What is the value of |x|?

[cschramke]

from CAT #3...
What is the value of |x|?

(1) |x2 + 16| - 5 = 27

(2) x2 = 8x - 16

I understand why (2) is sufficient, but in the explanation for (1) they state: "Since the value of x2 must be non-negative, the value of (x2 + 16) is always positive, therefore |x2 + 16| can be written x2 +16. Using this information, we can solve for x..."
I am having trouble understanding how you know x2 must be a non-negative and therefore you only need to solve for the value of (x2 + 16) rather than also evaluating (-x2 -16). Please advise!

[RonPurewal]

x^2 is non-negative because that's a fundamental property of squares (and all other even powers, too).
Try squaring something and getting a negative result -- Not gonna happen.

Because x^2 must be either 0 or positive, the expression (x^2 + 16) must be at least 16.
By similar reasoning, the other expression, (-x^2 - 16), is always -16 or less.
So, the absolute value is always going to be the first one.