Problem 127 - College Algebra Essentials

Question

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. A company manufactures and sells personalized stationery. The weekly fixed cost is \(\$ 3000\) and it costs \(\$ 3.00\) to produce each package of stationery. The selling price is \(\$ 5.50\) per package. How many packages of stationery must be produced and sold each week for the company to generate a profit?

Step-by-Step Solution

Answer

The company must produce and sell more than 1200 packages of stationery each week in order to make a profit.

Step 1: Define Variables

Let's denote the number of packages by \(x\). Therefore, the cost of production is \(3x\) dollars and the revenue from selling is \(5.5x\) dollars.

Step 2: Set Up Inequality

The company will make a profit if the revenue from selling is greater than the total cost (fixed cost + cost of production). So, we can write the inequality as \(5.5x > 3000 + 3x\).

Step 3: Solve The Inequality

First, simplify the inequality by subtracting \(3x\) from both sides: \(2.5x > 3000\). Then, to solve for \(x\), divide both sides by 2.5: \(x > \frac{3000}{2.5}\).

Step 4: Calculate Result

After calculating the right-hand side we obtain: \(x > 1200\). This means the company must produce and sell more than 1200 packages weekly to make a profit.