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Home » \( 4a + 2b = 12 \) - Nurtured Nest

\( 4a + 2b = 12 \) - Nurtured Nest

📅 May 8, 2026 👤 scraface

May 08, 2026

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📌 Let \( x, y, z \) be positive real numbers. Find the minimum value of

📌 \frac{x + y}{x - y} + \frac{y + z}{y - z} + \frac{z + x}{z - x}

📌 given that \( x > y > z \).

📌 We are to minimize the expression

📌 S = \frac{x + y}{x - y} + \frac{y + z}{y - z} + \frac{z + x}{z - x}

📌 under the condition \( x > y > z > 0 \). Note that all denominators are positive since \( x > y > z \), so each fraction is positive and unbounded as any variable approaches another.

📌 We analyze the structure of each term. Consider the identity:

📌 \frac{x + y}{x - y} = \frac{(x - y) + 2y}{x - y} = 1 + \frac{2y}{x - y}

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