To solve the equation \(\sqrt{x+1} - \sqrt{x-4} = 1\), we start by isolating one of the square root terms. We can rearrange the equation as follows:
\[ \sqrt{x+1} = \sqrt{x-4} + 1 \]
Next, we square both sides to eliminate the square roots:
\[ (\sqrt{x+1})^2 = (\sqrt{x-4} + 1)^2 \]
This gives us:
\[ x + 1 = (x - 4) + 2\sqrt{x-4} + 1 \]
Simplifying the right-hand side:
\[ x + 1 = x - 4 + 2\sqrt{x-4} + 1 \]
\[ x + 1 = x - 3 + 2\sqrt{x-4} \]
Now we can subtract \(x\) from both sides:
\[ 1 = -3 + 2\sqrt{x-4} \]
Adding 3 to both sides gives us:
\[ 4 = 2\sqrt{x-4} \]
Now, divide both sides by 2:
\[ 2 = \sqrt{x-4} \]
Next, we square both sides again to remove the square root:
\[ 2^2 = (x-4) \]
This simplifies to:
\[ 4 = x - 4 \]
Adding 4 to both sides:
\[ x = 8 \]
Now, we need to check if this solution satisfies the original equation. We substitute \(x = 8\) back into the original equation:
\[ \sqrt{8 + 1} - \sqrt{8 - 4} = \sqrt{9} - \sqrt{4} = 3 - 2 = 1 \]
Since the left-hand side equals the right-hand side, our solution is valid.
Thus, the solution to the equation \(\sqrt{x+1} - \sqrt{x-4} = 1\) is
\[ \boxed{8} \]