To solve the inequality \(|x| + 5 \geq 9\), we first isolate the absolute value:
\[ |x| \geq 4 \]
This inequality can be split into two cases:
- \(x \geq 4\)
- \(x \leq -4\)
Now, let's analyze the options:
- The solutions are \(x \geq 4\) and \(x \leq -4\), which means we have two separate solution sets: one extending to the right from 4 and one extending to the left from -4.
- Since the inequality includes \(\geq\), we use closed circles at the points -4 and 4 because those values satisfy the inequality.
Based on this analysis, the correct option is:
Option #3: closed circles on the end values; two separate solution sets.