Systems Of Linear Inequalities Word Problems

Delving into the world of systems of linear inequalities word problems opens a fascinating window into the practical applications of mathematics. These problems challenge us to translate real-world scenarios into mathematical models, providing a framework for optimizing decisions under various constraints.

Understanding the Basics

Before tackling complex word problems, it’s essential to grasp the fundamentals of linear inequalities. A linear inequality is a mathematical statement that compares two expressions using inequality symbols such as < (less than), > (greater than), ≤ (less than or equal to), and ≥ (greater than or equal to). A system of linear inequalities involves two or more linear inequalities considered simultaneously.

The solution to a system of linear inequalities is the set of all points that satisfy all the inequalities in the system. Graphically, this solution is represented by the overlapping region of the individual inequalities' graphs. This region is often referred to as the feasible region.

Steps to Solve Systems of Linear Inequalities Word Problems

Solving these types of problems involves a structured approach that combines algebraic manipulation with logical reasoning. Here's a step-by-step guide:

1. Read and Understand: Carefully read the problem statement. Identify the key quantities, constraints, and the objective you need to achieve.
2. Define Variables: Assign variables to represent the unknown quantities. For example, let x represent the number of units of product A and y represent the number of units of product B.
3. Translate into Inequalities: Convert the word problem's constraints into mathematical inequalities. Look for keywords like "at least," "at most," "minimum," "maximum," "no more than," and "no less than" to help you determine the correct inequality symbol.
4. Graph the Inequalities: Graph each inequality on the coordinate plane. Remember to use a solid line for ≤ and ≥, and a dashed line for < and >. Shade the region that satisfies each inequality.
5. Identify the Feasible Region: The feasible region is the area where all the shaded regions overlap. This region represents all the possible solutions that satisfy all the constraints.
6. Find the Corner Points: Determine the coordinates of the corner points of the feasible region. These points are crucial for optimization problems.
7. Define the Objective Function (if applicable): If the problem asks you to maximize or minimize a certain quantity (e.g., profit, cost), define an objective function that expresses this quantity in terms of the variables.
8. Evaluate the Objective Function: Substitute the coordinates of each corner point into the objective function. The point that yields the maximum or minimum value, depending on the problem's objective, is the optimal solution.
9. Interpret the Solution: Translate the mathematical solution back into the context of the original word problem. State your answer in a clear and concise manner.

Illustrative Examples

Let's work through some examples to solidify your understanding:

Example 1: Production Planning

A small furniture company produces chairs and tables. Each chair requires 2 hours of assembly and 1 hour of finishing. Each table requires 3 hours of assembly and 2 hours of finishing. The company has 24 hours available for assembly and 16 hours available for finishing per day. They want to determine the number of chairs and tables they can produce to maximize their profit, assuming the profit is $80 per chair and $120 per table.

1. Define Variables:

   • Let x = number of chairs
   • Let y = number of tables

2. Translate into Inequalities:

   • Assembly constraint: 2x + 3y ≤ 24
   • Finishing constraint: x + 2y ≤ 16
   • Non-negativity constraints: x ≥ 0, y ≥ 0 (The company cannot produce a negative number of chairs or tables)

3. Graph the Inequalities: Graph the inequalities on the coordinate plane.

4. Identify the Feasible Region: Find the overlapping region that satisfies all the inequalities.

5. Find the Corner Points: Determine the coordinates of the corner points of the feasible region. These points will be (0, 0), (0, 8), (12, 0), and the intersection of the two lines 2x + 3y = 24 and x + 2y = 16. To find the intersection, solve the system of equations:

   • 2x + 3y = 24
   • x + 2y = 16 => x = 16 - 2y Substitute x in the first equation:
   • 2(16 - 2y) + 3y = 24
   • 32 - 4y + 3y = 24
   • -y = -8
   • y = 8 Substitute y = 8 in x = 16 - 2y:
   • x = 16 - 2(8) = 0 However, this is incorrect. Solving the system properly: Multiply the second equation by -2: -2x - 4y = -32 Add this to the first equation: 2x + 3y = 24 This gives: -y = -8 y = 8 Substituting y = 8 into x + 2y = 16: x + 2(8) = 16 x + 16 = 16 x = 0 This is one corner point. We made an error previously. Let's find the correct intersection point. Using elimination:

   Multiply the second equation (x + 2y = 16) by -2: -2x - 4y = -32 Add this to the first equation (2x + 3y = 24): -y = -8 y = 8

   Now substitute y = 8 into the equation x + 2y = 16: x + 2(8) = 16 x + 16 = 16 x = 0

   It seems the lines intersect on an axis, where x = 0 and y = 8. This means other intersection points must exist at the axes. Corner points: (0,0), (12,0), (0,8). Let's check another possible point.

