Is xy > 0? (1) |3 - x| < x + 5 (2) |2 - 2y| < y - 1

To find: Is xy>0?
OR is the product of x and y positive?
OR are BOTH x and y positive or negative numbers?

Constraint: None, i.e. x and y can be integers, fractions, decimals etc.

Statement 1: |3 - x| < x + 5

This is an absolute value inequality with variables on both sides. Hence, on removing the Modulus symbol, we get two inequalities: (i) 3-x < x+5 and (ii) x-3 < x+5

On further simplifying to find out the range of variable x, we get
(i) x>-1 and
(ii) -3<5 (however, this doesn't tell us anything about the values of x)

Hence, from statement 1 we can say that the range of x which will satisfy the given inequality is x > -1 i.e. x can be a positive number, a negative number between 0 and -1, OR Zero.

If x = 0 then xy will be 0, which would give us a definite no for "is xy > 0?". However, if x is a negative number between 0 and -1, for instance, -0.25 or a positive number then to get a definite no for "is xy>0?" we would need information about the sign of y. Therefore, statement 1 is INSUFFICIENT.

Statement 2: |2 - 2y| < y - 1

Again, this is an absolute value inequality with variables on both sides. On solving we get two inequalities:
(i) 2-2y < y-1 and (ii) -2+2y < y-1

On further simplification, we get (i) 1 < y and (ii) y < 1
Hence, statement 2 isn't sufficient.

On further combining Statement 1 and 2, we still don't know anything about the sign of y.

Hence, answer is (E) Both statements are insufficient.