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Question

$$\dfrac{x-1}{2x+1}+\dfrac{2x+1}{x-1}=\dfrac{5}{2}$$.


Solution

$$\dfrac{x-1}{2x+1}+\dfrac{2x+1}{x-1}=\dfrac{5}{2}$$ (equation is given)
$$(x-1)^{2}+(2x+1)^{2}=\dfrac{5}{2}(2x+1)(x-1)$$
$$x^{2}-2x+1+4x^{2}+4x+1$$
$$=\dfrac{5}{2}(2x^{2}-2x+x-1)$$
$$5x^{2}+2x+2=5x^{2}-\dfrac{5}{2}x-\dfrac{5}{2}$$
$$2x+\dfrac{5}{2}x=-2-\dfrac{5}{2}$$
$$\dfrac{9}{2}x=\dfrac{-9}{2}$$
$$\boxed {x=-1}$$

1213116_1072416_ans_272d56b6a6094aa59bacd44b1a61daa5.jpg

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