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If x^3 < 16x which of the following includes at least some [#permalink]
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27 Sep 2012, 18:12
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If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions? A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0
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Last edited by Bunuel on 28 Sep 2012, 02:13, edited 1 time in total.
Renamed the topic and edited the question.



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Re: inequality [#permalink]
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27 Sep 2012, 19:55
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D is right.
Plugging in is the key here.
Given : x^3 < 16x Answer should include at least some of the possible solutions for x, but no values that are not solutions
So, even if for one value of x, the given statement fails, we will rule out that option.
A) x<4
x < 4 or x < 4 i.e x > 4. 4 < x < 4
say x = 0 , x^3 < 16x doesn't satisfy (0 = 0)
B) X <4
say x = 0 . Explained above
C) X>4
say x = 5, x^3 < 16x doesn't satisfy (125 > 80)
D) X<4
Say x = 5, x^3 < 16x satisfies (125 < 80)
E) X>0
say x = 1, x^3 < 16x satisfies (1 < 16) say x = 5, x^3 < 16x doesn't satisfy (125 > 80)



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Re: inequality [#permalink]
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27 Sep 2012, 20:36
alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A) x<4 B) X <4 C) X>4 D) X<4 E) X>0
totally confused !!! solving inequality  x^3<16 x x^3  16x < 0 x{x^2  16} < 0 This gives us 2 inequalities {A} x < 0  this cannot be solved further {B} x^2  16 < 0 (solving it further) x^2 < 16  Now solving for x 4<x<4  We got the range for X from {B} thus x can have 2 solutions Either [a] x <0 Or 4<x<4
We have to find  equations that contain at least some of the possible solutions for x, but no values that are not solutions?
A) x<4 > implies x lie only between 0 and 4  Which is not correct as per {a} and {b} solved above B) X <4 > x can be 3 or 10 or 100 not conclusive C) X>4 > x can be 5 or 100 or 400  Out of range for solutions of X D) X<4 > [b]PERFECT  as it satisfies Only {a} above and not any other value which cannot be a solution for X E) X>0 > X ranges from 1 to 100 to 1000  not conclusive as per the question. Hope it helps.
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0 \(x^3 < 16x\) > \(x^316x<0\) > \(x(x^216)<0\) > \(x(x+4)(x4)<0\). Roots are 4, 0, and 4. This gives us 4 ranges: \(x<4\), \(4<x<0\), \(0<x<4\), and \(x>4\). Now, test some extreme value: for example if \(x\) is very large number then the whole expression is positive. Here comes the trick: since in the fourth range, when \(x>4\), the expression is positive, then in third range it'll be negative, in the second positive, and in the first range it'l be negative again: ++. Thus, the ranges when the expression is negative are: \(x<4\) and \(0<x<4\). Only answer choice D does not include values of x that are not the solutions of given inequality. Answer: D. Solving inequalities: x24x94661.html#p731476inequalitiestrick91482.htmleverythingislessthanzero108884.html?hilit=extreme#p868863xyplane71492.html?hilit=solving%20quadratic#p841486P.S. Please read and follow: rulesforpostingpleasereadthisbeforeposting133935.html Pay attention to the rule #3: the name of the topic (subject field) MUST be the first 40 characters (~the first two sentences) of the question.
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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18 Mar 2014, 22:57
I mean can we write x^3<16x as x^316x<0 without knowing X is +ve or ve?
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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19 Mar 2014, 02:14



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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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15 Jan 2015, 17:01
Bunuel wrote: alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0 \(x^3 < 16x\) > \(x^316x<0\) > \(x(x^216)<0\) > \(x(x+4)(x4)<0\). Roots are 4, 0, and 4. This gives us 4 ranges: \(x<4\), \(4<x<0\), \(0<x<4\), and \(x>4\). Now, test some extreme value: for example if \(x\) is very large number then the whole expression is positive. Here comes the trick: since in the fourth range, when \(x>4\), the expression is positive, then in third range it'll be negative, in the second positive, and in the first range it'l be negative again: ++. Thus, the ranges when the expression is negative are: \(x<4\) and \(0<x<4\). Only answer choice D does not include values of x that are not the solutions of given inequality. Answer: D. Solving inequalities: x24x94661.html#p731476inequalitiestrick91482.htmleverythingislessthanzero108884.html?hilit=extreme#p868863xyplane71492.html?hilit=solving%20quadratic#p841486P.S. Please read and follow: rulesforpostingpleasereadthisbeforeposting133935.html Pay attention to the rule #3: the name of the topic (subject field) MUST be the first 40 characters (~the first two sentences) of the question.Bunuel  can you explain how you found the roots? I got it down to X(X+4)(X4) < 0...so x < 0, X < 4, and X < 4...I know this isn't correct but am unsure where my logic is flawed.



