Solved problems: Interpreting systems of linear inequalities in application contexts

📚 Understanding Systems of Linear Inequalities in Context

Systems of linear inequalities are used to model situations with multiple constraints. Interpreting their solutions involves understanding what the feasible region represents within the given context. Let's explore this concept in detail.

📜 History and Background

The study of inequalities dates back to ancient times, but the systematic use of linear inequalities for modeling and optimization emerged in the 20th century with the development of linear programming. George Dantzig is considered the father of linear programming, which heavily relies on systems of linear inequalities.

🔑 Key Principles

• 🧮 Defining Variables: Clearly define the variables representing the quantities in the problem. For example, let $x$ be the number of hours worked at job A and $y$ be the number of hours worked at job B.
• ✍️ Formulating Inequalities: Translate the constraints into linear inequalities. These constraints often involve limitations on resources, minimum requirements, or maximum capacities.
• 📈 Graphing the Inequalities: Graph each inequality on the coordinate plane. The region that satisfies all inequalities simultaneously is the feasible region.
• 🎯 Interpreting the Feasible Region: The feasible region represents all possible combinations of the variables that satisfy all the constraints. Any point within this region is a valid solution to the problem.
• 💡 Understanding Corner Points: In optimization problems, the optimal solution often occurs at one of the corner points (vertices) of the feasible region.

🌍 Real-World Examples

Example 1: Production Planning

A factory produces two types of products, A and B. To produce one unit of product A requires 2 hours of labor and 1 unit of raw material. To produce one unit of product B requires 3 hours of labor and 0.5 units of raw material. The factory has a maximum of 120 hours of labor and 40 units of raw material available.

Let $x$ be the number of units of product A and $y$ be the number of units of product B.

• ⏱️ Labor Constraint: $2x + 3y \leq 120$
• 📦 Raw Material Constraint: $x + 0.5y \leq 40$
• ➕ Non-negativity Constraints: $x \geq 0$, $y \geq 0$

The feasible region represents all possible production plans that satisfy the labor and raw material constraints. For example, producing 20 units of A and 20 units of B ($x=20, y=20$) would require $2(20) + 3(20) = 100$ hours of labor and $20 + 0.5(20) = 30$ units of raw material, which is within the limits.

Example 2: Dietary Requirements

A person needs to consume at least 60 grams of protein and 40 grams of fat per day. Food A contains 10 grams of protein and 5 grams of fat per serving. Food B contains 6 grams of protein and 8 grams of fat per serving.

Let $x$ be the number of servings of food A and $y$ be the number of servings of food B.

• 💪 Protein Constraint: $10x + 6y \geq 60$
• 🥑 Fat Constraint: $5x + 8y \geq 40$
• ➕ Non-negativity Constraints: $x \geq 0$, $y \geq 0$

The feasible region represents all possible combinations of servings of food A and food B that meet the minimum protein and fat requirements. For example, consuming 4 servings of food A and 4 servings of food B ($x=4, y=4$) would provide $10(4) + 6(4) = 64$ grams of protein and $5(4) + 8(4) = 52$ grams of fat, satisfying the requirements.

Example 3: Investment Portfolio

An investor wants to allocate funds between two investments, X and Y. Investment X requires a minimum investment of $2,000, and Investment Y requires a minimum investment of $3,000. The investor has a total of $15,000 to invest.

Let $x$ be the amount invested in X, and $y$ be the amount invested in Y.

• 💰 Minimum Investment X: $x \geq 2000$
• 🏦 Minimum Investment Y: $y \geq 3000$
• 💵 Total Investment Constraint: $x + y \leq 15000$

The feasible region represents all possible investment allocations that satisfy the minimum investment requirements and the total investment limit. For example, investing $5,000 in X and $7,000 in Y ($x=5000, y=7000$) meets the minimum investment requirements and stays within the $15,000 limit.

📊 Table: Summary of Examples

🔑 Conclusion

Interpreting systems of linear inequalities in application contexts involves understanding the constraints, defining appropriate variables, and recognizing that the feasible region represents all possible solutions that satisfy those constraints. By carefully analyzing the problem and formulating the inequalities, you can gain valuable insights into real-world scenarios.