To solve the compound inequality \(-t + 14 \leq -2 + t\) or \(t + 12 + 2t \leq 12\), we will solve each inequality separately.
1. Solve \(-t + 14 \leq -2 + t\):
First, rearrange the inequality:
\[ -t - t \leq -2 - 14 \]
Combine like terms:
\[ -2t \leq -16 \]
Now, divide by -2 and remember to flip the inequality sign:
\[ t \geq 8 \]
2. Solve \(t + 12 + 2t \leq 12\):
Combine like terms:
\[ 3t + 12 \leq 12 \]
Subtract 12 from both sides:
\[ 3t \leq 0 \]
Now divide by 3:
\[ t \leq 0 \]
3. Combine the results:
The results from the two parts give us:
- \(t \geq 8\)
- \(t \leq 0\)
Since we have an "or" condition, the solution is:
\[ t \leq 0 \quad \text{or} \quad t \geq 8 \]
In interval notation, this can be expressed as:
\[ (-\infty, 0] \cup [8, \infty) \]
In terms of integer solutions, we can state that:
\[ t \in { \ldots, -3, -2, -1, 0} \cup {8, 9, 10, \ldots} \]
Thus, the final solution in terms of integers is:
\[ t \leq 0 \quad \text{or} \quad t \geq 8 \]