To solve the inequality \( |x| + 5 \geq 9 \), we can start by isolating the absolute value:
\[ |x| + 5 \geq 9 \]
Subtract 5 from both sides:
\[ |x| \geq 4 \]
The absolute value inequality \( |x| \geq 4 \) means that \( x \) is either less than or equal to -4, or greater than or equal to 4. This can be expressed as:
\[ x \leq -4 \quad \text{or} \quad x \geq 4 \]
When we graph this solution on a number line, we use closed circles at -4 and 4 (because the inequality is "greater than or equal to"), indicating that these points are included in the solution. The graph consists of two separate parts: one extending to the left from -4 and the other extending to the right from 4.
Based on this reasoning, the correct representation of the graphed solution corresponds to:
Option #3: closed circles on the end values; two separate solution sets.