Question

Solve this linear equation and find x: 2÷x+1=4-x÷x-1

Answer 1

Solution:

2/(x+1) = (4-x)/(x-1)

Do the cross multiplication,we

get

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

=> 2x-2 = 4x+4-x²-x

=>2x-2-4x-4+x²+x = 0

=> x²-x-4=0

=> x²-x = 4

=> x²-2*x*(1/2)=4

=> x²-2*x*(1/2)+(1/2)²=4+(1/2)²

= (x-1/2)² = 4+1/4

=> (x-1/2)² = 17/4

=> x -1/2 = ±√(17/4)

=> x = 1/2±√17/2

=> x = (1±√17)/2

••••

Answer 2

$\bf\huge\textbf{\underline{\underline{According\:to\:the\:Question}}}$

$\bf\huge{\implies\dfrac{2}{x+1} = \dfrac{4-x}{x-1}}$  

⇒ 2(x - 1) = (4 - x)(x + 1)

⇒ 2x - 2 = 4x + 4 - x² - x

⇒2x - 2- 4x - 4 + x² + x = 0

⇒x² - x - 4 = 0

⇒ x² - x = 4

$\bf\huge{\implies x^2 -2\times x\times\dfrac{1}{2} = 4}$          

$\bf\huge{\implies x^2 -2\times x\times\dfrac{1}{2} +(\dfrac{1}{2})^2 = 4+(\dfrac{1}{2})^2}$          

$\bf\huge{\implies(\dfrac{x-1}{2})^2 = 4 + \dfrac{1}{4}}$  

$\bf\huge{\implies(\dfrac{x-1}{2})^2 = \dfrac{17}{4}}$  

$\bf\huge{\implies\dfrac{x-1}{2} = \sqrt{\dfrac{17}{4}}}$ 

$\bf\huge{\implies x = \dfrac{1}{2} = \dfrac{\sqrt{17}}{2}}$  

Both the + and - is your answer :-

$\bf\huge{\implies x =\dfrac{1+\sqrt{17}}{2}}$  

$\bf\huge{\implies x =\dfrac{1-\sqrt{17}}{2}}$