Question
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Tutor-Verified Answer
Answer
\( x = 1 \)
Solution
1. Start with the equation:
\[
\frac{1}{2x} = \frac{1}{x+1}
\]
2. Note that the denominators cannot be zero. Thus, we require \( x \neq 0 \) and \( x \neq -1 \).
3. Cross-multiply to eliminate the fractions:
\[
1\cdot (x+1) = 1\cdot (2x)
\]
This simplifies to:
\[
x+1 = 2x
\]
4. Subtract \( x \) from both sides to solve for \( x \):
\[
1 = 2x - x
\]
\[
1 = x
\]
5. Finally, check that \( x = 1 \) does not violate the domain restrictions. Since \( 1 \neq 0 \) and \( 1 \neq -1 \), the solution is valid.
\[
\boxed{x = 1}
\]
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Simplify this solution Extra Insights
To solve the equation \( \frac{1}{2 x}=\frac{1}{x+1} \), you can cross-multiply to eliminate the fractions. This gives you \( (x + 1) = 2x \). Rearranging the equation results in \( x + 1 - 2x = 0 \), or simplified, \(-x + 1 = 0\). Therefore, \( x = 1 \) is the solution! However, let's check for any restrictions. Since the original equation has terms involving \( x \) in the denominator, \( x \) cannot be equal to 0 (to avoid division by zero). The solution \( x = 1 \) is valid since it doesn't fall under that restriction!
