To solve the compound inequality \( 10 \geq m + 11 \geq -5 \), we can break it down into two separate inequalities:
- \( 10 \geq m + 11 \)
- \( m + 11 \geq -5 \)
Step 1: Solve the first inequality
Starting with the first inequality:
\[ 10 \geq m + 11 \]
We can isolate \( m \) by subtracting 11 from both sides:
\[ 10 - 11 \geq m \]
\[ -1 \geq m \]
This can be rewritten as:
\[ m \leq -1 \]
Step 2: Solve the second inequality
Now, let's solve the second inequality:
\[ m + 11 \geq -5 \]
Again, we will isolate \( m \) by subtracting 11 from both sides:
\[ m \geq -5 - 11 \]
\[ m \geq -16 \]
Step 3: Combine the inequalities
Now we combine the results from both parts. We have:
\[ -16 \leq m \leq -1 \]
Final Answer
The solution as a compound inequality is:
\[ -16 \leq m \leq -1 \]
This represents all integer values of \( m \) that satisfy the inequality. Therefore, the integer solutions are:
\[ m \in {-16, -15, -14, -13, -12, -11, -10, -9, -8, -7, -6, -5, -4, -3, -2, -1} \]