   Multiply the second equation (x + 2y = 16) by -3/2: -3/2 x - 3y = -24 Add to the first equation (2x + 3y = 24). (2 - 3/2)x = 0 (1/2) * x = 0 x = 0. This again yields y=8!

   Let's try substitution properly:

   Solve x + 2y = 16 for x: x = 16 - 2y

   Substitute into the first equation, 2x + 3y = 24: 2(16 - 2y) + 3y = 24 32 - 4y + 3y = 24 -y = -8 y = 8

   x = 16 - 2(8) = 0. So (0,8) is confirmed.

   Now find where 2x + 3y = 24 crosses the x axis: 2x + 3(0) = 24 x = 12

   And where x + 2y = 16 crosses the x axis: x + 2(0) = 16 x = 16

   Now find where 2x + 3y = 24 crosses the y axis: 2(0) + 3y = 24 y = 8

   And where x + 2y = 16 crosses the y axis: (0) + 2y = 16 y = 8

   The crucial intersection of the lines is (0,8). The other corner points are (0,0) and (12,0).

6. Define the Objective Function:

   • Profit = 80x + 120y

7. Evaluate the Objective Function:

   • At (0, 0): Profit = 80(0) + 120(0) = $0
   • At (12, 0): Profit = 80(12) + 120(0) = $960
   • At (0, 8): Profit = 80(0) + 120(8) = $960

8. Interpret the Solution:

   The company can maximize its profit by producing either 12 chairs and 0 tables or 0 chairs and 8 tables. In both cases, the maximum profit is $960.

Example 2: Dietary Requirements

A nutritionist is creating a meal plan with two food items: A and B. Each unit of food A contains 2 grams of protein and 5 grams of carbohydrates. Each unit of food B contains 3 grams of protein and 2 grams of carbohydrates. The nutritionist wants the meal to contain at least 10 grams of protein and at least 15 grams of carbohydrates. Food A costs $2 per unit, and Food B costs $3 per unit. The nutritionist wants to minimize the cost.

1. Define Variables:

   • Let x = number of units of food A
   • Let y = number of units of food B

2. Translate into Inequalities:

   • Protein constraint: 2x + 3y ≥ 10
   • Carbohydrate constraint: 5x + 2y ≥ 15
   • Non-negativity constraints: x ≥ 0, y ≥ 0

3. Graph the Inequalities: Graph the inequalities on the coordinate plane.

4. Identify the Feasible Region: Find the overlapping region that satisfies all the inequalities.

5. Find the Corner Points: Determine the coordinates of the corner points of the feasible region. These points will be the intersections of the lines 2x + 3y = 10, 5x + 2y = 15, x=0 and y=0. When x=0 in 2x + 3y = 10, y = 10/3. When y=0 in 5x + 2y = 15, x = 3.

   We solve the system 2x + 3y = 10 5x + 2y = 15

   Multiply the first equation by -5 and the second by 2: -10x - 15y = -50 10x + 4y = 30 Adding them gives -11y = -20 y = 20/11

   Substituting into 2x + 3y = 10 2x + 3(20/11) = 10 2x = 10 - 60/11 = 50/11 x = 25/11

   So the corner points are (3,0), (0, 10/3) and (25/11, 20/11).

6. Define the Objective Function:

   • Cost = 2x + 3y

7. Evaluate the Objective Function:

   • At (3, 0): Cost = 2(3) + 3(0) = $6
   • At (0, 10/3): Cost = 2(0) + 3(10/3) = $10
   • At (25/11, 20/11): Cost = 2(25/11) + 3(20/11) = 50/11 + 60/11 = 110/11 = $10

8. Interpret the Solution:

   The nutritionist can minimize the cost to $6 by using 3 units of food A and 0 units of food B.

Example 3: Investment Strategy

An investor wants to invest in two types of bonds: A and B. Bond A yields 5% interest per year, and Bond B yields 8% interest per year. The investor wants to invest at least $10,000 in total, and they want the annual interest income to be at least $600.

1. Define Variables:

   • Let x = amount invested in Bond A
   • Let y = amount invested in Bond B

2. Translate into Inequalities:

   • Total investment constraint: x + y ≥ 10000
   • Interest income constraint: 0.05x + 0.08y ≥ 600
   • Non-negativity constraints: x ≥ 0, y ≥ 0

3. Graph the Inequalities: Graph the inequalities on the coordinate plane.

4. Identify the Feasible Region: Find the overlapping region that satisfies all the inequalities.

5. Find the Corner Points: Determine the coordinates of the corner points of the feasible region. When x=0 in x+y = 10000, then y=10000. When y=0, x=10000. When x=0 in .05x + .08y = 600, .08y = 600, y=7500. When y=0, .05x=600, x=12000.

