If xyz ≠ 0, is x(y + z) >= 0?

I think the question should be:

If xyz ≠ 0, is x (y + z) >= 0?

xyz ≠ 0 means that neither of unknowns is 0.

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

(2) |x + y| = |x| + |y| --> the same here: either both \(x\) and \(y\) are positive or both are negative. Not sufficient, as no info about \(z\).

(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.

Answer: C.

For theory on absolute values check: math-absolute-value-modulus-86462.html
For practice check: search.php?search_id=tag&tag_id=37 (DS), search.php?search_id=tag&tag_id=58 (PS), inequality-and-absolute-value-questions-from-my-collection-86939.html (700+ DS).

Hope it helps.