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![\frac{a}{x + 1} = \frac{b}{x - 1 } \\ \\ \frac{a}{b} = \frac{x + 1}{x - 1} \\ \\ \frac{a}{b} = \frac{(x - 1) + 2}{x - 1} \\ \\ \frac{a}{b} = \frac{x - 1}{ x - 1 } + \frac{2}{x - 1} \\ \\ \frac{a}{b} = 1 + \frac{2}{x - 1} \\ \\ \frac{a}{b} - 1 = \frac{2}{x - 1} \\ \\ \frac{ \frac{a}{b} - 1 }{2} = \frac{1}{x - 1} \\ \\ \frac{ \frac{a - b}{b} }{2} = \frac{1}{x - 1} \\ \\ \frac{a - b}{2b} = \frac{1}{x - 1} \\ \\ x - 1 = \frac{2b}{a - b} \\ \\ x = 1 + \frac{2b}{a - b} \\ \\ x = \frac{a + b}{a - b} \frac{a}{x + 1} = \frac{b}{x - 1 } \\ \\ \frac{a}{b} = \frac{x + 1}{x - 1} \\ \\ \frac{a}{b} = \frac{(x - 1) + 2}{x - 1} \\ \\ \frac{a}{b} = \frac{x - 1}{ x - 1 } + \frac{2}{x - 1} \\ \\ \frac{a}{b} = 1 + \frac{2}{x - 1} \\ \\ \frac{a}{b} - 1 = \frac{2}{x - 1} \\ \\ \frac{ \frac{a}{b} - 1 }{2} = \frac{1}{x - 1} \\ \\ \frac{ \frac{a - b}{b} }{2} = \frac{1}{x - 1} \\ \\ \frac{a - b}{2b} = \frac{1}{x - 1} \\ \\ x - 1 = \frac{2b}{a - b} \\ \\ x = 1 + \frac{2b}{a - b} \\ \\ x = \frac{a + b}{a - b}](https://tex.z-dn.net/?f=+%5Cfrac%7Ba%7D%7Bx+%2B+1%7D++%3D++%5Cfrac%7Bb%7D%7Bx+-+1+%7D++%5C%5C++%5C%5C++%5Cfrac%7Ba%7D%7Bb%7D++%3D++%5Cfrac%7Bx+%2B+1%7D%7Bx+-+1%7D++%5C%5C++%5C%5C++%5Cfrac%7Ba%7D%7Bb%7D++%3D++%5Cfrac%7B%28x++-+1%29+%2B+2%7D%7Bx+-+1%7D++%5C%5C++%5C%5C++%5Cfrac%7Ba%7D%7Bb%7D++%3D++%5Cfrac%7Bx+-+1%7D%7B+x+-+1+%7D++%2B++%5Cfrac%7B2%7D%7Bx+-+1%7D++%5C%5C++%5C%5C++%5Cfrac%7Ba%7D%7Bb%7D++%3D+1+%2B++%5Cfrac%7B2%7D%7Bx+-+1%7D++%5C%5C++%5C%5C++%5Cfrac%7Ba%7D%7Bb%7D++-+1+%3D++%5Cfrac%7B2%7D%7Bx+-+1%7D++%5C%5C++%5C%5C++%5Cfrac%7B+%5Cfrac%7Ba%7D%7Bb%7D+-+1+%7D%7B2%7D++%3D++%5Cfrac%7B1%7D%7Bx+-+1%7D++%5C%5C++%5C%5C++%5Cfrac%7B+%5Cfrac%7Ba+-+b%7D%7Bb%7D+%7D%7B2%7D++%3D++%5Cfrac%7B1%7D%7Bx+-+1%7D++%5C%5C++%5C%5C++%5Cfrac%7Ba+-+b%7D%7B2b%7D++%3D++%5Cfrac%7B1%7D%7Bx+-+1%7D++%5C%5C++%5C%5C+x+-+1+%3D++%5Cfrac%7B2b%7D%7Ba+-+b%7D++%5C%5C++%5C%5C+x+%3D+1+%2B++%5Cfrac%7B2b%7D%7Ba+-+b%7D++%5C%5C++%5C%5C+x+%3D++%5Cfrac%7Ba+%2B+b%7D%7Ba+-+b%7D+)
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