Question

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Answer 1

Step-by-step explanation:

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Answer 2

Given:

• $\: \: \: \: \: \tt \: \dfrac{ \sqrt{x + 1 }- \sqrt{x - 1} } { \sqrt{x + 1} - \sqrt{x - 1} } = \dfrac{1}{2}$

Solution:

$\tt \: \dfrac{ \sqrt{x + 1 }- \sqrt{x - 1} } { \sqrt{x + 1} - \sqrt{x - 1} } = \dfrac{1}{2}$

Applying Componendo and Dividendo we get:

$\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \implies \tt \: \dfrac{ \sqrt{x + 1 }- \sqrt{x - 1} + \sqrt{x + 1 }+ \sqrt{x - 1}} { \sqrt{x + 1 }- \sqrt{x - 1} - \sqrt{x + 1 } - \sqrt{x - 1} } = \dfrac{3}{ - 1} \: \\ \\ \\ \tt \implies \frac{2 \sqrt{x + 1} }{ - 2 \sqrt{x - 1} } = \frac{3}{ - 1} \\ \\ \\ \tt \implies \frac{2 \sqrt{x + 1} }{ 2 \sqrt{x - 1} } = 3 \\ \\ \\ \tt \implies \frac{x + 1}{x - 1} = 9 \\ \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \implies9(x - 1) = x + 1 \\ \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \implies9x - 9 = x + 1 \\ \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \implies9x - x = 9 + 1 \\ \\ \\\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \implies8x = 10 \\ \\ \\\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \implies \: x = \frac{10}{8} \\ \\ \\\: \: \: \: \: \: \: \: \: \: \: \: \:\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \implies \: x = \frac{ \cancel{10}}{ \cancel{8}} \\ \\ \\ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \ \: \: \: \ \: \: \: \: \:\:\implies \tt \: x = \frac{5}{4}$

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• Hence Solved!