1) $$ \frac{1}{2 x}=\frac{1}{x+1} $$

Question

UpStudy AI · Accepted Answer

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.

$$
\boxed{x = 1}
$$
$$ x = 1 $$

1) \( \frac{1}{2 x}=\frac{1}{x+1} \)