Let's solve the given inequality step by step:

\(\frac{x+2}{4} - \frac{2x-3}{3} < 4\)

To solve this inequality, let's first find a common denominator for the fractions, which is the least common multiple (LCM) of 4 and 3, which is 12. We need to adjust the fractions accordingly:

\(\frac{3(x+2)}{12} - \frac{4(2x-3)}{12} < 4\)

Now, combine the fractions on the left side:

\(\frac{3x + 6 - 8x + 12}{12} < 4\)

Combine the terms in the numerator:

\(\frac{-5x + 18}{12} < 4\)

Now, multiply both sides of the inequality by 12 to eliminate the fraction:

\(-5x + 18 < 48\)

Next, subtract 18 from both sides:

\(-5x < 48 - 18\)

Simplify:

\(-5x < 30\)

Finally, divide both sides by -5 (remembering to reverse the inequality when dividing by a negative number):

\(x > -6\)