The given equation is:

$\frac{1}{2}(4x + 2) = 2(x + \frac{1}{2})$

Let's solve this step by step:

Step 1: Distribute the constants on both sides.

• Left-hand side: Distribute $\frac{1}{2}$ to both terms inside the parentheses: $\frac{1}{2}(4x + 2) = \frac{1}{2} \cdot 4x + \frac{1}{2} \cdot 2 = 2x + 1$

• Right-hand side: Distribute $2$ to both terms inside the parentheses: $2(x + \frac{1}{2}) = 2 \cdot x + 2 \cdot \frac{1}{2} = 2x + 1$

Step 2: Simplified equation.

After distributing, we get: $2x + 1 = 2x + 1$

Step 3: Subtract $2x + 1$ from both sides:

$0 = 0$

This results in a true statement, which means the equation has infinite solutions.

Conclusion:

The correct answer is B. $1 = 1$; infinite real solutions.

Would you like further details on any step? Here are some related questions you might consider:

1. How do you determine if an equation has infinite solutions?
2. What are the steps to isolate variables when solving equations?
3. How do distribution rules work in solving equations?
4. Can a linear equation have no solution? If so, how can it be identified?
5. What other types of equations commonly have infinite solutions?

Tip: When solving an equation, if you end up with a true statement like $0 = 0$, it indicates infinite solutions.