If y 0, is |x| = 1 ? (1) x = y/|y| (2) |x| = -x

Target question: Is |x| = 1?

Statement 1: \(x = \frac{y}{|y|}\)
First recognize that |y| must positive.
With this in mind let's consider two possible cases: y is positive and y is negative.
Case a: If y is negative, then x = \(\frac{y}{|y|}\) = (negative y)/(positive y) = -1, in which case |x| = |-1| = 1. So, the answer to the target question is YES, |x| = 1
Case b: If y is positive, then x = \(\frac{y}{|y|}\) = (positive y)/(positive y) = 1, in which case |x| = |1| = 1. So, the answer to the target question is YES, |x| = 1
Since both possible cases yield the same answer to the target question, statement 1 is SUFFICIENT

Statement 2: |x| = -x
There are infinitely many values of x that satisfy the equation |x| = -x
Here are two:
Case a: x = -1 (notice that |x| = -x becomes |-1| = -(-1), which simplifies to be |-1| = 1 (which is true). In this case, the answer to the target question is YES, |x| = 1
Case b: x = -2 (notice that |x| = -x becomes |-2| = -(-2), which simplifies to be |-2| = 2 (which is true). In this case, the answer to the target question is NO, |x| does not equal 1
Since we can’t answer the target question with certainty, statement 2 is NOT SUFFICIENT

Answer: A

Cheers,
Brent