Solve by Factoring x/(2x+2)=(-2x)/(4x+4)+(2x-3)/(x+1)

Problem

x/(2*x+2)=(−2*x)/(4*x+4)+(2*x−3)/(x+1)

Solution

1. Factor the denominators to find the least common denominator.

2*x+2=2*(x+1)

4*x+4=4*(x+1)

x+1=1*(x+1)

2. Identify the least common denominator (LCD) as 4*(x+1) and state the domain restriction x≠−1

LCD=4*(x+1)

3. Multiply every term in the equation by the LCD to clear the fractions.

4*(x+1)⋅x/(2*(x+1))=4*(x+1)⋅(−2*x)/(4*(x+1))+4*(x+1)⋅(2*x−3)/(x+1)

4. Simplify the resulting linear equation.

2*x=−2*x+4*(2*x−3)

2*x=−2*x+8*x−12

5. Combine like terms on the right side.

2*x=6*x−12

6. Isolate the variable x by moving terms and dividing.

−4*x=−12

x=3

7. Verify that the solution x=3 does not violate the domain restriction x≠−1

3≠−1

Final Answer

x/(2*x+2)=(−2*x)/(4*x+4)+(2*x−3)/(x+1)⇒x=3

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