0$_ makes the inequality in the prompt true:
$_27 \geq -4x$_. Divide both sides by $_-4$_, and flip the inequality because we are dividing by a negative value:
$_-\frac{27}{4} \leq x$_.
So, when $_x > -\frac{27}{4}$_, the expression is greater than or equal to $_0$_. Remember that when $_x = 0$_ the value of the expression is $_0$_. Thus, the expression in the prompt must be greater than or equal to $_0$_, and so Statement 2 alone is sufficient.
We are asked whether the expression $_(\frac{x}{2})^2 + (\frac{x}{3})^3$_ is greater than or equal to $_0$_.
Statement 1 tells us that $_x < 0$_. When $_x$_ is negative, the term $_(\frac{x}{2})^2$_ will always be positive because squaring any non-zero number will produce a positive result. However, the term $_(\frac{x}{3})^3$_ will always be negative because cubing any negative number will always produce a negative result.
Therefore, when $_x < 0$_, $_\left(\frac{x}{2}\right)^2 + \left(\frac{x}{3}\right)^3$_ is the sum of a positive number and a negative number. The value of the expression will be positive or negative depending on the magnitudes of the two terms. Testing cases will be helpful here. We choose simple, intuitive values to test.
When $_x = -1$_, the expression becomes $_\left(\frac{-1}{2}\right)^2 + \left(\frac{-1}{3}\right)^3 = \frac14 + \frac{-1}{27}$_. The magnitude of $_\frac14$_ is greater than the magnitude of $_-\frac{1}{27}$_, so the value of the expression is positive.
Let's see if we can find a value of $_x$_ that makes the expression negative-that is, a value of $_x$_ that makes the magnitude of $_\left(\frac{x}{3}\right)^3$_ greater than the magnitude of $_\left(\frac{x}{2}\right)^2$_. Since the second term, $_(\frac{x}{3})^3$_, is being cubed, it will increase in magnitude much faster than $_(\frac{x}{2})^2$_ as $_x$_ increases in magnitude. So, we should test a value where $_x$_ has a large magnitude; $_x = -10$_ is a good candidate to test since it is easy to work with.
When $_x = -10$_, the expression becomes $_(\frac{-10}{2})^2 + (\frac{-10}{3})^3 = \frac{100}{4} + \frac{-1000}{27}$_. The magnitude of the term $_\frac{-1000}{27}$_ is greater than that of $_\frac{100}{4}$_, so the sum will be negative. Thus, we have found a case where the value of the expression is positive and a case where it is negative. Statement 1 alone is insufficient. Eliminate answer choices A and D. The correct answer choice is either B, C, or E.
Statement 2 tells us that $_x > -\frac{27}{4}$_. It would be too difficult to plug that value into the expression, so let's see what we can do to the expression itself.
We can solve the inequality $_(\frac{x}{2})^2 + (\frac{x}{3})^3 \geq 0$_ to see where the expression is positive. First, subtract $_(\frac{x}{3})^3$_ from both sides:
$_(\frac{x}{2})^2 \geq -(\frac{x}{3})^3$_. Apply the exponent to the numerator and denominator of the fraction:
$_\frac{x^2}{4} \geq -\frac{x^3}{27}$_. Cross multiply:
$_27x^2 \geq -4x^3$_. Cancel $_x^2$_ from both sides. Note that because x^2 must be positive, we don't have to worry about flipping the inequality; we can also ignore the case in which $_x=0$_, since $_x=0$_ makes the inequality in the prompt true:
$_27 \geq -4x$_. Divide both sides by $_-4$_, and flip the inequality because we are dividing by a negative value:
$_-\frac{27}{4} \leq x$_.
So, when $_x > -\frac{27}{4}$_, the expression is greater than or equal to $_0$_. Remember that when $_x = 0$_ the value of the expression is $_0$_. Thus, the expression in the prompt must be greater than or equal to $_0$_, and so Statement 2 alone is sufficient.
eagleeye wrote:What language is this in? ahahaha. Sorry, I am cracking up at reading this. Could you please post an image or something. Seems like something went wonky here.karthikpandian19 wrote:Is $_(\frac{x}{2})^2 + (\frac{x}{3})^3 \geq 0$_?
$_x < 0$_
$_x > -\frac{27}{4}$_
How to approach this problem....???? What to assume ?
Regards,
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
---If you find my post useful, click "Thank" ---
---Never stop until cracking GMAT---
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
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---Never stop until cracking GMAT---
- karthikpandian19
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I am not sure though whether these kind of problems appear in GMAT
Regards,
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
---If you find my post useful, click "Thank" ---
---Never stop until cracking GMAT---
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
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---Never stop until cracking GMAT---
- Bill@VeritasPrep
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Is that LaTeX?
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- eagleeye
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Ok. I've figured out the question.
