Lena is going to rent a truck for one day. There are two companies she can choose from, and they have the following prices.
Company A charges an initial fee of $39 and an additional 60 cents for every mile driven.
Company B charges an initial fee of $30 and an additional 70 cents for every mile driven.
For what mileages will Company A charge no more than Company B?
Write your answer as an inequality, using m for the number of miles driven.

Let's define the cost functions for both companies in terms of the number of miles driven, $m$.

- **Company A:**  
  Cost $A(m) = 39 + 0.60m$  
- **Company B:**  
  Cost $B(m) = 30 + 0.70m$  

We are asked to find the range of mileages $m$ where **Company A charges no more than Company B**. That translates to the inequality:

\[
39 + 0.60m \leq 30 + 0.70m
\]

### Solving the inequality:
1. Subtract 30 from both sides:
   \[
   9 + 0.60m \leq 0.70m
   \]

2. Subtract $0.60m$ from both sides:
   \[
   9 \leq 0.10m
   \]

3. Divide both sides by 0.10:
   \[
   m \geq 90
   \]

### Final Inequality:
The mileage $m$ for which Company A's cost is no more than Company B's is:

\[
m \geq 90
\]

This means that for any mileage **90 miles or more**, Company A will be cheaper or the same as Company B.

---

Would you like more details or have any questions?  

### 5 Related Questions:
1. How would the answer change if Company B’s per-mile rate was 60 cents instead of 70?
2. At what mileage do both companies charge the same amount?
3. How do you graph these cost functions to visualize the break-even point?
4. What would the cost functions look like if there were discounts for driving more than 100 miles?
5. How does the slope of each company's cost function affect the solution?

**Tip:** Always compare both the fixed and variable costs when analyzing such pricing models to find optimal solutions.

Lena is going to rent a truck for one day. There are two companies she can choose from, and they have the following prices. Company A charges an initial fee of $39 and an additional 60 cents for every mile driven. Company B charges an initial fee of $30 and an additional 70 cents for every mile driven. For what mileages will Company A charge no more than Company B? Write your answer as an inequality, using m for the number of miles driven.

Discover how to compare truck rental prices between Company A and Company B using algebraic linear inequalities. This detailed solution explains how to find the mileage where Company A charges no more than Company B, using key concepts like cost functions and inequalities. Suitable for students in Grades 8-10.

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