1. Aubrey is deciding between two landscaping co equipment fee and charges ines company A charges 555 per hour and a $150 equipment fee. Company B has a 5225 equipment fee and charges $40 per hour. For how many hours wil Company A be less expensive than Company B?
b. Write a model to represent the situation & solve.
a. Define a variable.
b. Explain in a complete sentence what your solution means.
c. When is Company B a better deal? Use mathematical vocabulary to explain your answer.

Let's break down the problem and solve it step-by-step.

### 1. Define a Variable:
Let $ h $ represent the number of hours of landscaping work.

### 2. Write a Model (Inequality) to Represent the Situation:
- **Company A** charges $55 per hour and a $150 equipment fee. So, the total cost for Company A can be written as:
  \[
  \text{Cost of A} = 55h + 150
  \]
- **Company B** charges $40 per hour and has a $225 equipment fee. So, the total cost for Company B is:
  \[
  \text{Cost of B} = 40h + 225
  \]

We want to find when Company A is **less expensive** than Company B. So, set up the inequality:
\[
55h + 150 < 40h + 225
\]

### 3. Solve the Inequality:
1. Subtract $ 40h $ from both sides:
   \[
   55h - 40h + 150 < 225
   \]
   \[
   15h + 150 < 225
   \]

2. Subtract 150 from both sides:
   \[
   15h < 75
   \]

3. Divide both sides by 15:
   \[
   h < 5
   \]

### 4. Conclusion (b. Explanation):
The solution $ h < 5 $ means that **Company A will be less expensive than Company B if the number of hours is less than 5**. So, if the job takes **fewer than 5 hours**, Company A is a better deal.

### 5. When is Company B a Better Deal (c. Explanation)?
Company B is a better deal when the job takes **5 or more hours**. This is because when $ h \geq 5 $, the total cost of Company B will be less than or equal to the total cost of Company A. Mathematically, we can verify this by comparing their costs for $ h = 5 $ and beyond.

### Verification:
- For $ h = 5 $:
  - **Company A**: $ 55(5) + 150 = 275 + 150 = 425 $
  - **Company B**: $ 40(5) + 225 = 200 + 225 = 425 $

So, at exactly 5 hours, the costs are equal. For any time greater than 5 hours, Company B will be less expensive.

---

### Recap and Expansion:
Would you like any further details or clarification on any steps?

Here are 5 follow-up questions to consider:
1. How would the inequality change if the equipment fees for both companies were different?
2. What happens to the solution if Company A decreases its hourly rate to $50?
3. How many hours would make both companies charge the same, if Company B reduced its equipment fee?
4. What if Company B charged $35 per hour—how would that affect the solution?
5. How does changing the equipment fee affect the balance point where one company becomes cheaper?

**Tip**: Always break down word problems by defining variables and writing equations or inequalities to represent the situation.

Aubrey is deciding between two landscaping companies. Company A charges $55 per hour and a $150 equipment fee. Company B has a $225 equipment fee and charges $40 per hour. For how many hours will Company A be less expensive than Company B? Write a model to represent the situation and solve. Define a variable, explain in a complete sentence what your solution means, and determine when Company B is a better deal using mathematical vocabulary.

Learn how to compare costs between two landscaping companies using linear inequalities. Company A charges $55 per hour plus a $150 equipment fee, while Company B charges $40 per hour with a $225 equipment fee. Discover how many hours make Company A cheaper and when Company B is the better deal, suitable for students in Grades 7-9.

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