If |x|<x^2 , which of the following must be true?

To start with :- For the ease of calculation look at the LHS = |x|

|x| can be essentially seen as +ve x because mod CANNOT yield a negative value.

Now look at the RHS = x^2

x^2 can also be essentially seen as a positive value because squaring always yield a positive value.

IN SUMMARY the stimulus is telling us a positive number is smaller than its square.

Don't we all already know that.

That a number will always be smaller than its square EXCEPT ... EXCEPT WHEN ???? Except when the number is a positive decimal between 0 to 1, then x will be greater than its square x^2

For example

0.5 is always greater than its square 0.25

NOTICE HOW +0.5 IS A DECIMAL THAT FALLS BETWEEN 0 AND 1

But this is not case here. The stimulus tells us that the number is always less than its square. |x|<x^2

See this

-4 is always less than its square 16

+4 is always less than it's square 16

-0.6 is always less than its square 0.36

BUT +0.6 is always greater than its square 0.36

I. x^2>1 (when the irrefutable condition is |x|<x^2)

OFFCOURSE THIS MUST ALWAYS BE TRUE ... We just proved it earlier that if any number x is less than its square x^2 then it means that x^2 is always be greater than 1 ALWAYS TRUE/ MUST BE TRUE.

II. x>0 (when the irrefutable condition is |x|<x^2)

This will falsify our stimulus for values of x between 0 and 1 therefore it CAN BE TRUE FOR VALUES MORE THAN 1 BUT NOT ALWAYS TRUE WHEN X LIES BETWEEN 0 and 1

III. x<-1 (when the irrefutable condition is |x|<x^2)

This does not cover our other possible values when x > 1 So IT IS TRUE. BUT ONLY HALF TRUTH AS IT EXCLUDES THE OTHER POSSIBILITY X>1

ANSWER IS A

I HOPE MY ANSWER WILL HELP THOSE PEOPLE WHO ARE STUCK IN THE -1>X>1 CONFUSION.

YOU HAVE TO KEEP IN MIND THE IRREFUTABLE STIMULS |x|<x^2 IN MIND WHEN EVALUATION THE GIVEN OPTIONS I, II, III

Thanks

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I and III only

mehdiov wrote:

If |x|<x^2 , which of the following must be true?

I. x^2>1

II. x>0

III. x<-1

(A) I only

(B) II only

(C) III only

(D) I and II only

(E) I and III only

PLEASE READ THE WHOLE THREAD AND FOLLOW THE LINKS PROVIDED IN ORDER TO UNDERSTAND THE QUESTION/SOLUTION CORRECTLY.

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