The usual toll charge is 50 cents. If you purchase a special sticker for $5.50, the toll is only 35 cents. At least how many trips through the tunnel are needed before the sticker costs less than paying for each trip separately?

To determine the minimum number of trips needed for the special sticker to become cost-effective, let's break the problem down.

Let:
- $ x $ be the number of trips.
- The regular cost per trip is 50 cents.
- The sticker reduces the cost per trip to 35 cents, but the sticker itself costs $5.50 upfront.

### Step 1: Write the total cost expressions.
- Without the sticker, the total cost for $ x $ trips is:
  \[
  \text{Regular cost} = 0.50x
  \]
  
- With the sticker, the total cost for $ x $ trips is:
  \[
  \text{Sticker cost} = 5.50 + 0.35x
  \]

### Step 2: Set up the inequality.
We want the total cost with the sticker to be less than the total cost without the sticker:
\[
5.50 + 0.35x < 0.50x
\]

### Step 3: Solve the inequality.
First, subtract $ 0.35x $ from both sides:
\[
5.50 < 0.15x
\]

Next, divide both sides by 0.15 to isolate $ x $:
\[
x > \frac{5.50}{0.15}
\]
\[
x > 36.67
\]

### Step 4: Interpret the result.
Since the number of trips must be a whole number, you need at least 37 trips for the sticker to be cost-effective.

Thus, **at least 37 trips** are needed for the sticker to save money compared to paying the regular toll.

---

Would you like a more detailed explanation, or have any questions?

Here are 5 related questions for practice:
1. What is the total cost for 37 trips with the sticker?
2. How much would 40 trips cost without the sticker?
3. If the sticker costs $6.00 instead, how many trips would be needed?
4. What if the toll without a sticker was 60 cents instead of 50 cents?
5. How does the break-even point change if the sticker reduces the toll to 30 cents?

**Tip:** When comparing two options with different fixed and variable costs, break-even analysis helps find the minimum point where one option becomes more favorable.

Learn how to solve a cost analysis problem comparing regular tolls with a special sticker that reduces the toll price. This step-by-step solution covers linear equations, inequalities, and the break-even point. Suitable for students in Grades 6-8.

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