We can solve the given inequality by following these steps:
1. Isolate the term that has the absolute value on one side of the inequality, we can do this by adding 3 and dividing by 9 on both sides:
9 |1 + 8n| - 3 ≥ 78
9 |1 + 8n| ≥ 81
|1 + 8n| ≥ 81/9
|1 + 8n| ≥ 9
2. Apply the definition of the absolute value function:
1 + 8n ≥ 9 and 1 + 8n ≤ -9
3. Solve for n:
8n ≥ 8 and 8n ≤ -10
n ≥ 1 and n ≤ - 5/4
We can graph the solution like this:
