To solve the inequality \( \frac{1}{4} x - \frac{3}{4} > 5 \frac{1}{4} \), we first convert \( 5 \frac{1}{4} \) into an improper fraction:
\[ 5 \frac{1}{4} = 5 + \frac{1}{4} = \frac{20}{4} + \frac{1}{4} = \frac{21}{4} \]
Now, we rewrite the inequality:
\[ \frac{1}{4} x - \frac{3}{4} > \frac{21}{4} \]
Next, we add \( \frac{3}{4} \) to both sides of the inequality:
\[ \frac{1}{4} x > \frac{21}{4} + \frac{3}{4} \]
Combining the fractions on the right side:
\[ \frac{1}{4} x > \frac{21 + 3}{4} = \frac{24}{4} = 6 \]
Now we multiply both sides by 4 to isolate \( x \):
\[ x > 6 \cdot 4 \]
\[ x > 24 \]
Thus, the solution to the inequality is:
\[ \boxed{x > 24} \]