   To solve: x+y = 10000 .05x + .08y = 600 Multiply the top by -.05: -.05x - .05y = -500 Add to the bottom: .03y = 100 y = 100/.03 = 10000/3 = 3333.33 x + 3333.33 = 10000 x = 6666.67

   So the corner points are (12000, 0), (0, 10000) and (6666.67, 3333.33)

6. Define the Objective Function (Assuming minimization of investment):

   • Total investment = x + y (we already have that inequality)

   If the goal is to minimize the total investment, this simplifies to choosing the leftmost solution in the region.

7. Evaluate the Objective Function:

   • At (12000, 0): Total investment = 12000
   • At (0, 10000): Total investment = 10000
   • At (6666.67, 3333.33): Total investment = 10000

   Since we need x + y >= 10000, all these points satisfy it.

   Let's assume a DIFFERENT objective. MAXIMIZE the investment in bond B.

8. • New objective function: MAXIMIZE y.

9. The maximum value of y within our feasible region occurs at (0, 10000).

10. Interpret the solution. To maximize investment in Bond B (and still satisfy all the constraints), invest $0 in Bond A and $10,000 in Bond B. This fulfills the total investment requirement and the minimum income requirement. Key Considerations and Common Mistakes

• Non-Negativity Constraints: Always remember to include non-negativity constraints (x ≥ 0, y ≥ 0) when dealing with real-world quantities that cannot be negative.
• Accurate Graphing: Precise graphing is crucial for identifying the correct feasible region and corner points. Use graph paper or graphing software to ensure accuracy.
• Correct Inequality Symbols: Pay close attention to the wording of the problem to determine the correct inequality symbols.
• Interpreting the Solution: Always translate the mathematical solution back into the context of the original word problem. Make sure your answer makes sense in the real world.
• Unbounded Feasible Regions: Sometimes the feasible region is unbounded. In these cases, the objective function may not have a maximum or minimum value.

Advanced Techniques and Applications

While the basic steps outlined above are sufficient for many problems, some situations may require more advanced techniques.

• Linear Programming: When dealing with more complex systems of inequalities and objective functions, linear programming techniques can be used to find optimal solutions. The simplex method is a common algorithm used in linear programming.
• Sensitivity Analysis: This involves examining how changes in the constraints or objective function affect the optimal solution.
• Integer Programming: In some cases, the variables must be integers (e.g., you can't produce half a chair). Integer programming techniques are used to solve these types of problems.

Systems of linear inequalities word problems have numerous applications in various fields:

• Business and Economics: Production planning, resource allocation, portfolio optimization, transportation logistics.
• Nutrition and Dietetics: Meal planning, optimizing nutrient intake.
• Engineering: Design optimization, resource management.
• Environmental Science: Resource management, pollution control.

FAQ

Q: What is the feasible region?

A: The feasible region is the area on a graph that represents all possible solutions to a system of linear inequalities. It's the area where the shaded regions of all the inequalities overlap.

Q: What are corner points?

A: Corner points are the vertices of the feasible region. They are the points where the boundary lines of the inequalities intersect. They are critical because the optimal solution (maximum or minimum) to the objective function always occurs at a corner point (or along an edge between two corner points).

Q: How do I know which inequality symbol to use?

A: Look for keywords in the problem statement. "At least" or "no less than" indicates ≥. "At most" or "no more than" indicates ≤. "Greater than" indicates >, and "less than" indicates <.

Q: What if the feasible region is empty?

A: If the feasible region is empty, it means there is no solution that satisfies all the inequalities in the system. This could indicate an error in the problem setup or that the constraints are incompatible.

Q: Can I use a calculator to solve these problems?

A: Yes, graphing calculators or online graphing tools can be helpful for graphing the inequalities and finding the corner points. However, it's important to understand the underlying concepts and be able to set up the problem correctly.

Conclusion

Mastering systems of linear inequalities word problems is not just about learning mathematical techniques; it's about developing critical thinking and problem-solving skills that are applicable in many aspects of life. By understanding the fundamentals, practicing regularly, and applying a structured approach, you can confidently tackle these problems and unlock their potential for real-world applications. Embrace the challenge, and you'll find that the world of linear inequalities offers a powerful tool for decision-making and optimization. These problems are a fantastic way to bridge the gap between abstract mathematical concepts and the tangible realities of our world, enabling us to make informed and effective choices in a variety of situations.