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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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15 Jan 2015, 22:03
Hi All, This question is written in a "quirky" way  it's asking for an inequality that includes SOME of the solutions to the prompt (but NOT all of the solutions) and NO values that are NOT solutions. Quirky language almost always implies a pattern of some kind  in this case, the pattern can be found by focusing on the ANSWERS.... The prompt gives us X^3 < 16X We COULD solve this inequality, but we know form the prompt that some of the work won't be useful  we're looking for SOME of the solutions, but not all of the solutions. From the answers, we know that the numbers 4, 0 and 4 are worth paying attention to. Let's start with X = 0..... 0^3 is NOT < 16(0) 0 is NOT a solution, so if it appears in an answer, then that answer is WRONG. Eliminate A and B. Next, let's try X = 4 4^3 = 64 16(4) = 64 64 is NOT < 64 4 is NOT a solution, so if it appears in an answer, then that answer is WRONG. Eliminate E. X = 4 has no effect on either of the remaining answers, so we have to TEST something else... If....X = 5 5^3 = 125 16(5) = 80 125 is NOT < 80 5 is NOT a solution, so if it appears in an answer, then that answer is WRONG. Eliminate C. Final Answer: GMAT assassins aren't born, they're made, Rich
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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19 Jan 2015, 03:26
alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0 x(x)(x)  16x < 0 x(x(x)  16) <0 x(x+4)(x4)<0 We can observe that the value of x from 1 to 3 will work above which eliminates answer option C. x > 4 x = 5 works which will eliminate answer option A and E x = 4.1 works which eliminates answer option B. Hence the answer is D.
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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07 Jul 2015, 08:37
Bunuel wrote: alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0 \(x^3 < 16x\) > \(x^316x<0\) > \(x(x^216)<0\) > \(x(x+4)(x4)<0\). Roots are 4, 0, and 4. This gives us 4 ranges: \(x<4\), \(4<x<0\), \(0<x<4\), and \(x>4\). Now, test some extreme value: for example if \(x\) is very large number then the whole expression is positive. Here comes the trick: since in the fourth range, when \(x>4\), the expression is positive, then in third range it'll be negative, in the second positive, and in the first range it'l be negative again: ++. Thus, the ranges when the expression is negative are: \(x<4\) and \(0<x<4\). Only answer choice D does not include values of x that are not the solutions of given inequality. Answer: D. Solving inequalities: x24x94661.html#p731476inequalitiestrick91482.htmleverythingislessthanzero108884.html?hilit=extreme#p868863xyplane71492.html?hilit=solving%20quadratic#p841486P.S. Please read and follow: rulesforpostingpleasereadthisbeforeposting133935.html Pay attention to the rule #3: the name of the topic (subject field) MUST be the first 40 characters (~the first two sentences) of the question.Hi Bunuel, Is there a easy way to find the 4 ranges mentioned above. Can you Pls. help. I checked the graphical approach for quadratic inequality but could not understand how the above ranges can be deduced.



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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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07 Jul 2015, 10:19
robinpallickal wrote: Bunuel wrote: alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0 \(x^3 < 16x\) > \(x^316x<0\) > \(x(x^216)<0\) > \(x(x+4)(x4)<0\). Roots are 4, 0, and 4. This gives us 4 ranges: \(x<4\), \(4<x<0\), \(0<x<4\), and \(x>4\). Now, test some extreme value: for example if \(x\) is very large number then the whole expression is positive. Here comes the trick: since in the fourth range, when \(x>4\), the expression is positive, then in third range it'll be negative, in the second positive, and in the first range it'l be negative again: ++. Thus, the ranges when the expression is negative are: \(x<4\) and \(0<x<4\). Only answer choice D does not include values of x that are not the solutions of given inequality. Answer: D. Solving inequalities: x24x94661.html#p731476inequalitiestrick91482.htmleverythingislessthanzero108884.html?hilit=extreme#p868863xyplane71492.html?hilit=solving%20quadratic#p841486P.S. Please read and follow: rulesforpostingpleasereadthisbeforeposting133935.html Pay attention to the rule #3: the name of the topic (subject field) MUST be the first 40 characters (~the first two sentences) of the question.Hi Bunuel, Is there a easy way to find the 4 ranges mentioned above. Can you Pls. help. I checked the graphical approach for quadratic inequality but could not understand how the above ranges can be deduced. After going through all the posts on inequalities, I figured out the reasoning. Thanks a lot to Bunuel, Karishma, and gurupreetsingh for excellent posts on inequalities...



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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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07 Jul 2015, 15:04
alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0 x^3 < 16x i.e. x^3  16x < 0 i.e. x(x^2  16) < 0 i.e. (x4)x(x+4) < 0 Case1: Either (x4) is Negative and both x and (x+4) are positive i.e. x<4 and x>0Case2: Or (x4), x and (x+4) are all Negative i.e. x < 4 Answer: Option D
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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14 Mar 2016, 18:48
alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0 we can easily eliminate all but D. first, suppose x is positive divide both sides by x x^2<16 square root both x<4 but what if x=5 2 2^3 = 8 but 16*2 = 32, and the equation is not true. B  out. C = x=5. 5^3 = 125. 16*5 = 80. so out. E = x=5. same as in C. so out. the choice is between A and D. suppose x=2. it satisfies the condition x<4. 2^3 = 8 2*16 = 32 now 8 is greater than 32. so A is out, and D is the answer.



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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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29 May 2017, 11:15
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So we have only two possible ranges where the equation is less than zero. [1]: x < 4 [2]: 0<x<4 Option D matches our one of the ranges
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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26 Sep 2017, 02:08
alchemist009 wrote: If x^3 < 16x which of the following includes at least some of the possible solutions for x, but no values that are not solutions?
A. x < 4 B. x < 4 C. x > 4 D. x < 4 E. x > 0 x^3 < 16x x(x^216) < 0 (x+4)x(x4) < 0 So, x<4 or 0<x<4 Only option D satisfies the condition. Answer D
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