The problem asks to find the inverse element for the operation $*$ defined as $x * y = \frac{2x + 2y + 1}{2}$. We first need to find the identity element $e$ such that $x * e = x$. Then, we need to find the inverse element $x'$ such that $x * x' = e$.

1. Problem Description

The problem asks to find the inverse element for the operation $*$ defined as $x * y = \frac{2x + 2y + 1}{2}$. We first need to find the identity element $e$ such that $x * e = x$. Then, we need to find the inverse element $x'$ such that $x * x' = e$.

2. Solution Steps

First, we find the identity element $e$. We require $x * e = x$ for all $x$. Thus,

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

$2x + 2e + 1 = 2x$

$2e = -1$

$e = -\frac{1}{2}$

Now we find the inverse element $x'$ such that $x * x' = e = -\frac{1}{2}$.

$x * x' = \frac{2x + 2x' + 1}{2} = -\frac{1}{2}$

$2x + 2x' + 1 = -1$

$2x' = -2x - 2$

$x' = -x - 1$

3. Final Answer

$x' = -x - 1$