Question

Please answer it its important

Answer 1

$\large\underline{\sf{Solution-}}$

Given equation is

$\rm :\longmapsto\:2\bigg(\dfrac{3x - 5}{x + 2} \bigg) - 5\bigg(\dfrac{x + 2}{3x - 5} \bigg) = 3$

To solve this equation, Let assume that

$\rm :\longmapsto\:\dfrac{3x - 5}{x + 2} = y$

So, given equation reduced to

$\rm :\longmapsto\:2y - \dfrac{5}{y} = 3$

$\rm :\longmapsto\:\dfrac{ {2y}^{2} - 5}{y} = 3$

$\rm :\longmapsto\: {2y}^{2} - 5 = 3y$

$\rm :\longmapsto\: {2y}^{2} - 3y - 5 =0$

$\rm :\longmapsto\: {2y}^{2} - 5y + 2y - 5 =0$

$\rm :\longmapsto\:y(2y - 5) + 1(2y - 5) = 0$

$\rm :\longmapsto\:(y + 1)(2y - 5) = 0$

$\rm :\longmapsto\:y + 1 = 0 \: \: \: or \: \: \: \: 2y - 5= 0$

$\bf\implies \:y = - 1 \: \: \: \: or \: \: \: \: \dfrac{5}{2}$

So,

$\rm :\longmapsto\:y = - 1$

On substituting the value of y, we get

$\rm :\longmapsto\:\dfrac{3x - 5}{x + 2} = - 1 \: \: \:$

$\rm :\longmapsto\:3x - 5 = - x - 2$

$\rm :\longmapsto\:3x + x = 5 - 2$

$\rm :\longmapsto\:4x = 3$

$\bf\implies \:x = \dfrac{3}{4}$

Also,

$\rm :\longmapsto\:y = \dfrac{5}{2}$

$\rm :\longmapsto\:\dfrac{3x - 5}{x + 2} = \dfrac{5}{2}$

$\rm :\longmapsto\:6x - 10 = 5x + 10$

$\rm :\longmapsto\:6x - 5x = 10 + 10$

$\bf\implies \:x = 20$

Hence,

The solution of equation

$\rm :\longmapsto\:2\bigg(\dfrac{3x - 5}{x + 2} \bigg) - 5\bigg(\dfrac{x + 2}{3x - 5} \bigg) = 3$

is

$\: \: \: \: \: \: \: \: \: \: \purple{ \underbrace{\purple{ \boxed{ \bf{ \: \: x = \dfrac{3}{4} \: \: }}} \: \: \: \: and \: \: \: \: \purple{ \boxed{ \bf{ \: \: x = 20 \: \: }}}}}$