The triangle inequality for absolute values states that abs(x+y+z) ≤ abs(x) + abs(y) + abs(z). This equality holds when x, y, and z are non-negative or when they are arranged such that one variable does not exceed the sum of the others in absolute terms. A detailed case analysis is required to prove this, typically involving four scenarios based on the signs of x, y, and z. The proof can be simplified by assuming x ≤ y ≤ z and examining the resulting inequalities.
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