If x^3

Question

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

Bunuel · Accepted Answer

x^3 < 16x ; x^3-16x<0 ; x(x^2-16)<0 ; x(x+4)(x-4)<0 . Roots are -4, 0, and 4. This gives us 4 ranges: x<-4 , -44 . 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 0https://gmatclub.com/forum/x2-4x-94661.html#p731476 https://gmatclub.com/forum/inequalities ... 91482.html https://gmatclub.com/forum/everything-i ... me#p868863 https://gmatclub.com/forum/xy-plane-714 ... ic#p841486

piyatiwari · Answer

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)

prasannajeet · Answer

I mean 
can we write x^3<16x as x^3-16x<0 without knowing X is +ve or -ve?

Rgds
Prasannjeet

Bunuel · Answer

Yes, we can. I think you are mixing adding/subtracting a value from both sides of the inequality with multiplying/dividing both sides by some value (which we cannot do if the sign of that value is not known).

So, we can safely subtract 16x from both sides to get x^3-16x<0.

Hope it's clear.

EMPOWERgmatRichC · Answer

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:  D

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

PerfectScores · Answer

x(x)(x) - 16x < 0

x(x(x) - 16) <0

x(x+4)(x-4)<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.

GMATinsight · Answer

x^3 < 16x
i.e. x^3 - 16x < 0
i.e. x(x^2 - 16) < 0
i.e. (x-4)x(x+4) < 0

Case-1: Either (x-4) is Negative and both x and (x+4) are positive
 i.e. x<4 and x>0

Case-2: Or (x-4), x and (x+4) are all Negative 
 i.e. x < -4  Answer: Option D

Leo8 · Answer

FullSizeRender (3).jpg So we have only two possible ranges where the equation is less than zero. : x < -4 : 0

BrushMyQuant · Answer

x^3 < 16x Taking 16x to left hand side we get x^3 - 16x < 0 => x ( x^2 - 16) < 0 => x ( x^2 - 4^2 ) < 0 => x (x-4)(x+4) < 0 a^2 - b^2 = (a-b)*(a+b) ] Using Wave Method to solve the inequality . Watch this video to learn the method image-3.jpg Since the problem is asking for < 0 so we will take the values in the negative range (-) so answer will be x < -4 and 0 < x < 4 So, Answer will be D Let's see all the choices A. |x| < 4 =< -4 < x < 4 => This contains a range which is not in the solution -4 < x <=0 Watch this video to learn about Absolute Value Basics B. x < 4 => Same as Option A C. x > 4 => Not a solution D. x < -4 => Part of the solution E. x > 0 => Contains x >= 4 which is not in the solution Watch the Following Video to Master How to Solve Inequality Problems https://www.youtube.com/watch?v=0-Z0CeyB8jY

pranavshah7887 · Answer

We can't just shift the variables to one side without knowing the sign of the variables.

Just put x= -1

after shifting the inequality becomes x^3-16x<0. Put x=-1, it becomes -1+16<0 that is +15<0, which is not true.
Therefore the variables can not be shifted just like that per my understanding.
Please correct me if i am wrong.
 Bunuel

Bunuel · Answer

1. The value x = -1 does not satisfy the inequality x^3 < 16x.

2. When dealing with an inequality, it is always valid to add or subtract the same value from both sides of the inequality.

3. Consider reviewing the theory of inequalities to improve your understanding of their properties:
  Inequalities Made Easy! 
 Inequalities: Tips and hints 
 Arithmetic with Inequalities 
 Wavy Line Method Application - Complex Algebraic Inequalities 
 Solving Quadratic Inequalities: Graphic Approach 
 Inequalities - Quadratic Inequalities 
 Graphic approach to problems with inequalities 
 Inequalities trick 
 INEQUATIONS(Inequalities) Part 1 
 INEQUATIONS(Inequalities) Part 2 
     Questions   
 Inequality and absolute value questions from my collection 
 DS Questions 
 PS Questions

Hope it helps.

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If x^3 < 16x which of the following includes at least some

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