The problem asks to simplify the expression $\frac{x+1}{y} \div \frac{2(x+1)}{x}$.

1. Problem Description

The problem asks to simplify the expression $\frac{x+1}{y} \div \frac{2(x+1)}{x}$.

2. Solution Steps

To simplify the given expression, we need to remember that dividing by a fraction is the same as multiplying by its reciprocal.

So we can rewrite the expression as:

$\frac{x+1}{y} \div \frac{2(x+1)}{x} = \frac{x+1}{y} \times \frac{x}{2(x+1)}$

Now we can multiply the fractions:

$\frac{x+1}{y} \times \frac{x}{2(x+1)} = \frac{(x+1)x}{2y(x+1)}$

We can cancel the common factor $(x+1)$ from the numerator and denominator:

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

3. Final Answer

The simplified expression is $\frac{x}{2y}$.