Question

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?

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:
\[
106 \text{ items}
\]