https://file.aiquickdraw.com/mathbot/file/17259174496578hfcpubj.jpg

To solve the problem, let's set up the inequality based on the information provided:

- **Start-up cost:** $125,000 (this is a fixed cost)
- **Cost per device:** $6.50 (this is the variable cost for each device)
- **Selling price per device:** $9.00

### Step 1: Define Variables
Let $ x $ represent the number of devices the company needs to sell to make a profit.

### Step 2: Set Up the Inequality
To make a profit, the total revenue from selling $ x $ devices must be greater than the total cost to produce them.

- **Total Revenue:** $ 9x $ (since each device is sold for $9.00)
- **Total Cost:** $ 125,000 + 6.5x $ (fixed cost plus variable cost per device)

The company makes a profit when:

\[
9x > 125,000 + 6.5x
\]

### Step 3: Solve the Inequality
Subtract $ 6.5x $ from both sides to isolate $ x $ on one side:

\[
9x - 6.5x > 125,000
\]

\[
2.5x > 125,000
\]

Now, divide both sides by 2.5:

\[
x > \frac{125,000}{2.5}
\]

\[
x > 50,000
\]

### Conclusion
The company must sell **more than 50,000** devices to make a profit.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:
1. What happens if the selling price is reduced to $8.00 per device?
2. How would the break-even point change if the start-up cost increased to $150,000?
3. How many devices need to be sold to cover just the start-up costs?
4. What is the profit if 60,000 devices are sold?
5. How would changes in the cost per device affect the number of units needed to be sold for profit?

**Tip:** Understanding how to set up inequalities is crucial for solving problems related to profit and loss in business scenarios.

An electronics company has developed a new hand-held device. The company predicts that the start-up cost to manufacture the new product will be $125,000, and the cost to make one device will be $6.50. If the company plans on selling the devices at a price of $9, write and solve an inequality to determine how many must be sold for the company to make a profit. Show your work.

Learn how to solve an inequality to determine the number of devices an electronics company must sell to make a profit. This problem involves key concepts in algebra, including revenue, cost, and profit. Suitable for students in Grades 9-11.

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