Lesson Plan: Systems of Inequalities

 

Lesson Summary

In this 50-minute lesson, students will learn to solve systems of inequalities graphically and identify feasible regions that satisfy all constraints. Students will use graphing techniques and real-world examples to solidify their understanding. Multimedia resources from Media4Math.com enhance the learning experience. The lesson concludes with a 10-question quiz accompanied by an answer key.

Lesson Objectives

  • Graph systems of inequalities in two variables.
  • Identify and interpret the feasible region for solutions.
  • Apply systems of inequalities to solve real-world problems.

Common Core Standards

  • CCSS.MATH.CONTENT.HSA.REI.D.12: Graph solutions to linear inequalities in two variables as a half-plane (excluding the boundary in the case of a strict inequality) and show that the solution set is the intersection of the corresponding half-planes.

Prerequisite Skills

  • Graphing linear inequalities.
  • Identifying solutions to single inequalities.
  • Basic understanding of systems of equations.

Key Vocabulary

  • Inequality: An inequality is a mathematical statement that compares two expressions using symbols such as <, >, ≤, or ≥. Unlike equations, inequalities show a range of possible solutions rather than a specific value.
  • Linear Inequality: A linear inequality is an inequality that involves a linear expression. It can have one or more variables, and its graph is a region of the coordinate plane, often bounded by a straight boundary line. For example, y > 2x + 3 is a linear inequality.
  • Feasible Region: In the context of graphing systems of inequalities, the feasible region is the area on the graph where the shaded regions of all inequalities overlap. This region represents all the possible solutions that satisfy all the inequalities in the system simultaneously.
  • Boundary Line: A boundary line is the line that represents the equation corresponding to an inequality. It divides the coordinate plane into two halves, indicating where the inequality is either true or false.
    • Solid Line: Used when the inequality includes equality (≤ or ≥), indicating that points on the line are included in the solution set.
    • Dashed Line: Used when the inequality is strict (< or >), indicating that points on the line are not included in the solution set.

Additional Multimedia Resources

 

 

Warm-Up Activities (10 minutes)

Choose one or more of these activities.

1. Graphing a Linear Inequality on Desmos

Objective: Students will practice graphing a linear inequality to review key concepts before tackling systems of linear inequalities.

  • Graph the following inequality: y ≤ 1/2x + 3.
  • Student Tasks:
    • Open Desmos Graphing Calculator.
    • Enter the inequality y ≤ 1/2x + 3 into Desmos.
    • Observe the shaded region and the boundary line on the graph.
  • Discussion Questions:
    • What does the shaded region represent?
    • Is the boundary line solid or dashed? Why?
    • Test a point not on the boundary line (e.g., (0, 0)) and determine if it satisfies the inequality.
  • Quick Challenge:
    • Modify the inequality to y > 1/2x - 2.
    • Graph the new inequality and compare the shading and boundary line to the previous graph.
    • Discuss the differences.
  • Classroom Debrief:
    • Bring students together for a 3-minute discussion to summarize:
      • The significance of solid vs. dashed boundary lines.
      • How to determine which side of the line is shaded.

2. Graphing Systems with Desmos

Objective: Refresh students' understanding of graphing lines and systems of equations.

Activity:

  • Students will use Desmos to explore and graph linear equations.
  • Provide the following set of equations for graphing:

y = 2x + 3

y = -x + 5

  • Instructions:
    • Graph each equation on Desmos.
    • Identify the point of intersection (if any) and discuss whether the system is consistent and independent, dependent, or inconsistent.
    • Experiment with changing the slopes and y-intercepts of each equation to observe how the graph and solution change.
  • Teacher’s Role: Facilitate exploration by prompting students with questions like:
    • "What happens to the intersection point when the slopes of both lines are the same?"
    • "How can we verify the solution using substitution or elimination?"

3. Class Discussion

Objective: Foster a deeper understanding of systems of inequalities and their solutions through collaborative discussion.

Scenario for Discussion:

  • Present the system of inequalities below:

y ≤ 2x + 1

y > -x + 3

  • Display the graph of the inequalities (using Desmos or a pre-made visual) and ask:
    • "What does the shaded region represent?"
    • "How can you determine the solution set visually?"
    • "What happens if we change the inequality from ≤ to <?"
  • Encourage students to analyze boundary lines, test points in the shaded region, and discuss the practical meaning of overlapping regions.

