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

(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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