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

Consider this: \(x^4 < x^2\) holds true if \(-1<x<0\) or \(0<x<1\).

Therefore, since given that \(-1<x<0\), then \(x^4 < x^2\) must be true.

Hope it's clear.

Make sure to format the question properly. When you select the tag "source-others please specify" , do mention at the end of the question.

As for the question look below.

You are told that \(-1<x<0\) ---> x is a negative fraction less -1.

Let us analyse the 3 options.

I. \(x^3 < x^2\)----> MUST BE TRUE as \(x^2\) of \('x'\) in this question >0 while \(x^3 <0\). Consider \(x=-0.5\), \(x^2=0.25 >0\) while \(x^3=-0.125 <0\)[/m].

II. \(x^5 < 1 – x\) ----> MUST BE TRUE. Odd power of a negative umber \(x\) (=\(x^5\))<0 while 1-x for a negative \(x\) >0. Consider \(x=-0.5\) ---> \(1-(-0.5)=1+0.5=1.5 >0\)

III. \(x^4 < x^2\) -----> MUST BE TRUE. For fractions -1<x<0 or 0>x>1, greater the power, smaller will be the number. Consider \(x=-0.5\), \(x^2=0.25\), \(x^4 = 0.0625\), thus\(x^4<x^2\)

Thus, I,II,III are all MUST BE TRUE ----> E is the correct answer.

Hope this helps.

Hi,
this Qs tests the properties of numbers/ fractions between 0 and 1..

so lets first check the properties, the answer will come automatically..

1) any even power will always be greater than any odd power..
x^even>x or x^odd..

2) within even powers--
as we keep increasing power the value will keep becoming lesser..
x^4<x^2..

3) Opposite will be true of ODD powers
although the numeric value will be same as in even powers but due to a -ive sign, opposite is true..
x^3>x

4) Roots--
even roots will be imaginary..
\sqrt{x} not real
Odd roots are possible and will be lesser for lower
3rd root will be more than 5th root of x.. or 3rd root of x<x

so lets see three choices now--
I. \(x^3 < x^2\)
TRUE as per point 1) above

II.\(x^5 < 1 – x\)
x is negative so -x will be positive and 1-x will be positive too..
whereas x^5 is -ive..
hence TRUE

III.\(x^4 < x^2\)
TRUE as per point 2) above

All three are correct
ans E

I. x^3 < x^2
---> x<1 [dividing by x^2 in both side]
---> for negative value of x is less than 1. So, true.

II. x^5 < 1 – x
---> x^5+x<1
---> negative result-negative result<1
---> negative result<1
---> So, true.

III. x^4 < x^2
---> x^2<1 [dividing by x^2 in both side]
---> for any value of x (according to question stem) it must be less than 1
---> So, true.
So, the correct choice is E to me.

Try by substituting a value in the equation.

Take x = -1/2 and cross check each option.

I. x^3 < x^2 - True
II. x^5 < 1 – x - True
III. x^4 < x^2[/b] - True

Hence, answer is E - All - I, II & III

We can see that x is a negative number between 0 and -1. Thus, x^odd = negative and x^even = positive. So, I is true.

Furthermore, 1 - x = 1 + |x|, which is positive, so it’s greater than x^5, which is negative. So, II is also true.

Lastly, when x is raised to an even integer power, the larger the even integer, the smaller the power. For example, if x = -1/2, then x^4 = 1/16 and x^2 = 1/4. Thus, x^4 < x^2. So, III is also true .

Answer: E

imo E
take a value of -0.2 then substitute in every equation .
hence all satisfy

Here it is given that -1<x<0 so x is a -ve number greater than -1.
I. x^3 < x^2 is true . X^2 is positive, x^3 is -ve
II. x^5 < 1-x i true. X^5 is -ve , 1-x is +ve.
III. X^4 < x^2 is true. X^4 and x^2 are both +ve. Also since x has a domain (-1,0), x^4 < x^2.

-1 < x < 0 which means that 'x' is negative and it is a fraction value or decimal value.

Condition I. \(x^3\) < \(x^2\) - a cube of a negative number is always negative whereas the square of a negative number is always positive. - This condition is TRUE.

Condition II. \(x^5\) < 1 – x - Odd power of a negative number will result in a negative number whereas 1- (negative number) will result in a positive number. -This condition is TRUE.

Condition III. \(x^4\) < \(x^2\) - 4\(^{th}\) power of an integer is greater than the square of the same integer. But, here 'x' is a fraction. So denominator of 4\(^{th}\) power will be more than denominator of squared power which will result in 4\(^{th}\) power of x less than squared of x. (In other words, decimal values gets smaller when raised to higher powers) - This condition is TRUE.

Answer E.

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.