To solve the equation \( \tan(x + \frac{\pi}{4}) = 1 \), we can start by recognizing that the tangent function equals 1 at certain angles. Specifically, we know that:

\[
\tan\left(\frac{\pi}{4}\right) = 1
\]

The general solutions for the equation \( \tan(\theta) = 1 \) are given by:

\[
\theta = \frac{\pi}{4} + n\pi
\]

where \( n \) is any integer.

Now, we can set \( \theta \) equal to \( x + \frac{\pi}{4} \):

\[
x + \frac{\pi}{4} = \frac{\pi}{4} + n\pi
\]

Next, we will subtract \( \frac{\pi}{4} \) from both sides:

\[
x = n\pi
\]

Thus, the general solution for the equation \( \tan(x + \frac{\pi}{4}) = 1 \) is:

\[
\boxed{x = n\pi}
\]

where \( n \) is any integer.