Using properties of proportion, solve each of the following for $x$ :
(iv) $\frac{\sqrt{12 x + 1} + \sqrt{2 x - 3}}{\sqrt{12 x + 1} - \sqrt{2 x - 3}} = \frac{3}{2}$ .

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Solution

Calculate the value of $x$ :
If $\frac{a}{b} = \frac{c}{d}$ , then the property of componendo and dividendo states that $\frac{a + b}{a - b} = \frac{c + d}{c - d}$ .
Now,
$\frac{\sqrt{12 x + 1} + \sqrt{2 x - 3} + \sqrt{12 x + 1} - \sqrt{2 x - 3}}{\sqrt{12 x + 1} + \sqrt{2 x - 3} - (\sqrt{12 x + 1} - \sqrt{2 x - 3})} = \frac{3 + 2}{3 - 2} (b y c o m p o n e n d o a n d d i v i d e n d o) \Rightarrow \frac{2 \sqrt{12 x + 1}}{2 \sqrt{2 x - 3}} = \frac{5}{1} \Rightarrow \frac{\sqrt{12 x + 1}}{\sqrt{2 x - 3}} = 5 \Rightarrow \frac{12 x + 1}{2 x - 3} = 25 (S q u a r i n g b o t h s i d e s) \Rightarrow 12 x + 1 = 25 (2 x - 3) \Rightarrow 12 x + 1 = 50 x - 75 \Rightarrow 50 x - 12 x = 75 + 1 (S e p a r a t i n g t h e v a r i a b l e a n d c o n s \tan t t e r m s) \Rightarrow 38 x = 76 ∴ x = 2 (D i v i d i n g b o t h s i d e s b y 38)$
Final Answer: The value of $x = 2$ .

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