Here's the question in human understandable form:
"
Is (x/2)^2 + (x/3)^3 >= 0
1. x<0
2. x>-27/4
"
We need to find whether (x/2)^2 + (x/3)^3 >= 0
I am going to rephrase the question by manipulation
(x/2)^2 + (x/3)^3 >= 0
=> x^2/4 + x^3/27 >= 0
=> (x^2/4)*(1+ 4x/27)>=0
Now x^2/4 is always >=0, (x^2 is never negative) so we really need to find whether (1+4x/27) >=0
Now, 1+4x/27 >=0
=> 4x/27 >= -1
=> x >= -27/4
Hence rephrased, the question stem becomes Is x >= -27/4
Now this is a simple question to figure out.
Statement 1:
x < 0 . We can't say whether x >= - 27/4. Insufficient
Statement 2:
x > -27/4. So x is definitely greater than -27/4. Sufficient.
Hence B is the right answer.
Here's the question in human understandable form:
"
Is (x/2)^2 + (x/3)^3 >= 0
1. x<0
2. x>-27/4
"
We need to find whether (x/2)^2 + (x/3)^3 >= 0
I am going to rephrase the question by manipulation
(x/2)^2 + (x/3)^3 >= 0
=> x^2/4 + x^3/27 >= 0
=> (x^2/4)*(1+ 4x/27)>=0
Now x^2/4 is always >=0, (x^2 is never negative) so we really need to find whether (1+4x/27) >=0
Now, 1+4x/27 >=0
=> 4x/27 >= -1
=> x >= -27/4
Hence rephrased, the question stem becomes Is x >= -27/4
Now this is a simple question to figure out.
Statement 1:
x < 0 . We can't say whether x >= - 27/4. Insufficient
Statement 2:
x > -27/4. So x is definitely greater than -27/4. Sufficient.
Hence B is the right answer.
Last edited by eagleeye on Fri Jul 06, 2012 8:42 pm, edited 1 time in total.
- karthikpandian19
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Bill i didnt get u...???LaTeX??
Bill@VeritasPrep wrote:Is that LaTeX?
Regards,
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
---If you find my post useful, click "Thank" ---
---Never stop until cracking GMAT---
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
---If you find my post useful, click "Thank" ---
---Never stop until cracking GMAT---
- karthikpandian19
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Awesome.....explanation.....and paraphrasing....gr8
Do you think these kind of questions will come in GMAT?
How much time did you took to solve the problem or mainly to paraphrase?
Do you think these kind of questions will come in GMAT?
How much time did you took to solve the problem or mainly to paraphrase?
eagleeye wrote:Ok. I've figured out the question.
Here's the question in human understandable form:
"
Is (x/2)^2 + (x/3)^3 >= 0
1. x<0
2. x>-27/4
"
We need to find whether (x/2)^2 + (x/3)^3 >= 0
I am going to rephrase the question by manipulation
(x/2)^2 + (x/3)^3 >= 0
=> x^2/4 + x^3/27 >= 0
=> (x^2/4)*(1+ 4x/27)>=0
Now x^2/4 is always >=0, (x^2 is never negative) so we really need to find whether (1+4x/27) >=0
Now, 1+4x/27 >=0
=> 4x/27 >= -1
=> x >= -27/4
Hence rephrased, the question stem becomes Is x >= -27/4
Now this is a simple question to figure out.
Statement 1:
x < 0 . We can't say whether x >= - 27/4. Insufficient
Statement 2:
x<-27/4. So x is definitely smaller than -27/4. Hence x is not greater than or equal to -27/4. Sufficient.
Hence B is the right answer.
Regards,
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
---If you find my post useful, click "Thank" ---
---Never stop until cracking GMAT---
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
---If you find my post useful, click "Thank" ---
---Never stop until cracking GMAT---
- eagleeye
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Will they? I don't know.karthikpandian19 wrote: Do you think these kind of questions will come in GMAT?
Can they? 100%, they can.
Because I had to figure out what the question was in this case, and while I was figuring out the question the answer sort of appeared to me, I couldn't put a number on it. But if I ever see a exponent inequality question, one of the first things that automatically runs through my mind is, can I simplify the question stem? If I think I can, I do it right away. If I saw this question on the GMAT, I would take me anywhere from a minute to max. 2 to do it.karthikpandian19 wrote: How much time did you took to solve the problem or mainly to paraphrase?
[/quote]
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It's a markup language often used for equations in various fields: https://en.wikipedia.org/wiki/LaTeXkarthikpandian19 wrote:Bill i didnt get u...???LaTeX??Bill@VeritasPrep wrote:Is that LaTeX?
I have only a passing familiarity with it, but some of the symbols used seemed consistent with it.
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- karthikpandian19
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Thanks for your info.
Bill@VeritasPrep wrote:It's a markup language often used for equations in various fields: https://en.wikipedia.org/wiki/LaTeXkarthikpandian19 wrote:Bill i didnt get u...???LaTeX??Bill@VeritasPrep wrote:Is that LaTeX?
I have only a passing familiarity with it, but some of the symbols used seemed consistent with it.
Regards,
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
---If you find my post useful, click "Thank" ---
---Never stop until cracking GMAT---
Karthik
The source of the questions that i post from JUNE 2013 is from KNEWTON
---If you find my post useful, click "Thank" ---
---Never stop until cracking GMAT---