Problem Solving

[RiyaR]

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

I. x 3 < x 2

II. x 5 < 1 - x

III. x 4 < x 2

[Brent@GMATPrepNow]

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

I. x 3 < x 2

II. x 5 < 1 - x

III. x 4 < x 2

RiyaR, please read: https://www.beatthegmat.com/read-this-fi ... tml#729339

I have formatted the question as follows:

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

I. x³ < x²
II. x� < 1 - x
III. x� < x²

USEFUL RULES:
(negative)^(ODD integer) = some negative value
(negative)^(EVEN integer) = some positive value

I. x³ < x²
x is NEGATIVE. So, the above rule tells us that x³ is negative, and x² is positive
So, it must be true that x³ < x²

II. x� < 1 - x
x is NEGATIVE. So, the above rule tells us that x� is negative.
Also, since x is negative, 1 - x must be positive
So, it must be true that x� < 1 - x

III. x� < x²
We can take this inequality and divide both sides by x² (since we can be certain that x² is positive).
When we do so, we get: x² < 1, and this is definitely true.
We know that x² is less than 1 because we're told that -1 < x < 0
So, it must be true that x� < x²

So all 3 statements must be true.

Cheers,
Brent

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Hi RiyaR,

Brent's solution focused on Number Property knowledge - knowing those rules and applying them correctly can lead to some quick solutions on Test Day. If you don't have the rules memorized though, sometimes you have to PROVE a rule by TESTing VALUES and thinking about patterns.

Here, we're told that -1 < X < 0, so we know that X has to be a "negative fraction." We're asked which of the following MUST be true.

Let's TEST VALUES and look for patterns

I. x³ < x²

If X = -1/2.....

(-1/2)³ = -1/8
(-1/2)² = +1/4

Notice how "squaring" a negative fraction makes it positive.
Notice how "cubing" a negative fraction makes it negative.
A negative is always less than a positive.
Realizing this pattern is enough to prove that Roman Numeral I is ALWAYS TRUE.

II. x� < 1 - x

If X = -1/2.....

(-1/2)� = -1/32
1 - (-1/2) = 1.5

Notice how raising a negative fraction to the "5th power" makes it negative.
Notice how subtracting a negative fraction from 1 gives us a BIGGER positive.
Again, a negative is always less than a positive.
Realizing this pattern is enough to prove that Roman Numeral II is ALWAYS TRUE.

III. x� < x²

If X = 1/2....

(-1/2)� = +1/16
(-1/2)² = +1/4

Notice how raising a negative fraction to the "4th power" makes it positive.
Notice how "squaring" a negative fraction also makes it positive.
The result of raising a fraction to the "4th power" will always be less than the result of "squaring" a power.
Realizing this pattern is enough to prove that Roman Numeral III is ALWAYS TRUE.

Thus, ALL 3 Roman Numerals are TRUE.

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