Let's solve the inequality step by step:

\(\frac{1}{3}(x + 1) - 1 > \frac{1}{5}(x + 4)\)

First, we'll clear the fractions by multiplying both sides by the least common multiple of 3 and 5, which is 15:

15 * \(\frac{1}{3}(x + 1) - 15 * 1 > 15 * \(\frac{1}{5}(x + 4)\)

5(x + 1) - 15 > 3(x + 4)

Now, let's simplify further:

5x + 5 - 15 > 3x + 12

Next, subtract 3x from both sides:

2x + 5 - 15 > 12

Now, subtract 5 from both sides:

2x - 10 > 12

Finally, add 10 to both sides to isolate 2x:

2x > 12 + 10

2x > 22

Now, divide by 2 to solve for x:

x > 11

So, the solution to the inequality is:

x > 11