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# If x is an integer, what is the value of x?

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If x is an integer, what is the value of x?
(1) |x - |x^2|| = 2
(2) |x^2 - |x|| = 2

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( The square of a number is always a positive, and the modulus of a number is also positive number)
Statement 1) |x - |x^2|| = 2.
so, |X^2| = x^2
We can square on both sides.
=> ( x - x^2)^2 = 4
=> ( x - x^2) = ±2
when
=> ( x - x^2 ) = +2 => has no integer solutions
when
=>  ( x - x^2 ) = -2
=> x^2 -x -2 => x = -1 or 2
This is not sufficient, we have two values
Now let us pick up statement 2) |x^2 - |x|| = 2
the |x| is confusing, but if we assume |x| as y, then x^2 = y^2
=> | y^2 - y | = 2
there can be two cases
Case 1)
=> y^2 - y = +2
=> y^2 - y -2 = 0 => y =  (y - 2) ( y + 1) = 0 = y = 2 or -1
But since y = |x| so y cannot be -1 , hence x can be +2, -2
Case 2)
=> y^2 - y = -2
=> No integer solutions
This is not sufficient, we have two values
But if we combine both these statements, we figure out a common value +2.

answered Apr 4, 2015 by Guru (5,628 points)
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