`(6x)/(x+4)+4=(2x+2)/(x-1)` Solve the equation by using the LCD. Check for extraneous solutions.

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`(6x)/(x+4)+4=(2x+2)/(x-1)`

LCD is `(x+4)(x-1)`

Multiply each term of the equation by LCD and simplify,

`(x+4)(x-1)((6x)/(x+4))+4(x+4)(x-1)=(x+4)(x-1)((2x+2)/(x-1))`

`6x(x-1)+4(x+4)(x-1)=(x+4)(2x+2)`

`6x^2-6x+4(x(x-1)+4(x-1))=x(2x+2)+4(2x+2)`

`6x^2-6x+4(x^2-x+4x-4)=2x^2+2x+8x+8`

`6x^2-6x+4(x^2+3x-4)=2x^2+10x+8`

`6x^2-6x+4x^2+12x-16=2x^2+10x+8`

`6x^2+4x^2-6x+12x-16=2x^2+10x+8`

`10x^2+6x-16=2x^2+10x+8`

Isolate the terms containing x,

`10x^2-2x^2+6x-10x=8+16`

`8x^2-4x=24`

`8x^2-4x-24=0`

Factorize ,

`4(2x^2-x-6)=0`

`4(2x^2-4x+3x-6)=0`

`4(2x(x-2)+3(x-2))=0`

`4(x-2)(2x+3)=0`

Use the zero product property,

`x-2=0`  or `2x+3=0`

`x=2`   or `2x=-3`

`x=2`  or `x=-3/2` ``

Let's check...

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`(6x)/(x+4)+4=(2x+2)/(x-1)`

LCD is `(x+4)(x-1)`

Multiply each term of the equation by LCD and simplify,

`(x+4)(x-1)((6x)/(x+4))+4(x+4)(x-1)=(x+4)(x-1)((2x+2)/(x-1))`

`6x(x-1)+4(x+4)(x-1)=(x+4)(2x+2)`

`6x^2-6x+4(x(x-1)+4(x-1))=x(2x+2)+4(2x+2)`

`6x^2-6x+4(x^2-x+4x-4)=2x^2+2x+8x+8`

`6x^2-6x+4(x^2+3x-4)=2x^2+10x+8`

`6x^2-6x+4x^2+12x-16=2x^2+10x+8`

`6x^2+4x^2-6x+12x-16=2x^2+10x+8`

`10x^2+6x-16=2x^2+10x+8`

Isolate the terms containing x,

`10x^2-2x^2+6x-10x=8+16`

`8x^2-4x=24`

`8x^2-4x-24=0`

Factorize ,

`4(2x^2-x-6)=0`

`4(2x^2-4x+3x-6)=0`

`4(2x(x-2)+3(x-2))=0`

`4(x-2)(2x+3)=0`

Use the zero product property,

`x-2=0`  or `2x+3=0`

`x=2`   or `2x=-3`

`x=2`  or `x=-3/2` ``

Let's check the solutions by plugging them in the original equation,

For x=2,

`(6*2)/(2+4)+4=(2*2+2)/(2-1)`

`(12)/6+4=6/1`

`2+4=6`

`6=6`

It's true.

For x=`-3/2` ,

`(6(-3/2))/(-3/2+4)+4=(2(-3/2)+2)/(-3/2-1)`

`-9/(5/2)+4=(-1)/(-5/2)`

`-18/5+4=2/5`

`(-18+20)/5=2/5`

`2/5=2/5`

It's true,

So, Solutions of the equation are 2 and `-3/2`

 

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