SOLUTION: Solve (2x-5)/(x)+(4x-1)/(x+2)=-(3x+8)/(x^2+2x) Thank you so much!!!!! God Bless and Happy Holidays!!!!

Question 388707: Solve (2x-5)/(x)+(4x-1)/(x+2)=-(3x+8)/(x^2+2x)
Thank you so much!!!!!
God Bless and Happy Holidays!!!!
Answer by haileytucki(390) (Show Source): You can put this solution on YOUR website!
(2x-5)/(x)+(4x-1)/(x+2)=-(3x+8)/(x^(2)+2x)
Factor out the GCF of x from each term in the polynomial.
(2x-5)/(x)+(4x-1)/(x+2)=-(3x+8)/(x(x)+x(2))
Factor out the GCF of x from x^(2)+2x.
(2x-5)/(x)+(4x-1)/(x+2)=-(3x+8)/(x(x+2))
Find the LCD (least common denominator) of ((2x-5))/(x)+((4x-1))/((x+2))-((3x+8))/(x(x+2)).
Least common denominator: x(x+2)
Multiply each term in the equation by x(x+2) in order to remove all the denominators from the equation.
(2x-5)/(x)*x(x+2)+(4x-1)/(x+2)*x(x+2)=-(3x+8)/(x(x+2))*x(x+2)
Simplify the left-hand side of the equation by canceling the common factors.
6x^(2)-2x-10=-(3x+8)/(x(x+2))*x(x+2)
Simplify the right-hand side of the equation by simplifying each term.
6x^(2)-2x-10=-3x-8
Since -3x contains the variable to solve for, move it to the left-hand side of the equation by adding 3x to both sides.
6x^(2)-2x-10+3x=-8
Since -2x and 3x are like terms, subtract 3x from -2x to get x.
6x^(2)+x-10=-8
To set the left-hand side of the equation equal to 0, move all the expressions to the left-hand side.
6x^(2)+x-2=0
In this problem (2)/(3)*-(1)/(2)=-2 and (2)/(3)-(1)/(2)=1, so insert (2)/(3) as the right hand term of one factor and -(1)/(2) as the right-hand term of the other factor.
(x+(2)/(3))(x-(1)/(2))=0
Remove the fraction by multiplying the first term of the factor by the denominator of the second term.
(3x+2)(2x-1)=0
Set each of the factors of the left-hand side of the equation equal to 0.
3x+2=0_2x-1=0
Since 2 does not contain the variable to solve for, move it to the right-hand side of the equation by subtracting 2 from both sides.
3x=-2_2x-1=0
Divide each term in the equation by 3.
(3x)/(3)=-(2)/(3)_2x-1=0
Simplify the left-hand side of the equation by canceling the common factors.
x=-(2)/(3)_2x-1=0
Set each of the factors of the left-hand side of the equation equal to 0.
x=-(2)/(3)_2x-1=0
Since -1 does not contain the variable to solve for, move it to the right-hand side of the equation by adding 1 to both sides.
x=-(2)/(3)_2x=1
Divide each term in the equation by 2.
x=-(2)/(3)_(2x)/(2)=(1)/(2)
Simplify the left-hand side of the equation by canceling the common factors.
x=-(2)/(3)_x=(1)/(2)
The complete solution is the set of the individual solutions.
x=-(2)/(3),(1)/(2)
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