Answer:To solve the equation x + 1/x + 1/x = x - 6/x + 2 for x, we can follow these steps:
1. Simplify the left side of the equation:
- The common denominator for the fractions is x, so we can rewrite the equation as (x^2 + 2x + 1)/x = x - 6/x + 2.
2. Simplify the right side of the equation:
- Combine the x terms by finding a common denominator, which is x.
- We have (x^2 - 6 + 2x)/x = x - 6/x + 2.
3. Set the left side equal to the right side:
- (x^2 + 2x + 1)/x = (x^2 - 6 + 2x)/x.
4. Cross multiply to eliminate the denominators:
- Multiply both sides of the equation by x.
- We get x^2 + 2x + 1 = x^2 - 6 + 2x.
5. Simplify the equation:
- Cancel out like terms by subtracting x^2 from both sides.
- We have 2x + 1 = -6.
6. Solve for x:
- Subtract 1 from both sides of the equation.
- We get 2x = -7.
- Divide both sides by 2 to isolate x.
- The solution is x = -7/2.
In summary, the solution to the equation x + 1/x + 1/x = x - 6/x + 2 for x is x = -7/2.