To solve the inequalities \( \frac{a - 4}{4} < 0 \) and \( a - 6 \geq 7 \), we'll solve each part separately and then combine the results.
- Solving the first inequality:
\[ \frac{a - 4}{4} < 0 \]
To eliminate the fraction, multiply both sides by 4 (note that 4 is positive, so the inequality direction stays the same).
\[ a - 4 < 0 \]
Now, add 4 to both sides:
\[ a < 4 \]
- Solving the second inequality:
\[ a - 6 \geq 7 \]
Add 6 to both sides:
\[ a \geq 13 \]
- Combining the results:
The solution to the inequalities is:
\[ a < 4 \quad \text{or} \quad a \geq 13 \]
In interval notation, this can be expressed as:
\[ (-\infty, 4) \cup [13, \infty) \]
Summarizing, the answer in terms of compound inequality with integers is:
\[ a < 4 \quad \text{or} \quad a \geq 13 \]