To solve the compound inequality \( a - \frac{4}{4} < 0 \) or \( a - 6 \geq 7 \), we can break it down into two separate inequalities.
Solve the first inequality:
\[ a - \frac{4}{4} < 0 \] \[ a - 1 < 0 \] \[ a < 1 \]
Solve the second inequality:
\[ a - 6 \geq 7 \] \[ a \geq 7 + 6 \] \[ a \geq 13 \]
Combining the results:
The compound inequality from both conditions is: \[ a < 1 \quad \text{or} \quad a \geq 13 \]
In interval notation, this can be represented as:
\[ (-\infty, 1) \cup [13, \infty) \]
In terms of integers, the answer indicates that \( a \) can be any integer less than 1 or any integer greater than or equal to 13. Thus: \[ a \in { \ldots, -3, -2, -1, 0 } \cup { 13, 14, 15, \ldots } \]
In conclusion, the solutions for \( a \) can be expressed as: \[ a < 1 \quad \text{or} \quad a \geq 13 \]