Question

$\frac{5}{x - 2} - \frac{3}{x + 6} = \frac{4}{x}$
​

Answer 1

${ \rm{ \huge{SOLUTION }}}$



${ \green{ \sf{ \frac{5}{x - 2} - \frac{3}{x + 6} = \frac{4}{x}}}}$



${ \implies{ \green{ \sf{ \frac{5(x + 6) - 3(x - 2)}{(x - 2)(x + 6)} = \frac{4}{x}}}}}$



${ \implies{ \green{ \sf{ \frac{(5x + 30) - (3x - 6)}{ {x}^{2} + 6x - 2x - 12 } = \frac{4}{x}}}}}$



${ \implies{ \green{ \sf{ \frac{5x + 30 - 3x + 6}{ {x}^{2} + 4x - 12} = \frac{4}{x}}}}}$



${ \implies{ \green{ \sf{ \frac{2x + 36}{ {x}^{2} + 4x - 12} = \frac{4}{x}}}}}$



${ \implies{ \green{ \sf{ \frac{2( x + 18)}{ {x}^{2} + 4x - 12} = \frac{4}{x}}}}}$



${ \implies{ \sf{ \green{ \frac{x + 18}{ {x}^{2} + 4x - 12} = \frac{2}{x}}}}}$



${ \implies{ \green{ \sf{x(x + 18) = 2( {x}^{2} + 4x - 12)}}}}$



${ \implies{ \green{ \sf{ {x}^{2} + 18x = 2 {x}^{2} + 8x - 24}}}}$



${ \implies{ \sf{ \green{2 {x}^{2} - {x}^{2} + 8x - 18x - 24 = 0}}}}$



${ \implies{ \green{ \sf{ {x}^{2} - 10x - 24 = 0}}}}$



${ \implies{ \green{ \sf{ {x}^{2} - (6x + 4x) - 24 = 0}}}}$



${ \implies{ \sf{ \green{ {x}^{2} - 6x - 4x - 24 = 0}}}}$



${ \implies{ \sf{ \green{x(x - 6) - 4(x - 6) = 0}}}}$



${ \implies{ \sf{ \green{(x - 4)(x - 6) = 0}}}}$



${ \implies{ \red{ \sf{ \boxed{ \boxed{x = 4 \: and \: 6}}}}}}$

Answer 2

$\sf{ \bold { Given \: - }} \\ \\ \sf{ \bold { \implies { \dfrac{5}{x - 2} - \dfrac{3}{x + 6} = \dfrac{4}{x} }}} \\ \\ \sf{ \bold { Solution \: - }} \\ \\ \sf{ \bold { \implies { \dfrac{5}{x - 2} - \dfrac{3}{x + 6} = \dfrac{4}{x} }}} \\ \\ \sf{ \bold { \implies { \dfrac{ 5(x + 6 ) - 3 ( x - 2 ) }{ ( x - 2 )( x + 6 ) } = \dfrac{ 4 }{ x } }}} \\ \\ \sf{ \bold { \implies { \dfrac{ 2 ( x + 18 ) }{ ( x - 2 )( x + 6 ) } = \dfrac{ 4 }{ x } }}} \\ \\ \sf{ \bold { By \: cancelling \: the \: multiple \: of \: 2 \: - }} \\ \\ \sf{ \bold { \implies { \dfrac{ ( x + 18 ) }{ ( x - 2 )( x + 6 ) } = \dfrac{ 2 }{ x } }}} \\ \\ \sf{ \bold { \implies { 2(x - 2)( x + 6 ) = x ( x + 18 ) }}} \\ \\ \sf{ \implies { \bold { 2( x^2 - 4x - 12) = x^2 + 18x }}} \\ \\ \sf{ \implies { \bold { 2x^2 - 8x - 24 = x^2 + 18x }}} \\ \\ \sf{ \bold { \implies { x^2 - 10x + 24 = 0 }}} \\ \\ \sf{ \bold { \implies { x^2 - 4x - 6x + 24 = 0 }}} \\ \\ \sf{ \bold { \implies { x ( x - 6 ) - 4 ( x - 6 ) = 0 }}} \\ \\ \sf{ \bold { \implies { ( x - 4 )( x - 6 ) = 0 }}} \\ \\ \sf{ \bold { \implies { x = 4 \: or \: 6 }}} \\ \\ \sf{ \bold { This \: is \: the \: answer \: . }}$