https://file.aiquickdraw.com/mathbot/file/1726190630512sp46lwtc.jpg

To solve this problem, let's analyze the graph provided in the image to determine the range of miles for which each company offers the cheapest cost.

### Step-by-step Analysis:

1. **Identify the companies' lines:**
   - The graph shows three lines for three different companies:
     - **Company A**: Red Line (horizontal)
     - **Company B**: Blue Line (steeper slope)
     - **Company C**: Green Line (less steep slope)

2. **Determine the points of intersection:**
   - The red (Company A) and green (Company C) lines intersect at **120 miles**.
   - The green (Company C) and blue (Company B) lines intersect at **400 miles**.

3. **Analyze the cost behavior between intersections:**
   - From 0 to 120 miles:
     - **Company A** is the cheapest since the red line is below the others.
   - From 120 to 400 miles:
     - **Company C** is the cheapest since the green line is below the others.
   - From 400 to 500 miles:
     - **Company B** is the cheapest since the blue line is below the others.

4. **Fill in the blanks:**
   - **Company A is cheapest if you drive between 0 and 120 miles.**
   - **Company B is cheapest if you drive between 400 and 500 miles.**
   - **Company C is cheapest if you drive between 120 and 400 miles.**

Would you like more details about this solution or have any questions?

### Related Questions:
1. How can you find the exact equations for each line to determine costs mathematically?
2. What would happen if the intersection points changed?
3. What assumptions are we making about the cost per mile in this scenario?
4. How does the slope of each line affect the range in which each company is the cheapest?
5. Can we apply a similar analysis to other types of cost comparison problems?

**Tip:** When comparing costs, always look for points of intersection to determine switching points between different options.

Complete the boxes to report under what circumstances each company is the cheapest, assuming the maximum number of miles driven is 500. Company A is cheapest if you drive between ____ and ____ miles. Company B is cheapest if you drive between ____ and ____ miles. Company C is cheapest if you drive between ____ and ____ miles.

Learn how to determine which company offers the cheapest cost depending on the number of miles driven, by analyzing the intersection points of cost lines on a graph. This example uses linear equations and covers key mathematical concepts in cost analysis, suitable for students in Grades 8-10.

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