Final answer:
To solve the equation x + 1/(x - 2) = 4/7, we combine fractions, use algebra to maintain the equation’s equality as we perform operations, and then isolate and solve for the variable x.
Step-by-step explanation:
The solution to the equation x + 1/(x - 2) = 4/7 involves manipulating the equation to isolate the variable x. By using rules of algebra, we can keep the equation as an equality while we perform operations on both sides. Let's solve this step by step.
Firstly, we find a common denominator to combine the fractions on the left side of the equation:
- Multiply both sides of the equation by the denominator (x - 2) to eliminate the fraction.
- This will result in (x - 2)(x + 1)/(x - 2) = (4/7)(x - 2).
- Simplify the left side to x + 1 since the x - 2 terms cancel each other out.
- Distribute the (4/7) across the right side to get x + 1 = 4x/7 - 8/7.
- Now, we can solve for x by moving all terms containing x to one side and the constant terms to the other side.
- Multiply all terms by 7 to eliminate the fraction and simplify further to find the value of x.
By understanding the properties of fractions, equality, and manipulation of equations, we can solve this problem in a systematic way.