To solve the compound inequality \(-16n - 20 \leq n - 20 \leq -16n - 3\), we can break it down into two separate inequalities and solve them one at a time.
Step 1: Solve the left part of the compound inequality
We start with the inequality: \[ -16n - 20 \leq n - 20 \]
First, we will add \(20\) to both sides: \[ -16n \leq n \] Now, we will subtract \(n\) from both sides: \[ -16n - n \leq 0 \] This simplifies to: \[ -17n \leq 0 \] Dividing by \(-17\) (note that the inequality sign will flip): \[ n \geq 0 \]
Step 2: Solve the right part of the compound inequality
Now, we consider the second part: \[ n - 20 \leq -16n - 3 \]
We start by adding \(16n\) to both sides: \[ n + 16n - 20 \leq -3 \] This simplifies to: \[ 17n - 20 \leq -3 \] Next, we will add \(20\) to both sides: \[ 17n \leq 17 \] Dividing both sides by \(17\): \[ n \leq 1 \]
Step 3: Combine the results
Combining both results, we have: \[ 0 \leq n \leq 1 \]
Conclusion
In interval notation, the solution is: \[ [0, 1] \]
Thus, the compound inequality with integers is: \[ n = 0 \text{ or } n = 1 \]