Question

\frac { x } { x - 2 } + \frac { x - 1 } { x + 1 } = - 1
I cant firgure it out​

Answer 1

`\huge{\color{magenta}{\fbox{\textsf{\textbf{Answer}}}}}`

x = -1

Explanation:

`\sf(x)/(x - 2) + (x - 1)/(x + 1) = - 1`

Taking LCM

`\sf \hookrightarrow (x(x + 1) + (x - 1)(x - 2))/((x - 2)(x + 1)) = - 1 \\ \\ \sf \hookrightarrow \frac{ {x}^(2) + x + {x}^(2) - 3x + 2 }{ {x}^(2) - x - 2} = - 1 \\ \\ \sf \hookrightarrow \frac{2 {x}^(2) - 2x + 2 }{ {x}^(2) - x - 2 } = - 1`

Cross multiplying

`\sf \hookrightarrow 2 {x}^(2) - 2x + 2 = - 1( {x}^(2) - x -2) \\ \\ \sf \hookrightarrow 2 {x}^(2) - 2x + \cancel2 = - {x}^(2) + x + \cancel2 \\ \\ \sf \hookrightarrow 3 {x}^(2) - 3x = 0`

Taking 3 common

`\sf \hookrightarrow {x}^(2) - x = 0 \\ \\ \sf \hookrightarrow x(x - 1) = 0 \\ \\ \sf \hookrightarrow x - 1 = 0 \\ \\ \\ \green{\boxed{ \hookrightarrow x = 1}}`

• Verifying

Taking x as 1

`\sf \implies(x)/(x - 2) + (x - 1)/(x + 1) = - 1 \\ \\ \sf \implies (1)/(1 - 2) + (1 - 1)/(1 + 1) = - 1 \\ \\ \sf \implies - 1 + 0 = - 1 \\ \\ \sf \implies - 1 = - 1`