To solve the linear equation
\[ \frac{4(x + 5)}{2} - 5 = \frac{4x + 1}{2}, \]
we start by simplifying both sides.
First, simplify the left-hand side:
\[ \frac{4(x + 5)}{2} = \frac{4x + 20}{2} = 2x + 10. \]
So, the left-hand side becomes:
\[ 2x + 10 - 5 = 2x + 5. \]
For the right-hand side, we simplify \(\frac{4x + 1}{2}\) as it is:
\[ \frac{4x + 1}{2} = 2x + \frac{1}{2}. \]
Now, we rewrite the equation with these simplifications:
\[ 2x + 5 = 2x + \frac{1}{2}. \]
Next, we can eliminate \(2x\) from both sides:
\[ 5 = \frac{1}{2}. \]
This statement is clearly false, indicating that there are no \(x\) values that satisfy the original equation.
Therefore, the conclusion is that there are no solutions to the linear equation.