Question

(3x-3)/(x+1)+(x+1)/(x-1)+3/1 =(2x-1)/(2x+1)
prove that LHS is equal to RHS​

Answer 1

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$\: \: \circ \underline {\boxed{ \bf{given \:}}}$

LHS:

$\: \to\bf{ \frac{3x - 3}{x + 1} + \frac{x + 1}{x - 1} + \frac{3}{1} }$

RHS:

$\: \to \boxed{ \bf{ \frac{2x - 1}{2x + 1} }}$

$\divideontimes \displaystyle \sf{to \: verify \to \: lhs = rhs}$

$\: \bullet \underline{ \boxed{ \bf{solution}}}$

LHS:

$\: \implies\displaystyle \tt{ \frac{3x - 3}{x + 1} + \frac{x + 1}{x - 1} + \frac{3}{1} }$

$\: \implies \displaystyle \tt{by \: cross \: multiplication}$

$\: \implies \displaystyle \tt{ \frac{3x(x - 1) - 3(x - 1) + x(x + 1) + 1(x + 1)}{x(x - 1) + 1(x - 1) }} + \frac{3}{1}$

$\: \implies \\ \displaystyle \tt{ \frac{3 {x}^{2} - 3x - 3x + 3 + {x}^{2} + x + x + 1 }{ {x}^{2} - x + x - 1} + \frac{3}{1} }$

$\implies \displaystyle \tt{ \frac{ {3x}^{2} - 6x + 3 + {x}^{2} + 2x + 1 }{ {x}^{2} - 1 } + \frac{3}{1} }$

$\implies \displaystyle \tt{ \frac{ {4x}^{2} - 4x + 4}{ {x}^{2} - 1 } + \frac{3}{1} }$

$\: \implies \displaystyle \tt{ \frac{4( {x}^{2} - x + 1) }{ {x}^{2} - 1} + \frac{3}{1} }$

$\: \implies \displaystyle \tt{ \frac{4 {x}^{2} - 4x + 4}{3 {x}^{2} - 3} }$

$\implies \displaystyle \tt{ \frac{4( {x}^{2} - x + 1) }{3( {x}^{2} - 1)} }$

$\: \implies \displaystyle \tt{ \frac{4( {x}^{2} - x + 1) }{3( {x}^{2} - 1) } }$

$\: \implies \displaystyle \tt{ \frac{4(x(x - 1) + 1)}{3(x - 1)(x + 1)} }$

$\: \: \implies \displaystyle \tt{ \frac{4(x + 1)(x - 1)}{3(x - 1)(x + 1)} }$

$\: \: \implies \displaystyle \tt{\cancel \frac{(x + 1)(x - 1)}{(x + 1)(x - 1)} \frac{4}{3} }$

$\: \implies \displaystyle \tt{lhs = \frac{4}{3} }$

$\: \: \implies \displaystyle \tt{rhs = \frac{2x - 1}{2x + 1} }$

$\: \dashrightarrow \displaystyle \bf{lhs = rhs}$

$\: \: \implies \displaystyle \tt{ \frac{4}{3} = \frac{2x - 1}{2x + 1} }$

$\: \implies \displaystyle \tt{4(2x + 1) = 3(2x - 1)}$

$\: \implies \displaystyle \tt{8x + 4 = 6x - 3}$

$\implies \displaystyle \tt{8x + 4 - 6x + 3 = 0}$

$\: \implies \displaystyle \tt{2x + 7 = 0}$

$\implies \displaystyle \tt{x = \frac{ - 7}{2} }$