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

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Re: If x^3 < 16x which of the following includes at least some [#permalink]
I mean
can we write x^3<16x as x^3-16x<0 without knowing X is +ve or -ve?

Rgds
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
prasannajeet wrote:
I mean
can we write x^3<16x as x^3-16x<0 without knowing X is +ve or -ve?

Rgds
Prasannjeet

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.
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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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.

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.

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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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)(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.

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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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. (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
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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Attachment:

FullSizeRender (3).jpg [ 27.6 KiB | Viewed 7921 times ]
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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If x^3 < 16x which of the following includes at least some [#permalink]
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$$x^3$$ < 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 [ Using $$a^2$$ - $$b^2$$ = (a-b)*(a+b) ]

Using Wave Method to solve the inequality. Watch this video to learn the method

Attachment:

image-3.jpg [ 12.52 KiB | Viewed 4050 times ]

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

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

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If x^3 < 16x which of the following includes at least some [#permalink]
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

Originally posted by pranavshah7887 on 01 Apr 2023, 10:13.
Last edited by pranavshah7887 on 01 Apr 2023, 11:19, edited 1 time in total.
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
pranavshah7887 wrote:
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.
Please correct me if i am wrong.
Bunuel

The right way to solve this problem is to use the concept of higer order inequalitiy range.

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:

Hope it helps.
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If x^3 < 16x which of the following includes at least some [#permalink]
Ok thanks. let me check these out.
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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Re: If x^3 < 16x which of the following includes at least some [#permalink]
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