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

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

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

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:

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

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

Assuming X#0, taking the 6th root on both sides,

X<1

Smash C

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