To solve the inequality \( \frac{1}{4}x - \frac{3}{4} > 5 \frac{1}{4} \), we first need to 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, our inequality becomes:
\[ \frac{1}{4}x - \frac{3}{4} > \frac{21}{4} \]
Next, we will add \( \frac{3}{4} \) to both sides to isolate the term with \( x \):
\[ \frac{1}{4}x > \frac{21}{4} + \frac{3}{4} \]
Combine the fractions on the right-hand side:
\[ \frac{21}{4} + \frac{3}{4} = \frac{21 + 3}{4} = \frac{24}{4} = 6 \]
Now, we have:
\[ \frac{1}{4}x > 6 \]
Next, we will multiply both sides of the inequality by \( 4 \) to solve for \( x \):
\[ x > 6 \times 4 \] \[ x > 24 \]
Thus, the solution to the inequality is:
\[ \boxed{x > 24} \]
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