If $x,y,z$ are three distinct positive real numbers and are in H.P., then $\frac{3x + 2y}{2x - y} +

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If $x,y,z$ are three distinct positive real numbers and are in H.P., then $\frac{3x + 2y}{2x - y} + \frac{3z + 2y}{2z - y}$ is greater than

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Sol. [Key Concept: $a,b,c$ distinct positive in H.P. ⇒ A.M.–G.M.–H.M. inequalities give the bounds.]$$\frac{3x + \frac{4xz}{x + z}}{2x - \frac{2xz}{x + z}} + \frac{3z + \frac{4xz}{x + z}}{2z - \frac{2xz}{z + x}} = \frac{3x^{2} + 7xz}{2x^{2}} + \frac{7xz + 3z^{2}}{2z^{2}}$$$$= 3 + \frac{7}{2}\left( \frac{z}{x} + \frac{x}{z} \right) \in (10,\infty)$$✔ (a) expression $\in (10,\infty)$, so always $> 9$✔ (b) always $> 10$✘ (c) can be just above 10, $< 12$✘ (d) can be $< 15$

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