Question

Solve for x :
X-1/2x-1+2x+1/c-1=2,wherex=-1/2,1

Answer 1

I think, your question needs a correction.

Correct Question :$\dfrac{x-1}{2x+1}+\dfrac{2x+1}{x-1}=2$


Now,
Let the value of 2x + 1 be b and the value of x - 1 be a.


Then,
Equation is $\dfrac{a}{b}+\dfrac{b}{a}=2$


Now, solving the equation for x :

$\implies \dfrac{a {}^{2} + b {}^{2} }{ab} = 2 \\ \\ \\ \implies {a}^{2} + {b}^{2} = 2ab \\ \\ \implies {a}^{2} + {b}^{2} - 2ab = 0$


From factorization, we know : -
• a² + b² - 2ab = ( a - b )²


Therefore,
$\implies(a - b) {}^{2} = 0 \\ \implies a - b = 0 \\ \implies a = b$


Now, substituting the values which where assumed .

= > 2x + 1 = x - 1
= > 2x - x = - 1 - 1
= > x = - 2


Thus,
The numeric value of the used variable 'x' is 2.

Answer 2

x-1/2x+1+2x+1/x-1=2

and
x=-1/2,1

(x-1/2x+1)+(2x+1/x-1)=2
L.C.M. is (2x+1)(x-1)

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→(x-1)(x-1)+(2x+1)(2x+1) / (2x+1)(x-1)=2
→(x-1)²+(2x+1)²/ (2x+1)(x-1)=2
→(x-1)²+(2x+1)² =2(2x+1)(x-1)

→(x²-2x+1)+(4x²+4x+1)=2(2x²-2x+x-1)
→(x²-2x+1)+(4x²+4x+1)=2(2x²-x-1)
→x²-2x+1+4x²+4x+1=4x²-2x-2

→x²+4x²-4x²-2x+4x+2x+1+1+2=0

→x²+4x+4=0
→(x+2)²=0
→Hence value of x=-2