To determine how many miles Katie can travel without exceeding her spending limit of $23, we can set up an equation based on the taxi's fare structure.
The total cost \( C \) of the taxi ride can be expressed as:
\[ C = 1.95 + 0.54m \]
where \( m \) is the number of miles driven.
Katie wants the cost \( C \) to be less than or equal to her spending limit of $23:
\[ 1.95 + 0.54m \leq 23 \]
Now, we can solve for \( m \):
- Subtract the flat rate from both sides:
\[ 0.54m \leq 23 - 1.95 \]
- Calculate \( 23 - 1.95 \):
\[ 23 - 1.95 = 21.05 \]
So now we have:
\[ 0.54m \leq 21.05 \]
- Divide both sides by 0.54 to isolate \( m \):
\[ m \leq \frac{21.05}{0.54} \]
- Calculating this gives:
\[ m \leq 38.94 \]
Since \( m \) must be a whole number (you can't travel a fraction of a mile in this context), we take the largest whole number less than or equal to 38.94, which is 38.
Thus, the maximum number of miles Katie can travel without exceeding her spending limit is:
\[ \boxed{38} \]