5.3Writing Systems of Inequalities
Translate real-world constraints into systems of linear inequalities. Learn to identify variables, extract conditions, choose the correct inequality symbol, and model situations from ticket sales to budgets to geometry.
Why This Matters
Translating real-world constraints into inequalities is a skill used in business planning, resource allocation, and engineering design. This is the bridge between abstract algebra and practical decision-making.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
How do you translate the constraints of a real-world situation into a system of linear inequalities that can be solved mathematically?
How Do You Write a System of Inequalities?
To write a system of inequalities, identify each boundary line, determine whether the boundary is included, and decide which side of each line should be shaded. Write one inequality for each condition. The solution to the system is the region where all shaded areas overlap.
Lesson Overview
In Chapters 1 and 2 you learned to check solutions and graph systems of inequalities. In this chapter you will learn to write them from scratch. Real-world problems rarely hand you a ready-made inequality — you must read carefully, identify what is unknown, find every constraint, and translate each one into mathematical language. This skill — called mathematical modeling — is one of the most important in all of algebra.
Budget region: 2x + 4y ≤ 16, y ≥ 0 (first quadrant)
Define Variables
Always define variables before writing inequalities.
Solid Lines and Dashed Lines
Solid Line — ≤ or ≥
Use a solid boundary line when the inequality uses ≤ or ≥. Points on the line are included in the solution. A solid line means "equal to" is included.
Example: y ≤ x + 1 → solid line at y = x + 1
Dashed Line — < or >
Use a dashed boundary line when the inequality uses < or >. Points on the line are not included in the solution. A dashed line means "equal to" is not included.
Example: y > −2x + 4 → dashed line at y = −2x + 4
How Do You Know Which Side to Shade?
Shading Rules for y = mx + b Form
y > mx + b or y ≥ mx + b
Shade above the boundary line.
y < mx + b or y ≤ mx + b
Shade below the boundary line.
Not solved for y? Use a test point.
Substitute a point not on the boundary line — (0, 0) works unless the line passes through the origin. If the test point makes the inequality true, shade the side containing that point. If it makes the inequality false, shade the opposite side.
⚠ Important: If you multiply or divide an inequality by a negative number, reverse the inequality sign.
Example: −2y > 6 → divide by −2 → y < −3 (sign flips)
Inequality Keywords Reference
Inequality Keywords → Symbols
| Keyword / Phrase | Symbol | Example |
|---|---|---|
| at most | ≤ | "You can spend at most $50" → cost ≤ 50 |
| no more than | ≤ | "No more than 10 items" → items ≤ 10 |
| maximum | ≤ | "Maximum capacity of 200" → people ≤ 200 |
| at least | ≥ | "At least 3 hours of study" → hours ≥ 3 |
| no less than | ≥ | "No less than $15 per hour" → wage ≥ 15 |
| minimum | ≥ | "Minimum order of 5" → order ≥ 5 |
| more than | > | "More than 100 points" → points > 100 |
| greater than | > | "Greater than 0" → x > 0 |
| less than | < | "Less than 60 minutes" → time < 60 |
| fewer than | < | "Fewer than 20 students" → students < 20 |
| exceeds | > | "Exceeds the budget" → cost > budget |
| does not exceed | ≤ | "Does not exceed 500" → total ≤ 500 |
| is within | ≤ and ≥ | "Temperature is within 5° of 70°" → 65 ≤ T ≤ 75 |
| between (inclusive) | ≤ and ≥ | "Between 2 and 8 inclusive" → 2 ≤ x ≤ 8 |
| cannot exceed | ≤ | "Cannot exceed 30 mph" → speed ≤ 30 |
Decision Tree — Choosing the Right Symbol
Decision Tree — Which Inequality Symbol?
LIMIT / MAXIMUM
"at most", "no more than", "maximum", "cannot exceed", "up to"
less than or equal to
REQUIREMENT / MINIMUM
"at least", "no less than", "minimum", "must have", "need at least"
greater than or equal to
STRICT LESS THAN
"less than", "fewer than", "below", "under", "before"
strictly less than
STRICT GREATER THAN
"more than", "greater than", "above", "over", "exceeds"
strictly greater than
The 8-Step Modeling Process
Read the problem
Read it twice. Identify what is being asked.
Identify the unknowns
What quantities are changing? These become your variables.
Define variables
Write "Let x = ..." and "Let y = ..." with units.
Extract constraints
Find every condition: limits, totals, minimums, maximums.
Identify inequality keywords
at most → ≤, at least → ≥, more than → >, less than → <
Write each inequality
Translate each constraint into an inequality using your variables.
Check reasonableness
Do the inequalities make sense? Are there implied constraints (x ≥ 0)?
Solve and interpret
Graph or solve the system. State the answer in context.
Words → Variables → Inequalities
Words
You can spend at most $80 on supplies.
Variable
c = total cost ($)
Inequality
c ≤ 80
Words
The club needs at least 15 members.
Variable
m = number of members
Inequality
m ≥ 15
Words
The car travels more than 60 miles per hour.
Variable
s = speed (mph)
Inequality
s > 60
Words
Fewer than 200 people can fit in the room.
Variable
p = number of people
Inequality
p < 200
Words
The temperature must not exceed 98.6°F.
Variable
T = temperature (°F)
Inequality
T ≤ 98.6
Error Analysis
Error Analysis
Student's Work ✗
'At most $50' → cost ≥ 50
The Error
'At most' means the value is a MAXIMUM — the cost cannot exceed $50. The correct symbol is ≤, not ≥.
Correct Version ✓
cost ≤ 50
Error Analysis
Student's Work ✗
'At least 10 items' → items < 10
The Error
'At least' means the value is a MINIMUM — you need 10 or more. The correct symbol is ≥, not <.
