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If x^7 < x^6 , which of the following could be the value of x ?

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If x^7 < x^6 , which of the following could be the value of x ?  [#permalink]

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New post 16 Apr 2018, 05:02
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A
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C
D
E

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Re: If x^7 < x^6 , which of the following could be the value of x ?  [#permalink]

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New post 16 Apr 2018, 05:18
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Bunuel wrote:
If x^7 < x^6, which of the following could be the value of x?

I. −3
II. 2/3
III. 3/2

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III


This question is designed to test properties of negative numbers and fractions.

1. -3 ...........always true. coz we have the even power.
2. It is also true. proper fraction
3. improper fraction.....while solving this one , reduce the power to verify how this kind of fraction reacts. this violates the given inequality.

thus , the correct answer is C.
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If x^7 < x^6 , which of the following could be the value of x ?  [#permalink]

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New post 16 Jun 2018, 11:18
1
x^7 < x^6
x^7-x^6<0
x(x^6-1)<0

I. −3
x=-ve, x^6-1 = +ve
overall <-ve, X can be -3

II. 2/3
x=+ve,x^6-1= (2/3)^6-1 , will always be negative
overall -ve, X can be 2/3

III. 3/2
x=+ve, x^6-1+ (3/2)^6-1, will always be positive, cannot be value of X

Hence i and ii ARE CORRECT

Hence C answer
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If x^7 < x^6 , which of the following could be the value of x ?  [#permalink]

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New post 18 Jun 2018, 12:35
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1
Bunuel wrote:
If x^7 < x^6, which of the following could be the value of x?

I. −3
II. 2/3
III. 3/2

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

Number properties often can be verified with testing.

And if questions have numbers with large exponents, usually the question can be replicated on a smaller scale.

We can test values using smaller exponents as long as we stay true to the original problem (inequality).

The prompt gives an inequality that means "a number raised to an odd power is smaller than the number raised to an even power."
Thus: \(x^3 < x^2\)

Which of the following COULD be the value of x?

I. −3
Test: \((-3)^3=-27\) and \((-3)^2=9\)
\(-27<9\) ... Possible

Number property tested is that
-- A negative number raised to an odd power is negative
-- Raised to an even power, the number is positive

II. 2/3
Test the much easier number \(\frac{1}{2}\) INSTEAD (any fraction between 0 and 1 works)
\((\frac{1}{2})^3=\frac{1}{8}\), and
\((\frac{1}{2})^2=\frac{1}{4}\)

\(\frac{1}{8}<\frac{1}{4}\). ...Possible

Property: For any fraction between 0 and 1, the value of the fraction decreases as the exponent increases.

III. 3/2 - Test:
\((\frac{3}{2})^3=\frac{27}{8}=3.xx\), and
\((\frac{3}{2})^2=\frac{9}{4}=2.xx\)

LHS is not < RHS. Not possible

Property: For any number greater than 1, the value of the number increases as the exponent increases

I and II only

Answer C
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Re: If x^7 < x^6 , which of the following could be the value of x ?  [#permalink]

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New post 20 Jun 2018, 17:21
Bunuel wrote:
If x^7 < x^6, which of the following could be the value of x?

I. −3
II. 2/3
III. 3/2

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III


Since x^7 is less than x^6, x could be either, a positive number between 0 and 1 or a negative number. Thus x could be -3 or 2/3.

Alternate Solution:

Let’s consider each Roman numeral:

If x = -3, then we see that x^7 is a negative number, and x^6 is a positive number. Thus, Roman numeral I works.

If x = 2/3, then x is a proper fraction. For a positive proper fraction, the greater the exponent, the smaller the result. Thus, (2/3 )^7 is less than (2/3 )^6. Roman numeral II works.

If x = 3/2, then (3/2)^7 is greater than (3/2)^6. This outcome would hold for any value of x greater than 1.

Answer: C
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If x^7 < x^6 , which of the following could be the value of x ?  [#permalink]

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New post 21 Jun 2018, 01:21

Solution



Given:
    • \(x^7 < x^6\)

To find:
    • Among the three given values, which is a possible value of x.

Approach and Working:
    • \(x^7 < x^6\)
      o Subtracting x^6 from both the sides, we get: \(x^6(x-1)<0\)
      o We can solve this by the wavy line below:

Image
Hence, for x<1, \(x^6(x-1)<0\)
Hence, the correct answer is option C.

Answer: C

To read more about the wavy line: https://gmatclub.com/forum/wavy-line-me ... l#p1727247
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Re: If x^7 < x^6 , which of the following could be the value of x ?  [#permalink]

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New post 22 Jul 2018, 21:29
Given: x^7<x^6

Divide both side by x^6
(x^7/x^6)<(x^6/x^6) ..........Since x^6 will always be positive, the sign of inequality doesn't change
x<1

Sincle x<1, only option I(-3) and option II (2/3) is correct. Hence option C
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Re: If x^7 < x^6 , which of the following could be the value of x ?  [#permalink]

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New post 25 Nov 2019, 05:23
Assuming X#0, taking the 6th root on both sides,

X<1

Smash C
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Re: If x^7 < x^6 , which of the following could be the value of x ?   [#permalink] 25 Nov 2019, 05:23
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