Question

Solve: (x-1)/(x-2) + (x-3)(x-4) = 3 1/3 (mixed fraction)​

Answer 1

Given=(x−2)(x−1)+(x−4)(x−3)=331

Converting the mixed fraction to improper fraction form, we get:

$\sf\implies \dfrac{(x-1)}{(x-2)} + \dfrac{(x-3)}{(x-4)} = \dfrac{10}{3}$

Taking LCM on the RHS, we get:

$\begin{gathered}\sf\implies \dfrac{(x-1)(x-4) + (x-3)(x-2)}{(x-2)(x-4)} = \dfrac{10}{3}\\\\\\\sf\implies \dfrac{(x^2 -4x - x + 4) + (x^2-2x-3x+6)}{(x^2 - 4x - 2x + 8)} = \dfrac{10}{3}\\\\\\\ \sf\implies \dfrac{(x^2 - 5x + 4 + x^2 - 5x + 6)}{(x^2 - 6x + 8)} = \dfrac{10}{3}\\\\\\\sf\implies \dfrac{(2x^2 -10x + 10)}{(x^2 - 6x +8)} = \dfrac{10}{3}\\\\\\\text{Cross multiplying we get:}\end{gathered}$

$\begin{gathered}\sf\implies 3 ( 2x^2 - 10x + 10 ) = 10 ( x^2 - 6x + 8 )\\\\\\\sf\implies 6x^2 - 30x + 30 = 10x^2 - 60x + 80\\\\\\\sf\implies 10x^2 - 6x^2 -60x + 30x + 80 - 30 = 0\\\\\\\sf\implies 4x^2 - 30x + 50 = 0\\\\\\\sf\implies 4x^2 - 20x - 10x + 50 = 0\\\\\\\sf\implies 4x ( x - 5 ) - 10 ( x - 5 ) = 0\\\\\\\sf\implies ( 4x - 10 ) ( x - 5 ) = 0\\\\\\\boxed{\sf{\implies x = \dfrac{10}{4} = \dfrac{5}{2} \:\:(and)\;\: x = 5}}\end{gathered}$