- The equation \(\frac{x}{x+2} + \frac{1}{x} = 1\) is restricted by \(x
eq -2\) and \(x
eq 0\).
- Multiplying by the common denominator \(x(x+2)\) leads to the linear equation \(x+2=2x\).
- Solving the linear equation yields the single potential solution \(x=2\), which is within the allowed domain.
- The equation has one valid solution and no extraneous solutions.
- \(\boxed{\text{The equation has one valid solution and no extraneous solutions.}}\)
Explanation
- Problem Analysis and Domain Restrictions
The given equation is \(\frac{x}{x+2} + \frac{1}{x} = 1\). To identify potential solutions, we must first note the domain restrictions. The denominators cannot be zero, which means \(x+2
eq 0\) and \(x
eq 0\). Thus, the domain is all real numbers \(x\) except \(x = -2\) and \(x = 0\). If any solution we find falls into this restricted set, it is considered extraneous.
- Solving the Rational Equation
To solve the equation, we multiply all terms by the common denominator, \(x(x+2)\), to clear the fractions:
\[x(x+2) \left( \frac{x}{x+2} \right) + x(x+2) \left( \frac{1}{x} \right) = x(x+2)(1).\]
This simplifies to:
\[x^2 + (x+2) = x(x+2).\]
Expanding the right side gives:
\[x^2 + x + 2 = x^2 + 2x.\]
Subtracting \(x^2\) from both sides yields the linear equation:
\[x + 2 = 2x.\]
Solving for \(x\) by subtracting \(x\) from both sides, we find:
\[x = 2.\]
- Verification and Conclusion
We must now check if the candidate solution \(x=2\) is valid. As determined in step 1, our domain restrictions are \(x
eq -2\) and \(x
eq 0\). Since \(x=2\) does not violate these conditions, it is a valid solution. Substituting it back into the original equation:
\[\frac{2}{2+2} + \frac{1}{2} = \frac{2}{4} + \frac{1}{2} = \frac{1}{2} + \frac{1}{2} = 1.\]
Since the equation holds true, \(x=2\) is indeed a valid solution. There were no other candidate solutions generated, and thus there are no extraneous solutions.
Examples
Rational equations like this appear in rate problems, such as determining how long it takes for two workers to complete a task together. For instance, if one person paints a wall in \(x+2\) hours and another in \(x\) hours, their combined work rate is represented by the sum of their individual rates, \(\frac{1}{x+2} + \frac{1}{x} = \frac{1}{T}\), where \(T\) is the total time. Understanding how to clear denominators and verify solutions ensures that you correctly identify the time taken for such real-world tasks, while avoiding mathematically invalid results.