Solve the following equation using the method | vedclass

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Question

(D) Given equation: $\frac{x}{x+1} + \frac{x+1}{x} = \frac{5}{2}$.Let $y = \frac{x}{x+1}$. Then the equation becomes $y + \frac{1}{y} = \frac{5}{2}$.M

Vedclass · Accepted Answer

$\{1, -2\}$

Vedclass · Answer

(D) Given equation: $\frac{x}{x+1} + \frac{x+1}{x} = \frac{5}{2}$.Let $y = \frac{x}{x+1}$. Then the equation becomes $y + \frac{1}{y} = \frac{5}{2}$.Multiplying by $2y$,we get $2y^2 + 2 = 5y$,which simplifies to $2y^2 - 5y + 2 = 0$.Factoring the quadratic: $2y^2 - 4y - y + 2 = 0 \implies 2y(y - 2) - 1(y - 2) = 0$.So,$(2y - 1)(y - 2) = 0$,giving $y = 1/2$ or $y = 2$.Case $1$: $\frac{x}{x+1} = \frac{1}{2} \implies 2x = x + 1 \implies x = 1$.Case $2$: $\frac{x}{x+1} = 2 \implies x = 2x + 2 \implies x = -2$.The solution set is $\{1, -2\}$.

Solve the following equation using the method of factorization and write its solution set: $\frac{x}{x+1} + \frac{x+1}{x} = \frac{5}{2}$

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