5.4.4 Practice Modeling Two-variable Systems Of Inequalities

5.4.4 Practice: Modeling Two-Variable Systems of Inequalities

This comprehensive guide delves into the practice of modeling two-variable systems of inequalities, a crucial concept in algebra and real-world problem-solving. We'll explore the process step-by-step, from understanding the fundamentals to tackling complex scenarios. We'll also incorporate various examples and strategies to solidify your understanding and prepare you for advanced applications.

Understanding Two-Variable Inequalities

Before diving into modeling, let's revisit the core concepts. A two-variable inequality involves two variables (typically x and y) and an inequality symbol (<, >, ≤, ≥). The solution to a single inequality is a region on a coordinate plane.

Key Components:

• Variables: Represent unknown quantities.
• Inequality Symbols: Indicate the relationship between the expressions:
  • < (less than)
  • > (greater than)
  • ≤ (less than or equal to)
  • ≥ (greater than or equal to)
• Coordinate Plane: A two-dimensional plane used to visualize the solution region.

Graphing Inequalities:

Graphing a two-variable inequality involves these steps:

1. Treat it as an equation: First, graph the corresponding equation (replace the inequality symbol with an equals sign). This forms the boundary line of the solution region.
2. Determine the boundary line type:
   • If the inequality includes = (≤ or ≥), the boundary line is solid, indicating that points on the line are included in the solution.
   • If the inequality does not include = (< or >), the boundary line is dashed, indicating that points on the line are not included in the solution.
3. Test a point: Choose a point not on the boundary line (usually (0,0) for simplicity). Substitute the coordinates into the original inequality.
   • If the inequality is true, shade the region containing the test point.
   • If the inequality is false, shade the region not containing the test point.

Modeling Two-Variable Systems of Inequalities

A system of inequalities involves two or more inequalities with the same variables. The solution to a system is the region where the solution regions of all inequalities overlap.

Steps to Model a Real-World Scenario:

1. Define Variables: Identify the unknown quantities and assign them variables (usually x and y).
2. Identify Constraints: Determine the limitations or restrictions in the problem. These constraints translate into inequalities.
3. Write Inequalities: Express the constraints mathematically using inequalities. Pay close attention to the inequality symbols.
4. Graph the System: Graph each inequality on the same coordinate plane. The overlapping shaded region represents the solution to the system.
5. Interpret the Solution: Analyze the solution region to answer the question posed in the problem.

Example Problems and Solutions:

Let's work through several examples to illustrate the modeling process:

Example 1: Budgeting for a Party

You are planning a party and can spend at most $200 on food and drinks. Food costs $15 per person, and drinks cost $5 per person. You expect at least 10 people to attend. Model this situation using a system of inequalities.

Solution:

1. Variables:

   • x: Number of people attending the party.
   • y: Total cost of the party.

2. Constraints:

   • Total cost ≤ $200
   • Number of people ≥ 10
   • Cost = 15x + 5x (cost of food + cost of drinks)

3. Inequalities:

   • 15x + 5x ≤ 200 (Budget constraint)
   • x ≥ 10 (Attendance constraint)

4. Graphing: Graph both inequalities on the same coordinate plane. The solution region will be the area where both shaded regions overlap. Remember to consider only the positive quadrant, as the number of people and cost cannot be negative.

5. Interpretation: The overlapping region shows the combinations of the number of people and the total cost that satisfy both constraints. Any point within this region represents a feasible party plan.

Example 2: Production Constraints

A factory produces two types of products, A and B. Product A requires 2 hours of machine time and 1 hour of labor, while product B requires 1 hour of machine time and 3 hours of labor. The factory has a maximum of 12 hours of machine time and 15 hours of labor available daily. Model the possible production levels of A and B.

Solution:

1. Variables:

   • x: Number of units of product A produced.
   • y: Number of units of product B produced.

2. Constraints:

   • Machine time: 2x + y ≤ 12
   • Labor time: x + 3y ≤ 15
   • x ≥ 0, y ≥ 0 (Non-negativity constraints: you can't produce a negative number of products)

3. Inequalities:

   • 2x + y ≤ 12
   • x + 3y ≤ 15
   • x ≥ 0
   • y ≥ 0

4. Graphing: Graph all four inequalities. The solution region is the polygon formed by the intersection of the shaded regions. This represents the feasible production combinations.

5. Interpretation: Any point (x, y) within the solution region represents a feasible production plan that doesn't exceed the available resources.

Example 3: Grade Requirements

A student needs to achieve an average score of at least 80% on two exams to pass the course. The first exam has a weight of 40%, and the second exam has a weight of 60%. Model the possible scores on both exams that will ensure the student passes.

Solution:

1. Variables:

   • x: Score on the first exam (percentage).
   • y: Score on the second exam (percentage).

2. Constraints:

   • Weighted average ≥ 80
   • 0 ≤ x ≤ 100
   • 0 ≤ y ≤ 100

3. Inequalities:

   • 0.4x + 0.6y ≥ 80
   • 0 ≤ x ≤ 100
   • 0 ≤ y ≤ 100

4. Graphing: Graph these inequalities. The solution region will be bounded by the lines x=0, x=100, y=0, y=100 and the inequality representing the weighted average.

5. Interpretation: Any point within the solution region represents a combination of scores on both exams that will result in a passing grade.

Advanced Applications and Considerations:

Modeling two-variable systems of inequalities extends far beyond these basic examples. Here are some advanced applications and considerations:

• Linear Programming: This optimization technique uses systems of inequalities to find the maximum or minimum value of a linear objective function subject to constraints. This is widely used in business and operations research.
• Integer Programming: Extends linear programming by requiring the variables to be integers, reflecting scenarios where fractional solutions are not feasible (e.g., number of cars produced).
• Non-linear Inequalities: While this guide focuses on linear inequalities, systems involving non-linear inequalities (e.g., involving quadratic or exponential functions) can also be modeled and solved graphically, though the solution regions become more complex.
• Sensitivity Analysis: In real-world problems, the constraints might be uncertain. Sensitivity analysis explores how changes in the constraint values affect the solution region and the optimal solution.

Conclusion:

Mastering the art of modeling two-variable systems of inequalities is a fundamental skill for tackling diverse real-world problems. This process involves careful variable definition, constraint identification, inequality formulation, graphical representation, and solution interpretation. By understanding the underlying principles and practicing with various examples, you'll develop a strong foundation for solving complex problems across numerous disciplines. Remember to always check your work and ensure the solution aligns logically with the context of the problem. The more practice you get, the more comfortable and proficient you'll become in applying this powerful mathematical tool.