mpany’s profits are modeled by the inequality:

Understanding the Inequality

1. Components of the Inequality:
   • Lower Bound:
   • Upper Bound:
   • Profit : Satisfies
2. Interpretation:
   • The profit is not a single fixed value but lies within a range determined by .
   • Variable : Represents an independent variable that affects the profit. The exact meaning of depends on the context (e.g., number of units sold, hours worked, etc.).

Possible Scenarios and Applications

1. Determining Profit Range for a Given :
   • For a specific value of , you can calculate the minimum and maximum possible profits.
   Example:
   If ,
   So, would be between $97 and $155.
2. Solving for Given a Desired Profit Range:
   • If you have a target profit and want to find the range of that achieves it, you can solve the inequalities accordingly.
   Example:
   Suppose you want the profit to be at least $50.
   For :
   and
   Since must satisfy both conditions, ensures .
3. Graphical Representation:
   • You can graph the inequalities on a coordinate plane with on the horizontal axis and on the vertical axis.
   • The area between the lines and represents all the possible profit values for each .
   (Replace with an actual graph if needed)
4. Analyzing Sensitivity:
   • Understanding how changes in affect the profit range.
   • For instance, increasing expands both the minimum and maximum possible profits.

Additional Considerations

• Contextual Meaning of : To provide more tailored assistance, knowing what represents in your scenario (e.g., production units, sales volume, time) would be helpful.
• Potential Restrictions on : Depending on the real-world situation, there might be logical constraints on (e.g., must be a non-negative integer).
• Optimization:
  • If you’re looking to optimize profit, understanding whether minimizing or maximizing leads to desirable outcomes within the given bounds can be useful.

Next Steps