mpany's profits \( P \) are modeled by the inequality: \[ 10 x-3 \leq P \leq 15 x+5 \]

It looks like you've provided an inequality that models a company's profits \( P \) based on some variable \( x \): \[ 10x - 3 \leq P \leq 15x + 5 \] To better assist you, I'll break down what this inequality represents and explore possible interpretations and applications. If you have a specific question or context in mind, please let me know! ### Understanding the Inequality 1. **Components of the Inequality:** - **Lower Bound:** \( 10x - 3 \) - **Upper Bound:** \( 15x + 5 \) - **Profit \( P \):** Satisfies \( 10x - 3 \leq P \leq 15x + 5 \) 2. **Interpretation:** - The profit \( P \) is not a single fixed value but lies within a range determined by \( x \). - **Variable \( x \):** Represents an independent variable that affects the profit. The exact meaning of \( x \) depends on the context (e.g., number of units sold, hours worked, etc.). ### Possible Scenarios and Applications 1. **Determining Profit Range for a Given \( x \):** - For a specific value of \( x \), you can calculate the minimum and maximum possible profits. **Example:** If \( x = 10 \), \[ \begin{align*} \text{Minimum Profit} &= 10(10) - 3 = 100 - 3 = 97 \\ \text{Maximum Profit} &= 15(10) + 5 = 150 + 5 = 155 \\ \end{align*} \] So, \( P \) would be between \$97 and \$155. 2. **Solving for \( x \) Given a Desired Profit Range:** - If you have a target profit and want to find the range of \( x \) that achieves it, you can solve the inequalities accordingly. **Example:** Suppose you want the profit \( P \) to be at least \$50. \[ 10x - 3 \leq P \leq 15x + 5 \\ \] For \( P \geq 50 \): \[ 10x - 3 \geq 50 \implies 10x \geq 53 \implies x \geq 5.3 \] and \[ 15x + 5 \geq 50 \implies 15x \geq 45 \implies x \geq 3 \] Since \( x \) must satisfy both conditions, \( x \geq 5.3 \) ensures \( P \geq 50 \). 3. **Graphical Representation:** - You can graph the inequalities on a coordinate plane with \( x \) on the horizontal axis and \( P \) on the vertical axis. - The area between the lines \( P = 10x - 3 \) and \( P = 15x + 5 \) represents all the possible profit values for each \( x \).  *(Replace with an actual graph if needed)* 4. **Analyzing Sensitivity:** - Understanding how changes in \( x \) affect the profit range. - For instance, increasing \( x \) expands both the minimum and maximum possible profits. ### Additional Considerations - **Contextual Meaning of \( x \):** To provide more tailored assistance, knowing what \( x \) represents in your scenario (e.g., production units, sales volume, time) would be helpful. - **Potential Restrictions on \( x \):** Depending on the real-world situation, there might be logical constraints on \( x \) (e.g., \( x \) must be a non-negative integer). - **Optimization:** - If you're looking to optimize profit, understanding whether minimizing or maximizing \( x \) leads to desirable outcomes within the given bounds can be useful. ### Next Steps If you have a specific question related to this inequality—such as solving for \( x \), interpreting results in a particular context, or applying it to a real-world scenario—please provide more details, and I'd be happy to help further!