5.5 Applications of Systems of Inequalities
Model real-world constraints with systems of inequalities, graph feasible regions, and interpret solutions in context.
Why This Matters
Applying systems of inequalities to real problems — like maximizing profit within budget and resource constraints — is the core of linear programming, used in supply chain management, finance, and operations research.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How can systems of inequalities model real-world constraints, and how do we use the feasible region to make decisions?
Lesson Overview
In previous chapters you learned to graph systems of inequalities and identify feasible regions. Now you will apply those skills to real-world modeling — translating word problems into systems of inequalities, graphing the constraints, and interpreting the solution in context. Real-world problems always come with constraints — limits on resources, time, money, or quantities. The feasible region is the set of all combinations that satisfy every constraint simultaneously. Any point inside the feasible region is a valid solution.
The 6-Step Modeling Process
Read & Identify
Underline key quantities and constraints
Define Variables
Let x = ___ and y = ___
Write Inequalities
Translate each constraint into an inequality
Graph the System
Draw boundary lines; shade each half-plane
Identify Feasible Region
Find the overlap of all shaded regions
Interpret the Solution
Answer the question in context
Budget constraint region: 4x + 2y ≤ 60, x ≥ 5 (scaled)
Corner Point Method
To find the optimal solution in a bounded feasible region:
- Find all vertices (corner points) of the feasible region.
- Substitute each vertex into the objective function.
- The maximum or minimum value occurs at a vertex.
Used in linear programming to optimize real-world decisions.
Keyword → Inequality Symbol Reference
| Words / Context | Variable | Inequality |
|---|---|---|
| at most / no more than / does not exceed | quantity | ≤ |
| at least / no less than / minimum | quantity | ≥ |
| more than / exceeds / greater than | quantity | > |
| fewer than / less than / under | quantity | < |
| exactly / is / equals | quantity | = |
Worked Examples
Fundraising feasible region: 3x + 2y ≥ 24 and x + y ≤ 15
A student has at most $60 to spend on notebooks ($4 each) and pens ($2 each). She wants at least 5 notebooks. Write and graph the system. Identify the feasible region.
Define variables: x = number of notebooks, y = number of pens.
Budget constraint: 4x + 2y ≤ 60.
Minimum notebooks: x ≥ 5.
Non-negativity: x ≥ 0, y ≥ 0.
Graph both boundary lines; shade below 4x+2y=60 and to the right of x=5.
The feasible region is the overlap — bounded on the left by x=5 and above by 4x+2y=60.
Budget Constraint
Shade below 4x+2y=60
Both Constraints
Feasible region: overlap
A factory makes chairs (x, 2 hrs each) and tables (y, 5 hrs each). At most 40 labor hours per day. Must make at least 3 chairs and at least 2 tables. Write and graph the system.
Define: x = chairs per day, y = tables per day.
Labor: 2x + 5y ≤ 40.
Minimum chairs: x ≥ 3. Minimum tables: y ≥ 2.
Non-negativity: x ≥ 0, y ≥ 0.
Graph all boundary lines and shade the feasible overlap.
Test vertex (3, 2): 2(3)+5(2)=16 ≤ 40 ✓; 3 ≥ 3 ✓; 2 ≥ 2 ✓ — minimum production plan.
| Constraint | Words | Inequality |
|---|---|---|
| Labor hours | At most 40 hours total | 2x + 5y ≤ 40 |
| Minimum chairs | At least 3 chairs | x ≥ 3 |
| Minimum tables | At least 2 tables | y ≥ 2 |
| Non-negativity | Cannot make negative items | x ≥ 0, y ≥ 0 |
A school club sells cookies (x, $3 each) and brownies (y, $2 each). They need to raise at least $24 and can bake at most 15 items total. Write and graph the system. Is (4, 6) a solution?
Revenue constraint: 3x + 2y ≥ 24.
Total items: x + y ≤ 15.
Non-negativity: x ≥ 0, y ≥ 0.
Test (4, 6): 3(4)+2(6)=12+12=24 ≥ 24 ✓; 4+6=10 ≤ 15 ✓.
Both constraints satisfied — (4, 6) is in the feasible region.
A delivery company uses vans (x, 500 kg each) and trucks (y, 2000 kg each). They need to transport at least 6000 kg using at most 8 vehicles. Write and graph the system.
Define: x = vans, y = trucks.
Capacity: 500x + 2000y ≥ 6000.
Vehicle limit: x + y ≤ 8.
Non-negativity: x ≥ 0, y ≥ 0.
Graph both lines; shade above capacity line and below vehicle limit line.
Test (4, 4): 500(4)+2000(4)=10000 ≥ 6000 ✓; 4+4=8 ≤ 8 ✓ — valid fleet.
Capacity ≥ 6000 kg
Shade above capacity line
Both Constraints
Overlap = valid fleet combos
Error analysis: A student wrote x + y ≤ 10 and 8x + 5y ≥ 100 for 'sell at least 10 tickets and collect at most $100.' Identify and correct both errors.
Error 1: "at least 10 tickets" → x + y ≥ 10 (student wrote ≤ — reversed).
Error 2: "at most $100" → 8x + 5y ≤ 100 (student wrote ≥ — reversed).
Correct system: x + y ≥ 10 and 8x + 5y ≤ 100.
Verify with (5, 6): 5+6=11 ≥ 10 ✓; 8(5)+5(6)=40+30=70 ≤ 100 ✓.
