Notes on Systems of Linear Inequalities

Learning Outcomes

  • Understand how to write and graph a system of linear inequalities.

  • Model real-world situations using inequalities, particularly in sales scenarios.

Example Scenario: Cathy’s Ice Cream Sale

  • Cathy is selling two sizes of ice cream cones at a school fundraiser:

    • Small Cone: 1 scoop

    • Large Cone: 2 scoops

  • Price:

    • Small cone: $3

    • Large cone: $5

  • Sales Goal: Cathy aims to earn at least $120.

  • Scoop Limit: Maximum available scoops = 70.

Variables

  • Define the variables for the system:

    • s = number of small cones sold

    • l = number of large cones sold

System of Inequalities

  1. Scoop Constraint:
    The total scoops used for small (1 scoop each) and large cones (2 scoops each) cannot exceed 70 scoops:

    • Inequality: s + 2l \leq 70
      (Total scoops used must be less than or equal to 70)

  2. Sales Constraint:
    The total sales must be at least $120:

    • Inequality: 3s + 5l \geq 120
      (Total revenue from cones must be greater than or equal to 120)

  3. Non-negativity Constraints:
    Both quantities of cones sold must be non-negative:

    • Inequalities: s \geq 0, l \geq 0

Complete System

  • The complete system reflects all restrictions:
    \begin{cases} \ s + 2l \leq 70 \ 3s + 5l \geq 120 \ s \geq 0 \ l \geq 0 \end{cases}

Graphing the System

  • Axes Setup:

    • Graph the variable s (small cones) along the x-axis.

    • Graph the variable l (large cones) along the y-axis.

Step-by-Step Graphing:

  1. Graph the line for s + 2l = 70

    • Find the intercepts and plot points, then draw the line.

  2. Test points to determine which side of the line satisfies the inequality s + 2l ≤ 70.

  3. Graph the line for 3s + 5l = 120

    • Again, determine the intercepts and plot.

  4. Test points to shade the region satisfying 3s + 5l ≥ 120.

  5. Consider the non-negativity constraints (shading above the axes).

Finding Feasible Region

  • The solution to the system will lie in the overlapping shaded areas of the graphs.

  • Identify the region where both inequalities hold:

    • The purple region represents valid (s, l) combinations where Cathy meets her sales goals without exceeding her scoop limit.

Conclusion

  • As long as Cathy sells ice cream combinations represented by points in the purple region, she will successfully reach her financial goal and stay within scoop constraints.

  • Systems of linear inequalities can effectively model real-life scenarios such as sales targets and resource limits.