To solve the inequality \( \frac{1}{4}x - \frac{3}{4} > 5 \frac{1}{4} \), we first convert \( 5 \frac{1}{4} \) to 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:
\[ \frac{1}{4}x > \frac{21}{4} + \frac{3}{4} \]
\[ \frac{1}{4}x > \frac{24}{4} \]
\[ \frac{1}{4}x > 6 \]
Now, we multiply both sides by 4 to solve for \(x\):
\[ x > 24 \]
Thus, the solution is:
\[ \boxed{x > 24} \]
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