 

 

Teach

Example 1: Algebraic System of Inequalities

Problem: Solve the following system of inequalities:

y > 2x - 1 

y ≤ -x + 4

Solution:

  1. Graph the first inequality: \( y > 2x - 1 \)
    • Graph the boundary line \( y = 2x - 1 \) as a dashed line.
    • Test the point \( (0, 0) \): \( 0 > -1 \) (True). Shade above the line.

 

Systems of Inequalities

 

Graph the second inequality: \( y \leq -x + 4 \)

  • Graph the boundary line \( y = -x + 4 \) as a solid line.
  • Test the point \( (0, 0) \): \( 0 \leq 4 \) (True). Shade below the line.

 

Systems of Inequalities

 

Solution region: The overlapping shaded region is the solution.

 

Systems of Inequalities

 

Example 2: Another Algebraic System of Inequalities

Problem: Solve the following system of inequalities:

y ≤ 3x + 2 

y > -2x - 1

Solution:

Graph the first inequality: \( y \leq 3x + 2 \)

  • Graph the boundary line \( y = 3x + 2 \) as a solid line.
  • Test the point \( (0, 0) \): \( 0 \leq 2 \) (True). Shade below the line.

 

System of Inequalities

 

Graph the second inequality: \( y > -2x - 1 \)

  • Graph the boundary line \( y = -2x - 1 \) as a dashed line.
  • Test the point \( (0, 0) \): \( 0 > -1 \) (True). Shade above the line.Solution region: The overlapping shaded region is the solution.

 

System of Inequalities

 

Solution region: The overlapping shaded region is the solution.

 

System of Inequalities

 

Example 3: Real-World Scenario

Problem: A bakery produces two types of gourmet cookies: chocolate chip and oatmeal raisin. Each batch of chocolate chip cookies requires 2 cups of flour and 1 cup of sugar. Each batch of oatmeal raisin cookies requires 1 cup of flour and 1.5 cups of sugar. The bakery has at most 10 cups of flour and 9 cups of sugar. Write and solve a system of inequalities to determine how many of each type of cookie the bakery can produce.

Solution:

Define variables:

  • Let \( x \) represent a batch of chocolate chip cookies and \( y \) represent a batch of oatmeal raisin cookies.

Write the inequalities:

2x + y ≤ 10 (Flour constraint) 

x + 1.5y ≤ 9 (Sugar constraint) 

x ≥ 0, y ≥ 0 (Non-negativity)

Graph the inequalities:

  • Graph \( 2x + y ≤ 10 \) and \( x + 1.5y ≤ 9 \), shading below the lines.
  • Restrict the solution to the first quadrant (\( x ≥ 0, y ≥ 0 \)).

Find vertices of the feasible region:

  • Solve the system for intersections, e.g., \( (3, 4) \).

 

Systems of Inequalities

 

Solution: The bakery can produce up to 3 batches of chocolate chip cookies and 4 batches of oatmeal raisin cookies.

 

 

Review

Key Vocabulary

  • System of Linear Inequalities: A set of two or more linear inequalities that define constraints on the same variables.
  • Solution of a System of Inequalities: The overlapping shaded region on the graph, representing all possible solutions that satisfy all inequalities in the system.
  • Boundary Line: The line representing an inequality when the inequality symbol is replaced with an equal sign, used to divide the plane into regions.
  • Shading: The area on a graph that represents all possible solutions for a single inequality, determined based on test points or the inequality symbol.

Explanation of the Examples in the Teach Section

In the Teach section, the three examples solved systems of linear inequalities graphically, focusing on accurately shading regions and identifying the solution set:

  1. First Example: Introduced the basics of graphing a system of inequalities. Each inequality's boundary line was graphed (dashed for < or >, solid for ≤ or ≥), and the shaded regions were drawn. The solution set was identified as the area where all shaded regions overlapped.
  2. Second Example: Showed a system with inequalities that involved different slopes and intercepts. This example emphasized careful shading and the importance of testing points to confirm the correct regions.
  3. Third Example (Real-World Context): Presented a scenario involving baking cookies, where two types of cookies were constrained by ingredient availability. Each inequality represented a constraint, such as the maximum amount of flour or sugar. Graphing the system revealed the feasible region for the number of each type of cookie that could be baked. The solution set demonstrated how to optimize the baking process within given constraints, offering a practical application of systems of inequalities.

This progression of examples built understanding from foundational graphing skills to interpreting and solving real-world problems.

Additional Multimedia Resources

 

 

Quiz

  1. Solve and Graph: Graph the system of inequalities y ≤ 2x + 1 and y > -x - 3. Identify the solution region and describe its boundaries.
     
