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If xyz ≠ 0, xy > 0, and x2 + xy + xz < 0, then which of the following [#permalink]
Can someone please simplify why it cannot be E

Aren't I II III independent statements?
If yes can we make use of the 3 of them to imply whether x is negative or no?
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Re: If xyz ≠ 0, xy > 0, and x2 + xy + xz < 0, then which of the following [#permalink]
CAMANISHPARMAR wrote:
If xyz ≠ 0, xy > 0, and x^2 + xy + xz < 0, then which of the following must be true?

I. x(y + z) < 0

II. x + y + z < 0

III. If x < 0, then z > 0.

A) I only
B) I and II only
C) I and III only
D) II and III only
E) I, II, and III


Conditions:
a) xyz ≠ 0
b) xy > 0 (it means xy is either both positive or both negative)
c) x^2 + xy + xz < 0 (it means x^2 positive, xy must be positive from the second condition, and xz has to be negative)

I. x(y + z) < 0, => xy + xz <0, from condition c, we can see xz is negative and greater than positive x^2 + xy combined, so this statement is true
II. x + y + z < 0, we can not say this true. Maybe, maybe not.
III. If x < 0, then z > 0. Here given x is negative.

Condition c: x^2 + xy + xz < 0
x(x+y+z) <0, now to make x(x+y+z) negative/less than 0, we will need to make (x+y+z) positive. From condition b, we already know when x is negative, y has to be negative. So to make (x+y+z) positive, z has to be positive since x and y both are negative.

So this statement is true.


C) I and III only
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Re: If xyz 0, xy > 0, and x2 + xy + xz < 0, then which of the following [#permalink]
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Re: If xyz 0, xy > 0, and x2 + xy + xz < 0, then which of the following [#permalink]
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