Question

solve for x 1/x-2 + 2/x-1 = 6/x

Answer 1

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$\large\mathsf{Given \ expression \ -}$

$\mathsf\red{1/(x - 2) + 2/(x - 1) \ = \ 6/x}$

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$\large\mathsf{On \ solving \ further,}$

$\frac{1}{x - 2} + \frac{2}{x - 1} = \frac{6}{x} \\ \\ \\ \frac{1(x - 1) + 2(x - 2)}{(x - 2)(x - 1)} = \frac{6}{x} \\ \\ \\ \frac{x - 1 + 2x - 4}{x(x - 1) - 2(x - 1)} = \frac{6}{x} \\ \\ \\ \frac{3x - 5}{ {x}^{2} - x - 2x + 2 } = \frac{6}{x} \\ \\ \\ \frac{3x - 5}{ {x}^{2} - 3x + 2} = \frac{6}{x}$

$\large\mathsf{Cross \ Multiplication}$

$x(3x - 5) = 6( {x}^{2} - 3x + 2) \\ \\ \\ 3 {x}^{2} - 5x = 6 {x}^{2} - 18x + 12 \\ \\ \\ 6 {x}^{2} - 3 {x}^{2} - 18x + 5x + 12 = 0 \\ \\ \\ 3 {x}^{2} - 13x + 12 = 0$

$\large\mathsf{Using \ middle \ term \ factorisation}$

$3 {x}^{2} - 9x - 4x + 12 = 0 \\ \\ \\ 3x(x - 3) - 4(x - 3) = 0 \\ \\ \\ (3x - 4)(x - 3) = 0 \\ \\ \\ 3x - 4 = 0 \\ \\ And \\ \\ x - 3 = 0 \\ \\ \\ x = \frac{4}{3} \\ \\ And \\ \\ x = 3$

$\mathsf\red{The \ values \ of \ x \ = \ 4/3 \ and \ 3.}$

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