SOLUTION: If xy>0 and xz<0, which of the following statements must be true? I. x+y+z>0 II. xyz<0 III. xy+xz+yz>0 (A) I only (B) II only (C) III only (D) II and III only (E) None

Question 1202898: If xy>0 and xz<0, which of the following statements must be true?
I. x+y+z>0
II. xyz<0
III. xy+xz+yz>0
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) None of the above
Answer by ikleyn(52778)   (Show Source): You can put this solution on YOUR website!
.
If xy > 0 and xz < 0, which of the following statements must be true?
I. x + y + z > 0
II. xyz < 0
III. xy + xz + yz > 0
(A) I only
(B) II only
(C) III only
(D) II and III only
(E) None of the above
~~~~~~~~~~~~~~~~~

        The answer is (E).

        To prove it, it is enough to provide couner-examples to cases (I), (II) and (III) separately.

Counter-example to (I)  :  x= -1, y= -1, z= 1.


Counter-example to (II) :  x= -1, y= -1, z= 1.


Counter-example to (III):  x= -1, y= -1, z= 1.

Solved, with explanations.

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