Let $x$ be the total miles of the ride. What expression gives the cost if the ride is $x$ miles within the first 15 miles at $1.20 per mile plus a $2 flat fee?

Once the ride is longer than 15 miles, you must pay for the first 15 miles at $1.20 per mile, and only the extra miles at $0.80 per mile. Write a new expression that starts with the cost of 15 miles, then adds the cost for the miles beyond 15.

Set your cost expression (for the longer-than-15-miles case) less than or equal to the maximum he can spend, solve for $x$, and then decide what the largest whole number value of $x$ can be.

In Desmos, graph the two cost expressions as separate lines:

• Type y = 2 + 1.2x and visually focus on $x \le 15$.
• Type y = 8 + 0.8x and visually focus on $x \ge 15$. These together represent the ride cost for different distances.

Type y = 25 to represent the maximum amount Kai can spend on the ride. This is a horizontal line.

Look at where the line y = 8 + 0.8x (the cost for rides longer than 15 miles) intersects the line y = 25. Click the intersection point and note the $x$-value; this is the maximum distance (in miles) allowed in that region.

Use the $x$-value from the intersection as an upper bound. Take the largest integer less than or equal to this $x$-value; that is the greatest whole number of miles Kai can travel while keeping the cost at or below $25.

Kai has $30 in ride credit and must still have at least $5 left after the ride.

So the maximum he can spend on this ride is

Any cost for the ride must be at most $25.

Let $x$ be the total number of miles Kai travels.

For the first 15 miles, the cost is a flat $2 plus $1.20 per mile. If the ride were $x$ miles long and $x \le 15$, the cost would be

However, because the price changes after 15 miles, we need a separate expression for when $x$ is greater than 15.

If the ride goes more than 15 miles, then:

• The first 15 miles cost $1.20 per mile.
• Every mile after 15 costs $0.80 per mile.

First 15 miles cost:

Including the $2 flat fee, the starting cost at 15 miles is

For miles beyond 15, the additional cost is $0.80 per mile. So for $x > 15$ miles total, the cost is

Simplify this:

Now use the spending limit: this cost must be at most $25:

Solve for $x$:

So, when $x > 15$, the total miles must be at most $21.25$.

From the longer-ride case, Kai can travel up to 21.25 miles, but the problem asks for the greatest whole number of miles.

So we need the largest integer less than or equal to 21.25, which is 21.

Finally, check that this is consistent with the pricing:

• For 21 miles, use the $x > 15$ formula: $\text{cost} = 8 + 0.80 \cdot 21 = 8 + 16.8 = 24.8$, which is less than or equal to $25.

Thus, the greatest whole number of miles Kai can travel is 21.