To make a profit, a company’s revenue must be greater than its operating costs. The company’s revenue is modeled by the expression 7.5x – 100, where x represents the number of items sold. The company’s operation costs are modeled by the expression 79.86 + 5.8x. How many items does the company need to sell to make a profit?

Question

The inequality that will determine the number of items that need to be sold to make a profit is
.

The solution to the inequality is
.

The company must sell at least
items to make a profit.

Answer 1

To determine how many items the company needs to sell to make a profit, we set up the inequality based on the information given.

The revenue \( R \) is given by: \[ R = 7.5x - 100 \]

The operating costs \( C \) are given by: \[ C = 79.86 + 5.8x \]

To make a profit, the revenue must be greater than the operating costs: \[ 7.5x - 100 > 79.86 + 5.8x \]

Now, we can solve this inequality step by step.

1. Isolate terms involving \( x \) on one side: \[ 7.5x - 5.8x > 79.86 + 100 \] Simplifying: \[ 1.7x > 179.86 \]

2. Divide both sides by 1.7: \[ x > \frac{179.86}{1.7} \]

3. Calculate the division: \[ x > 105.2 \]

Since the number of items sold must be a whole number, we round up: \[ x \geq 106 \]

Thus, the company must sell at least 106 items to make a profit.

Summary:

The inequality that will determine the number of items that need to be sold to make a profit is: \[ 7.5x - 100 > 79.86 + 5.8x \]

The solution to the inequality is: \[ x > 105.2 \]

The company must sell at least: \[ 106 \text{ items} \]