If x is an integer, what is the value of x?
1)|x-|x^2||=2
2)|x^2 -|x||=2
In statement 1, |x²| is redundant: since x² cannot be negative, |x²| = x².
Statement 1: |x-x²|=2
x - x² = ±2
x(1-x) = ±2.
Since x must be an integer, x=±1 or x=±2.
Check which of these values are valid solutions for |x-x²| = 2.
If x=1, then |x-x²| = |1 - 1²| = 0.
If x=-1, then |x-x²| = |-1 - (-1)²| = 2.
If x=2, then |x-x²| = |2 - 2²| = 2.
If x=-2, then |x-x²| = |-2 - (-2)²| = 6.
Since it's possible that x=-1 or that x=2, INSUFFICIENT.
Statement 2: |x² -|x||=2
x²-|x| = ±2
Since x² = |x|*|x|, we can factor out |x|:
|x| (|x|-1) = ±2.
Since x must be an integer, |x|=1 or |x|=2, implying that x=±1 or x=±2.
Check which of these values are valid solutions for |x² -|x||=2.
If x=-1, then |x² -|x|| = |(-1)² - |-1|| = 0.
If x=1, then |x² -|x|| = |1² - |1|| = 0.
If x=2, then |x² -|x|| = |2² - |2|| = 2.
If x=-2, then |x² -|x|| = |(-2)² - |-2|| = 2.
Since it's possible that x=2 or that x=-2, INSUFFICIENT.
Statements 1 and 2 combined:
Both statements are satisfied only by x=2.
SUFFICIENT.
The correct answer is
C.
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