To determine the maximum number of hours Karl can rent the bike for at most $55, we can start by formulating the total cost of renting the bike.
The cost of renting the bike includes a flat fee and a variable fee based on the number of hours rented. The total cost \( C \) can be expressed as:
\[ C = 9.50 + 7h \]
where \( h \) is the number of hours Karl rents the bike.
Since Karl wants the total cost to be at most $55, we can set up the following inequality:
\[ 9.50 + 7h \leq 55 \]
Next, we isolate \( h \) in the inequality:
\[ 7h \leq 55 - 9.50 \] \[ 7h \leq 45.50 \] \[ h \leq \frac{45.50}{7} \] \[ h \leq 6.5 \]
Since any portion of an hour is charged as a full hour, Karl can only rent the bike for complete hours. Therefore, we take the greatest integer less than or equal to 6.5, which is 6.
Thus, the maximum number of hours Karl can rent the bike is:
\[ \boxed{6} \]
To confirm, we can compute the total cost for 6 hours:
\[ C = 9.50 + 7 \times 6 = 9.50 + 42 = 51.50 \]
Since $51.50 is less than $55, the calculation is valid.
If he were to rent for 7 hours:
\[ C = 9.50 + 7 \times 7 = 9.50 + 49 = 58.50 \]
Since $58.50 exceeds $55, renting for 7 hours is not possible.
The maximum number of hours Karl can rent the bike is indeed:
\[ \boxed{6} \]