`C=8n+522`
The equation above gives the cost, C, in dollars of manufacturing n items. A profit is made when the total revenue from selling a quantity of items is greater than the total cost of manufacturing the same quantity of items. If each items can be sold $14, which of the following inequalies gives all possible values of n that will produce a profit?

Options

A. `ngt87`
B. `ngt112`
C. `ngt522`
D. `ngt609`

Answer

To solve the problem, we need to determine the values of \( n \) that will produce a profit when manufacturing items. The profit condition is that the total revenue must be greater than the total cost.

### Step-by-step Solution:

1. **Identify the Cost Function**:
The cost \( C \) of manufacturing \( n \) items is given by the equation:
\[ C = 8n + 522 \]

2. **Determine the Revenue Function**:
Each item is sold for $14, so the total revenue \( R \) from selling \( n \) items is:
\[ R = 14n \]

3. **Set Up the Profit Inequality**:
To find when a profit is made, we set up the inequality:
\[ R > C \]
Substituting the expressions for revenue and cost, we have:
\[ 14n > 8n + 522 \]

4. **Simplify the Inequality**:
To simplify, we subtract \( 8n \) from both sides:
\[ 14n - 8n > 522 \]
This simplifies to:
\[ 6n > 522 \]

5. **Solve for \( n \)**:
Next, we divide both sides of the inequality by 6:
\[ n > \frac{522}{6} \]
Calculating the right side:
\[ n > 87 \]

6. **Conclusion**:
Therefore, the inequality that gives all possible values of \( n \) that will produce a profit is:
\[ n > 87 \]

### Final Answer:
The correct inequality is \( n > 87 \).

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