If x^3 < 16x Which Of The Following Includes at Least Some of the Possible Solutions for x GMAT Problem Solving

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?

1. |x| < 4
2. x < 4
3. x > 4
4. x < -4
5. x > 0

‘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?’ - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book “The Official Guide for GMAT Reviews”. To solve GMAT Problem Solving questions a student must have knowledge about a good amount of qualitative skills. The GMAT Quant topic in the problem-solving part requires calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

This question can be solved by only one approach.
It is given in the question that x^3 < 16x. It has asked which of the given options has some of the possible solutions for x, but no values that are not solutions of x.
Given that,
\(x^3\)< 16x
\(x^3\)-16x < 0
x(\(x^2\)-16) < 0
x(x-4)(x+4) < 0
The roots of the equation are x = -4,0, 4
This gives us four ranges
X < -4
-4 <= x < 0
0 <= x < 4
x>= 4
As the given equation is given to be less than zero, the sign for the four ranges will be
- + - +
The equation is negative for the range x < -4 and 0 < x < 4
Now checking the given options
In option 1,
|x| < 4
We get,
-4 < x <4
This contains a range (-4,0) which is not in the range of solution.
In option 2,
X < 4
Only this condition is not enough to define the range of solution.
In option 3,
X > 4
This region will give positive values for the given equation.
In option 4,
X < -4
This is the correct range
In option 5,
X > 0
But for x > 4 the value of x will not satisfy the equation
Therefore the correct answer is option D.

Correct Answer: D

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