Question

Solve 3/x-2 - 2/x-3 = 4/x-3 - 3/x-1

Answer 1

Answer:

$\frac{3}{x - 2} - \frac{2}{x - 3} = \frac{4}{x - 3} - \frac{3}{x - 1} \\ \frac{3(x - 3) - 2(x - 2)}{(x - 2)(x - 3)} = \frac{4(x - 1) - 3(x - 3)}{(x - 3)(x - 1)} \\ \frac{3x - 9 - 2x + 4}{(x - 2)(x - 3)} = \frac{4x - 4 - 3x + 9}{(x - 3)(x - 1)} \\ ( x- 3)(x - 1)(x - 5) = (x + 5)(x - 2)(x - 3) \\( x - 1)(x - 5) = (x + 5)(x - 2) \\ {x}^{2} - 5x - x + 5 = {x}^{2} - 2x + 5x - 10 \\ {x}^{2} - 6x + 5 = {x}^{2} + 3x - 10 \\ {x}^{2} - {x}^{2} - 6x - 3x = - 10 - 5 \\ - 9x = - 15 \\ x = \frac{15}{9} = \frac{5}{3}$

Answer 2

Given,

$\dfrac{3}{x-2}-\dfrac{2}{x-3}=\dfrac{4}{x-3}-\dfrac{3}{x-1}$

To find,

The value of x.

Solution,

We have,

$\dfrac{3}{x - 2} - \dfrac{2}{x - 3} = \dfrac{4}{x - 3} - \dfrac{3}{x - 1}$

Taking LCM on both the sides, we get :

$\dfrac{3(x - 3) - 2(x - 2)}{(x - 2)(x - 3)} = \dfrac{4(x - 1) - 3(x - 3)}{(x - 3)(x - 1)} \\\\ \dfrac{3x - 9 - 2x + 4}{(x - 2)(x - 3)} = \dfrac{4x - 4 - 3x + 9}{(x - 3)(x - 1)}$

Now, solving both sides such that,

$( x- 3)(x - 1)(x - 5) = (x + 5)(x - 2)(x - 3) \\\\( x - 1)(x - 5) = (x + 5)(x - 2) \\\\ {x}^{2} - 5x - x + 5 = {x}^{2} - 2x + 5x - 10 \\\\ {x}^{2} - 6x + 5 = {x}^{2} + 3x - 10 \\\\ {x}^{2} - {x}^{2} - 6x - 3x = - 10 - 5 \\\\ - 9x = -15\\\\x=\dfrac{5}{3}$

So, the value of x is 5/3.