If xyz ≠ 0, is x(y + z) >= 0? GMAT Data Sufficiency

Question: If xyz ≠ 0, is x(y + z) >= 0?

(1) |y + z| = |y| + |z|
(2) |x + y| = |x| + |y|

A) Statement (1) ALONE is sufficient, but statement (2) ALONE is not sufficient.
B) Statement (2) ALONE is sufficient, but statement (1) ALONE is not sufficient.
C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
D) EACH statement ALONE is sufficient.
E) Statements (1) and (2) TOGETHER are not sufficient.

Answer: C

Solution and Explanation:

Approach Solution 1:

xyz ≠ 0 means that neither of the unknowns is 0.

(1) |y + z| = |y| + |z| --> either both yy and zz are positive or both are negative, because if they have opposite signs then |y+z||y+z| will be less than |y|+|z||y|+|z| (|-3+1|<|-3|+1). Not sufficient, as no info about xx.
(2) |x + y| = |x| + |y| --> the same here: either both xx and yy are positive or both are negative. Not sufficient, as no info about zz.

(1)+(2) Either all three are positive or all three are negative --> but in both cases the product will be positive: x(y+z)=positive∗(positive+positive)=positive>0 and
x(y+z) = negative ∗ (negative + negative ) = negative ∗ negative = positive > 0 Sufficient.

C is the correct answer.

Approach Solution 2:

Keep in mind that |a+b|=|a|+|b| denotes that the signs of a and b are equal; either they are both positive or both negative.
This query provides us with None of those numbers that can be zero because XYZ is not a zero.
First, sign(y)=sign(z) Lacking (choose y=z=4, x=1, and x=-1)... the provided product can still be positive or negative
Secondly, sign(x)=sign(y)... Insufficient... the product may once more have an indicator (take x=1, y=1, then take z=1 or z=-2)

(1+2) Whether they are all positive or all negative, sign(x)=sign(y)=sign(z) will always result in a positive product. Sufficient
C is the correct answer.

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