Question

solve x-1/2x+1+2x+1/x-1=2 where x is not equal to -1/2,1

Answer 1

$\frac{x - 1}{2x + 1} + \frac{2x + 1}{x - 1} = 2$

$let \: \: \frac{x - 1}{2x + 1} = y$

$then \: \: y + \frac{1}{y} = 2 \\ \\ {y}^{2} + 1 = 2y$

${y}^{2} - 2y + 1 = 0 \\ \\ (y - 1)^{2} = 0$

$y - 1 = 0 \\ \\ y = 1 \\ \\ \frac{x - 1}{2x + 1} = 1$

$x - 1 = 2x + 1 \\ \\ 2x - x = - 1 - 1 \\ \\ x = - 2$

Answer 2

Answer:

The solution is x = -2.

Step-by-step explanation:

Here, the given expression is,

$\frac{x-1}{2x+1}+\frac{2x+1}{x-1}=2$

$\frac{(x-1)^2+(2x+1)^2}{(x-1)(2x+1)}=2$           ( Adding fractions )

$\frac{x^2-2x+1+4x^2+4x+1}{2x^2+x-2x-1}=2$  

$\frac{5x^2+2x+2}{2x^2-x-1}=2$

$5x^2+2x+2=4x^2-2x-2$    ( By cross multiplication )

$5x^2+2x+2-4x^2+2x+2=0$

$x^2+4x+4=0$            ( Combining like terms )

$x^2+2x+2x+4=0$

$x(x+2)+2(x+2)=0$

$(x+2)(x+2)=0$

$\implies x = -2$

Which is the required solution.