  2. True or False: The solution to a system of linear inequalities is always a single point. Explain your reasoning.
     
  3. Test a Point: For the system of inequalities y ≥ (1/2)x - 4 and x + y ≤ 3, is the point (2, -1) a solution? Show your work.
     
  4. Determine Overlap: Explain what happens if the shaded regions of two inequalities in a system do not overlap. What does this mean for the solution set?
     
  5. Rewrite in Standard Form: Rewrite the inequality y < -3x + 5 in standard form Ax + By < C, and describe the direction of the shading on the graph.
     
  6. Baking Constraints: A baker is making batches of brownies and cookies. Each batch of brownies uses 3 cups of flour, and each batch of cookies uses 2 cups. The baker has 18 cups of flour available. They need to make at least 2 batches of brownies and at least 3 batches of cookies. Write a system of inequalities to represent the constraints, graph the solution, and describe the feasible region.
     
  7. Gardening Constraints: A gardener is planting flowers in a rectangular garden bed. They want to plant at least 10 roses and at least 5 tulips, but the total number of flowers cannot exceed 20. Write and graph a system of inequalities to represent this situation.
     
  8. Time Management: A student is preparing for two exams. They can spend no more than 6 hours studying. They want to spend at least 2 hours on math and at least 1 hour on history. Write a system of inequalities to represent the student’s study plan and graph the solution region.
     
  9. Sports Practice: A basketball player practices shooting and dribbling each day. They want to spend at least 30 minutes on each activity, but their total practice time cannot exceed 2 hours. Write and graph a system of inequalities to represent this scenario, and describe a possible solution.
     
  10. Business Optimization: A bakery makes muffins and scones. Each muffin uses 2 cups of flour, and each scone uses 1 cup. The bakery has 16 cups of flour available. They want to make at least 4 muffins and at least 6 scones. Write a system of inequalities to represent the constraints, graph the solution, and describe the feasible region.

Answer Key

  1. Solve and Graph: The solution region is the overlapping area of the graphs. For y ≤ 2x + 1, shade below the solid line. For y > -x - 3, shade above the dashed line. The boundary lines intersect at a specific point, which is part of the solution set.
  2. True or False: False. The solution to a system of linear inequalities is a region, not a single point, as it includes all points that satisfy all inequalities in the system.
  3. Test a Point: For y ≥ (1/2)x - 4, substituting (2, -1) gives -1 ≥ (1/2)(2) - 4, which simplifies to -1 ≥ -3 (true). For x + y ≤ 3, substituting (2, -1) gives 2 + (-1) ≤ 3, which simplifies to 1 ≤ 3 (true). Therefore, (2, -1) is a solution.
  4. Determine Overlap: If the shaded regions of the inequalities do not overlap, the system has no solution. This indicates that no points satisfy all the inequalities simultaneously.
  5. Rewrite in Standard Form: Rewrite y < -3x + 5 as 3x + y < 5. The shading direction is below the dashed line because the inequality is "<".

  6. Baking Constraints: The system of inequalities is:

    • 3b + 2c ≤ 18 (flour constraint)
    • b ≥ 2 (minimum batches of brownies)
    • c ≥ 3 (minimum batches of cookies)

    Graph the constraints and identify the feasible region. The solution region includes all points satisfying these inequalities, representing valid combinations of batches of brownies and cookies.

  7. Gardening Constraints: The system of inequalities is:

    • r + t ≤ 20 (total flowers)
    • r ≥ 10 (minimum roses)
    • t ≥ 5 (minimum tulips)

    Graph the constraints and shade the region where all inequalities overlap. The feasible region represents valid flower combinations.

  8. Time Management: The system of inequalities is:

    • m + h ≤ 6 (total study time)
    • m ≥ 2 (minimum math time)
    • h ≥ 1 (minimum history time)

    Graph the constraints and identify the feasible region. A valid solution is any point within the overlap, such as spending 3 hours on math and 2 hours on history.

  9. Sports Practice: The system of inequalities is:

    • s + d ≤ 120 (total practice time in minutes)
    • s ≥ 30 (minimum shooting time)
    • d ≥ 30 (minimum dribbling time)

    Graph the constraints and identify the feasible region. A possible solution is spending 45 minutes on shooting and 45 minutes on dribbling.

  10. Business Optimization: The system of inequalities is:

    • 2m + s ≤ 16 (flour constraint)
    • m ≥ 4 (minimum muffins)
    • s ≥ 6 (minimum scones)

    Graph the constraints and identify the feasible region. A possible solution is baking 4 muffins and 6 scones.