If -1<x<0, which of the following must be true? 1. x^3

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alanforde800Maximus: Master | Next Rank: 500 Posts; Posts: 187; Joined: Tue Sep 13, 2016 12:46 am

If -1<x<0, which of the following must be true? 1. x^3

by alanforde800Maximus » Mon Sep 26, 2016 6:45 pm

If -1<x<0, which of the following must be true?
1. x^3 < x^2
2. x^5 < 1-x
3. x^4 < x^2

a) 1 only
b) 1 and 2 only
c) 2 and 3 only
d) 1 and 3 only
e) 1,2 and 3 only

Please assist with above problem.

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Source: Beat The GMAT — Problem Solving | Mon Sep 26, 2016 6:45 pm

GMAT/MBA Expert

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by [email protected] » Mon Sep 26, 2016 7:08 pm

Hi alanforde800Maximus,

This question is based on several Number Properties, so you can get to the solution knowing those NPs, TESTing VALUES or a combination of the two.

From the given range, we know that X is a NEGATIVE FRACTION between -1 and 0. We're asked which of the following MUST be true.

I. X^3 < X^2

(negative)^3 = negative
(negative)^2 = positive

(negative) is always less than a (positive)
Roman Numeral I is ALWAYS TRUE

II. X^5 < 1 - X

(negative)^5 = negative
1 - (negative) = postiive

(negative) is always less than a (positive)
Roman Numeral II is ALWAYS TRUE

III. X^4 < X^2

Both X^4 and X^2 will be positive in these circumstances, so we have to think about what happens when you raise a negative fraction to those two exponents.

(-1/2)^4 = +1/16
(-1/2)^2 = +1/4

1/16 < 1/4
This relationship will hold true for any value from the given range.
Roman Numeral III is ALWAYS TRUE

Final Answer: E

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

Contact Rich at [email protected]

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pm; Location: East Bay all the way; Thanked: 625 times; Followed by:119 members; GMAT Score:780

by Matt@VeritasPrep » Thu Sep 29, 2016 7:29 pm

I:

Divide both sides by xÂ², which gives

x < 1

True!

(Alternatively, notice that the left side is negative and the right side is positive, so this must be true. We'll use this same trick below in the next one.)

II:

Left side will be negative, right side will be positive. Neg < Pos is always true, so

True!

III:

Divide both sides by xÂ²:

xÂ² < 1

Since -1 < x < 0, squaring x will always give a result between 0 and 1, so

True!

And we're left with E.

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