To solve the inequality \( |x - 5| + 1 > 5 \), we first isolate the absolute value:
\[ |x - 5| > 5 - 1 \] \[ |x - 5| > 4 \]
This inequality indicates that the expression inside the absolute value, \( x - 5 \), is either greater than 4 or less than -4. Therefore, we can set up two separate inequalities:
- \( x - 5 > 4 \)
- \( x - 5 < -4 \)
Solving the first inequality:
\[ x - 5 > 4 \implies x > 9 \]
Solving the second inequality:
\[ x - 5 < -4 \implies x < 1 \]
So, the solution to the inequality \( |x - 5| + 1 > 5 \) is:
\[ x < 1 \quad \text{or} \quad x > 9 \]
Graphically, this means we have open circles at the points \( x = 1 \) and \( x = 9 \), and the solution set is represented by two separate intervals: \( (-\infty, 1) \) and \( (9, \infty) \).
Based on this analysis, the correct option is:
Option #1: open circles on the end values; two separate solution sets.