If -1

Question

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

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

Bunuel · Accepted Answer

The question should read: 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 A. I only B. I and II only C. II and III only D. I and III only E. I, II and III I. x^3 < x^2 --> from -1 < x < 0 it follows that LHS<0 x(x^4+1) < 1 --> negative*positive < 0 < 1, so this statement is also true. III. x^4 < x^2 --> reduce by x^2 (we can safely do that since from -1 < x < 0 is follows that x^2>0): x^2 < 1. Again, as -1 < x < 0, then x^2 must be less than 1. Hence, this statement is also true. Answer: E. Hope it's clear.

HarveyS · Answer

Bunuel, Did not understand the colored part.

As -1 < x < 0, then X will always be negative and X2 will always be positive, so how to derive that x<1.

Bunuel · Answer

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.

ENGRTOMBA2018 · Answer

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 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 . 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 -1x>1, greater the power, smaller will be the number. Consider x=-0.5 , x^2=0.25 , x^4 = 0.0625 , thus x^4 E is the correct answer. Hope this helps.

chetan2u · Answer

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^4x 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

TheUltimateWinner · Answer

I. x^3 < x^2 ---> x<1 ---> 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 ---> 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.

ydmuley · Answer

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  -   True

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

JeffTargetTestPrep · Answer

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

arvind910619 · Answer

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

shashankism · Answer

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.

Answer E.

MathRevolution · Answer

-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.

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

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

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