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Mathematics

Using properties of proportion solve for x, given :

5x+2x−65x−2x−6=4\dfrac{\sqrt{5x} + \sqrt{2x - 6}}{\sqrt{5x} - \sqrt{2x - 6}} = 4

Ratio Proportion

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Answer

Given,

5x+2x−65x−2x−6=4\dfrac{\sqrt{5x} + \sqrt{2x - 6}}{\sqrt{5x} - \sqrt{2x - 6}} = 4

Applying Componendo and Dividendo, we get :

⇒5x+2x−6+5x−2x−65x+2x−6−(5x−2x−6)=4+14−1⇒25x5x+2x−6−5x+2x−6=53⇒25x22x−6=53⇒5x2x−6=53\Rightarrow \dfrac{\sqrt{5x} + \sqrt{2x - 6} + \sqrt{5x} - \sqrt{2x - 6}}{\sqrt{5x} + \sqrt{2x - 6} - (\sqrt{5x} - \sqrt{2x - 6})} = \dfrac{4 + 1}{4 - 1} \\[1em] \Rightarrow \dfrac{2\sqrt{5x}}{\sqrt{5x} + \sqrt{2x - 6} - \sqrt{5x} + \sqrt{2x - 6}} = \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{2\sqrt{5x}}{2\sqrt{2x - 6}} = \dfrac{5}{3} \\[1em] \Rightarrow \dfrac{\sqrt{5x}}{\sqrt{2x - 6}} = \dfrac{5}{3}

Squaring both sides, we get :

⇒(5x2x−6)2=(53)2⇒(5x2x−6)=(259)⇒9(5x)=25(2x−6)⇒45x=50x−150⇒50x−45x=150⇒5x=150⇒x=1505=30\Rightarrow \Big(\dfrac{\sqrt{5x}}{\sqrt{2x - 6}}\Big)^2 = \Big(\dfrac{5}{3}\Big)^2 \\[1em] \Rightarrow \Big(\dfrac{5x}{2x - 6}\Big) = \Big(\dfrac{25}{9}\Big) \\[1em] \Rightarrow 9(5x) = 25(2x - 6) \\[1em] \Rightarrow 45x = 50x - 150 \\[1em] \Rightarrow 50x - 45x = 150 \\[1em] \Rightarrow 5x = 150 \\[1em] \Rightarrow x = \dfrac{150}{5} = 30

Hence, x = 30.

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