To solve the compound inequality \( k - 3 \leq -2k - 3 \) or \( k + 4 + 2k > 16 \), we will solve each inequality separately.
Solving the first inequality:
\[ k - 3 \leq -2k - 3 \] First, add \( 2k \) to both sides: \[ k + 2k - 3 \leq -3 \] This simplifies to: \[ 3k - 3 \leq -3 \] Next, add 3 to both sides: \[ 3k \leq 0 \] Now divide by 3: \[ k \leq 0 \]
Solving the second inequality:
\[ k + 4 + 2k > 16 \] Combine like terms: \[ 3k + 4 > 16 \] Now subtract 4 from both sides: \[ 3k > 12 \] Divide by 3: \[ k > 4 \]
Combine the results:
From the first inequality, we found \( k \leq 0 \), and from the second inequality, we found \( k > 4 \).
Since these inequalities are connected by "or," we can present our final result as: \[ k \leq 0 \quad \text{or} \quad k > 4 \]
Thus, the solution as a compound inequality is: \[ (-\infty, 0] \cup (4, \infty) \]
In terms of integers: \[ k \in { \ldots, -3, -2, -1, 0 } \quad \text{or} \quad k \in { 5, 6, 7, \ldots } \]