To solve the inequalities \( n - 2 + 5n \leq 10 \) and \( n - 9 \geq 6 - 2n \), we will solve each inequality separately.
First inequality:
\[ n - 2 + 5n \leq 10 \]
Combine like terms:
\[ 6n - 2 \leq 10 \]
Add 2 to both sides:
\[ 6n \leq 12 \]
Now, divide by 6:
\[ n \leq 2 \]
Second inequality:
\[ n - 9 \geq 6 - 2n \]
Add \( 2n \) to both sides:
\[ n + 2n - 9 \geq 6 \]
Combine like terms:
\[ 3n - 9 \geq 6 \]
Add 9 to both sides:
\[ 3n \geq 15 \]
Now, divide by 3:
\[ n \geq 5 \]
Conclusion:
The solution set for the first inequality is \( n \leq 2 \) and for the second inequality is \( n \geq 5 \). Since these two inequalities are connected by "or", the final solution in terms of a compound inequality is:
\[ n \leq 2 \quad \text{or} \quad n \geq 5 \]
Thus, the final answer in terms of integers is:
Answer: \( n \in (-\infty, 2] \cup [5, \infty) \)