"At most $25" means the total cost can be less than or equal to $25. What inequality symbol does that suggest when you write the equation for cost versus budget?

After you solve the inequality for $m$, ask: what does the solution tell you about the range of possible $m$ values, and which whole number at the upper end of that range is still allowed?

Try plugging in a candidate whole number of miles back into the cost formula to see if it stays within $25, and make sure any larger whole number would go over the budget.

In Desmos, type C(m) = 3.5 + 2.4m to represent the cost in dollars as a function of miles $m$.

In a new line, type 3.5 + 2.4m = 25 and look at the point where this line intersects the horizontal line $C = 25$ (or just read the solution for $m$ that Desmos gives). This value of $m$ is the maximum distance where the cost exactly equals the budget.

From the $m$-value you see, determine the largest whole number that is less than or equal to that value. That whole number of miles is the greatest distance the passenger can travel without going over the budget.

Let $m$ be the number of miles traveled.

The taxi cost is made of:

• A flat fee of $3.50, and
• $2.40 per mile, so $2.40m$.

So the total cost is $3.5 + 2.4m$. Because the passenger can spend at most $25, we write the inequality:

Now solve $3.5 + 2.4m \le 25$.

Subtract $3.5$ from both sides:

Now divide both sides by $2.4$:

So $m$ must be less than or equal to the value of $21.5 \div 2.4$.

We need to understand $\dfrac{21.5}{2.4}$.

Multiply top and bottom by $10$ to clear decimals:

Now compare $215$ to multiples of $24$:

• $24 \times 8 = 192$
• $24 \times 9 = 216$

So $\dfrac{215}{24}$ is between $8$ and $9$ (because it is just a bit less than $\dfrac{216}{24} = 9$).

The inequality tells us $m$ must be less than or equal to a number between $8$ and $9$.

The passenger can only travel a whole number of miles without exceeding the budget, so the largest whole number less than or equal to this upper limit is 8.

Answer: 8.