Question

Solve $\frac{x}{x - 1} + \frac{x - 1}{x} = 2 \frac{1}{2}$

Accepted Answer

The given equation is $\frac{x}{x - 1} + \frac{x - 1}{x} = 2 \frac{1}{2}$ .

We observe that the given equation is not quadratic, but it can be solved by reducing them to quadratic equations by suitable substitutions as follows:

Let $y = \frac{x}{x - 1}$ , then $\frac{1}{y} = \frac{x - 1}{x}$ .

Therefore, $\frac{x}{x - 1} + \frac{x - 1}{x} = 2 \frac{1}{2}$ becomes $y + \frac{1}{y} = \frac{5}{2}$

$\Rightarrow 2 y^{2} - 5 y + 2 = 0$

$\Rightarrow 2 y^{2} - 4 y - y + 2 = 0$

$\Rightarrow 2 y (y - 2) - (y - 2) = 0$

$\Rightarrow (2 y - 1) (y - 2) = 0$

$\Rightarrow y = \frac{1}{2}, 2$ .

When $y = \frac{1}{2}$ , we have, $\frac{x}{x - 1} = \frac{1}{2}$ $\Rightarrow 2 x = x - 1$ $\Rightarrow x = - 1$

When $y = 2$ , we have, $\frac{x}{x - 1} = 2$ $\Rightarrow x = 2 x - 2$ $\Rightarrow x = 2$ .

Therefore, the required roots are $x = - 1, 2$ .

Tamil Nadu Board Solutions for Chapter: Algebra, Exercise 30: Solved Example

Tamil Nadu Board Mathematics Solutions for Exercise - Tamil Nadu Board Solutions for Chapter: Algebra, Exercise 30: Solved Example

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