Solve for x 1/x+1/(x+1)=2/(3(x+1))

Problem

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

Solution

1. Identify the domain by setting the denominators to zero to find values that x cannot take.

x≠0

x+1≠0⇒x≠−1

2. Find the least common denominator (LCD) for all terms in the equation.

LCD=3*x*(x+1)

3. Multiply every term by the LCD to eliminate the fractions.

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

4. Simplify each term by canceling common factors.

3*(x+1)+3*x=2*x

5. Distribute the constant into the parentheses.

3*x+3+3*x=2*x

6. Combine like terms on the left side of the equation.

6*x+3=2*x

7. Isolate the variable by subtracting 6*x from both sides.

3=−4*x

8. Solve for x by dividing both sides by −4

x=−3/4

9. Verify the solution against the domain restrictions x≠0 and x≠−1 Since −3/4 is neither of those, it is a valid solution.

Final Answer

1/x+1/(x+1)=2/(3*(x+1))⇒x=−3/4

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