Guided Practice
Guided Practice Video: Applications of Systems of Inequalities
Watch the guided practice walkthrough for applications of systems of inequalities, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗A student has $30 to spend on apples ($1 each) and oranges ($2 each). She wants at least 5 apples. Write the system of inequalities and identify one valid combination.
Hint: Let x = apples, y = oranges. Budget: x + 2y ≤ 30. Min apples: x ≥ 5. Test (5, 10): 5+20=25 ≤ 30 ✓; 5 ≥ 5 ✓.
A gym offers yoga (x) and spin (y) classes. Each yoga class needs 1 instructor; each spin class needs 2 instructors. At most 10 instructors available. Must offer at least 2 of each. Write and graph the system.
Hint: Instructor constraint: x + 2y ≤ 10. Min yoga: x ≥ 2. Min spin: y ≥ 2. Graph all three and shade the feasible overlap.
A bakery makes muffins (x, 0.5 hrs each) and cupcakes (y, 1 hr each). At most 8 hours available. Must make at least 4 muffins. Write the system and test whether (6, 5) is feasible.
Hint: Time: 0.5x + y ≤ 8. Min muffins: x ≥ 4. Test (6, 5): 0.5(6)+5=3+5=8 ≤ 8 ✓; 6 ≥ 4 ✓ — feasible.
A company ships boxes (x, 20 kg each) and crates (y, 50 kg each). Truck capacity: at most 400 kg. Must ship at least 3 boxes and at least 2 crates. Write and graph the system.
Hint: Weight: 20x + 50y ≤ 400 → divide by 10: 2x + 5y ≤ 40. Min boxes: x ≥ 3. Min crates: y ≥ 2.
A farmer plants wheat (x acres, $200/acre) and barley (y acres, $150/acre). At most 20 acres total. Wants to earn at least $3000. Write the system and determine whether (10, 8) is feasible.
Hint: Acreage: x + y ≤ 20. Revenue: 200x + 150y ≥ 3000. Test (10, 8): 18 ≤ 20 ✓; 200(10)+150(8)=3200 ≥ 3000 ✓ — feasible.
Key Vocabulary
Constraint
A condition expressed as an inequality that limits the values of the variables in a real-world problem.
Feasible Region
The set of all ordered pairs that satisfy every inequality in the system — the overlapping shaded area on the graph.
Mathematical Model
A mathematical representation (equations, inequalities, graphs) of a real-world situation.
Objective
The goal of the problem — what you are trying to maximize, minimize, or simply satisfy.
Non-negativity Constraint
The constraints x ≥ 0 and y ≥ 0, which restrict solutions to the first quadrant in most real-world problems.
Vertex of the Feasible Region
A corner point of the feasible region, found by solving pairs of boundary equations simultaneously.
Bounded Region
A feasible region that is enclosed on all sides — it has a finite area.
Unbounded Region
A feasible region that extends infinitely in at least one direction.
Interactive Practice — 5 Questions
A problem states "at most 20 items." Which inequality correctly represents x + y?
Which point is in the feasible region of x + y ≤ 10 and x ≥ 3 and y ≥ 2?
A student earns $12/hr mowing (x) and $9/hr washing cars (y). She wants at least $54. Which inequality models this?
Why do most real-world systems include x ≥ 0 and y ≥ 0?
A feasible region is bounded. What does this mean in context?
Independent Practice
Independent Practice
A student buys sandwiches ($6) and drinks ($2). She has at most $24 and wants at least 2 sandwiches. Define variables, write the system, and identify one valid combination.
A factory makes product A (3 hrs each) and product B (2 hrs each). It has 18 labor hours and must make at least 2 of each. Write and graph the system.
A club sells adult tickets ($5) and student tickets ($3). They need to raise at least $45 and sell at most 20 tickets. Write the system and test whether (6, 9) is feasible.
A chef uses ingredient X (200 cal/cup) and ingredient Y (100 cal/cup). The dish must have at most 600 calories and at least 3 cups total. Write the system and identify two valid combinations.
A delivery service uses cars (100 kg capacity) and vans (300 kg capacity). They need to move at least 900 kg using at most 6 vehicles. Write the system and verify one feasible fleet combination.
Common Mistakes
Writing an equation (=) instead of an inequality for a constraint that says 'at most', 'at least', or 'no more than'.
'At most' → ≤. 'At least' → ≥. Use = only when the problem says 'exactly'.
Forgetting non-negativity constraints (x ≥ 0, y ≥ 0) for real-world quantities.
Physical quantities like items, hours, or dollars can't be negative. Always include x ≥ 0 and y ≥ 0 unless the context allows negatives.
Picking a test point and only checking one constraint instead of all of them.
A valid solution must satisfy every inequality in the system. Check the point in each constraint separately.
Reporting the feasible region as a single answer point instead of describing the region.
The solution to a system of inequalities is the entire feasible region — all points that satisfy every constraint.
Math Tips
Always define your variables before writing inequalities — state what x and y represent, including units.
Most real-world problems include x ≥ 0 and y ≥ 0 because you cannot have negative quantities of a physical item.
After graphing, test a point in the feasible region to verify it satisfies all inequalities.
The boundary line is solid (≤ or ≥) when the constraint allows equality; dashed (< or >) when it does not.
The Corner Point Method: in a bounded feasible region, the optimal value of an objective function always occurs at a vertex.