The daily cost for a publishing company to produce x books is `C(x)=4x+800`. The company sells each book for `$36`. Let `P(x)=R(x)-C(x)` where `R(x)` is the total income that the company gets for selling x books. The company takes a loss for the day if `P(x)lt0` . Which of the following inequalities gives all possible integer value of x that guarantee that the company will not take a loss on a given day?

Options

A. `x gt 24`
B. `xlt24`
C. `xgt144`
D. `xlt144`

Answer

To solve the problem, we need to determine the conditions under which the publishing company does not incur a loss. We will follow these steps:

1. **Define the Cost and Revenue Functions**:
- The cost function is given by \( C(x) = 4x + 800 \).
- The revenue function, which is the total income from selling \( x \) books, is given by \( R(x) = 36x \) (since each book is sold for $36).

2. **Define the Profit Function**:
- The profit function is defined as \( P(x) = R(x) - C(x) \).
- Substituting the expressions for \( R(x) \) and \( C(x) \), we have:
\[ P(x) = 36x - (4x + 800) \]

3. **Simplify the Profit Function**:
- Distributing the negative sign in the profit function:
\[ P(x) = 36x - 4x - 800 \]
- Combine like terms:
\[ P(x) = 32x - 800 \]

4. **Set Up the Inequality for No Loss**:
- The company takes a loss if \( P(x) < 0 \). To ensure no loss, we need:
\[ P(x) \geq 0 \]
- Thus, we set up the inequality:
\[ 32x - 800 \geq 0 \]

5. **Solve the Inequality**:
- Add 800 to both sides:
\[ 32x \geq 800 \]
- Divide both sides by 32:
\[ x \geq \frac{800}{32} \]
- Simplifying the right-hand side:
\[ x \geq 25 \]

6. **Determine the Integer Values**:
- Since \( x \) must be an integer, the smallest integer value for \( x \) that satisfies the inequality is 25. Therefore, the company will not take a loss if:
\[ x \geq 25 \]

7. **Rewrite the Inequality**:
- The inequality can also be expressed as:
\[ x > 24 \]
- This means that \( x \) must be strictly greater than 24.

**Final Answer**:
The inequality that guarantees the company will not take a loss is:
\[ x > 24 \]