Correct Version ✓
items ≥ 10
Error Analysis
Student's Work ✗
'x = number of tickets; total ≤ 200' written as: x ≤ 200 (only one variable)
The Error
The problem has TWO types of tickets (adult and student). Two unknowns require two variables: x + y ≤ 200.
Correct Version ✓
x + y ≤ 200 (where x = adult tickets, y = student tickets)
Worked Examples
Ticket sales feasible region: x + y ≤ 300, y ≥ 5 (example)
Budget: You have at most $60 to spend on notebooks (x) at $3 each and pens (y) at $1.50 each. You need at least 5 notebooks and at least 4 pens. Write a system of inequalities.
Define variables: x = number of notebooks; y = number of pens.
Constraint 1 (budget): 3x + 1.50y ≤ 60.
Constraint 2 (notebooks): x ≥ 5.
Constraint 3 (pens): y ≥ 4.
Verify (6, 4): 3(6)+1.5(4)=18+6=24 ≤ 60 ✓; 6 ≥ 5 ✓; 4 ≥ 4 ✓.
Problem
You have at most $60 to spend on notebooks ($3 each) and pens ($1.50 each). You need at least 5 notebooks and at least 4 pens.
Step 1 — Define Variables
Define Variables
Shopping scenario
Step 2 — Extract Constraints
Constraints
| Budget limit | ⭐at most $60 |
| Notebook price | ⭐$3 each |
| Pen price | ⭐$1.50 each |
| Min notebooks | ⭐at least 5 |
| Min pens | ⭐at least 4 |
Step 3 — Write the System
System of Inequalities
3x + 1.5y ≤ 60x ≥ 5y ≥ 43 constraints → 3 inequalities
Step 4 — Solve & Interpret
Verify (6, 4)
| Item | Qty | Price | Subtotal |
|---|---|---|---|
| Notebooks | 6 | $3.00 | $18.00 |
| Pens | 4 | $1.50 | $6.00 |
| Total cost | ≤ $60 ✓ | ||
Ticket sales: A school play sells adult tickets for $8 and student tickets for $5. The auditorium holds at most 300 people. The drama club needs to earn at least $1,500. Write a system of inequalities.
Define variables: x = number of adult tickets sold; y = number of student tickets sold.
Constraint 1 (capacity): x + y ≤ 300.
Constraint 2 (revenue): 8x + 5y ≥ 1500.
Implied constraints: x ≥ 0 and y ≥ 0.
Verify (100, 150): 100+150=250 ≤ 300 ✓; 8(100)+5(150)=800+750=1550 ≥ 1500 ✓.
Ticket Sales Model
| Ticket Type | Quantity | Price | Revenue |
|---|---|---|---|
| Adult | x | $8 | 8x |
| Student | y | $5 | 5y |
| Total revenue needed | ≥ $1,500 | ||
Example 1: Write a System from a Graph
A coordinate plane shows two boundary lines. Boundary 1 is a solid line along y = x + 1 with the region shaded below. Boundary 2 is a dashed line along y = −2x + 4 with the region shaded above. Write the system of inequalities.
Boundary 1 — solid line: points on the line are included → use ≤ or ≥.
The shading is below the line y = x + 1, so the inequality is y ≤ x + 1.
Boundary 2 — dashed line: points on the line are not included → use < or >.
The shading is above the line y = −2x + 4, so the inequality is y > −2x + 4.
The solution to the system is the overlapping region satisfying both inequalities.
Graph example: y ≤ x + 1 (solid, shaded below) and y > −2x + 4 (dashed, shaded above)
Example 2: Write a System from a Word Problem
A school club is selling student tickets for $5 and adult tickets for $8. The auditorium can hold no more than 200 people, and the club wants to earn at least $1,200. Write a system of inequalities.
Define variables: x = number of student tickets; y = number of adult tickets.
Constraint 1 (seating limit): x + y ≤ 200 — the total cannot exceed 200 people.
Constraint 2 (revenue goal): 5x + 8y ≥ 1200 — total revenue must be at least $1,200.
Constraint 3 (non-negativity): x ≥ 0 and y ≥ 0 — ticket quantities must be non-negative.
Note: In a real-world context, x and y should also be whole numbers.
Problem
Student tickets ($5) and adult tickets ($8). Auditorium holds at most 200 people. Club needs at least $1,200 revenue.
Step 1 — Define Variables
Define Variables
Ticket sales scenario
Step 2 — Extract Constraints
Constraints
| Seating limit | ⭐no more than 200 people |
| Student ticket price | ⭐$5 each |
| Adult ticket price | ⭐$8 each |
| Revenue goal | ⭐at least $1,200 |
Step 3 — Write the System
System of Inequalities
x + y ≤ 2005x + 8y ≥ 1200x ≥ 0, y ≥ 0Seating + revenue + non-negativity
Step 4 — Solve & Interpret
Verify (80, 100)
| Ticket Type | Quantity | Price | Revenue |
|---|---|---|---|
| Student | 80 | $5 | $400 |
| Adult | 100 | $8 | $800 |
| Total revenue | ≥ $1,200 ✓ | ||
Cell phone plans: Plan A costs $30/month plus $0.10 per text. Plan B costs $50/month with unlimited texts. You want to spend at most $60/month. Write inequalities for each plan in terms of number of texts (t).
Define variable: t = number of texts per month.
Plan A cost: 30 + 0.10t. Constraint: 30 + 0.10t ≤ 60.
Simplify Plan A: 0.10t ≤ 30 → t ≤ 300.
Plan B cost: $50 flat. Constraint: 50 ≤ 60 — always satisfied.
Plan A is cheaper when: 30 + 0.10t < 50 → t < 200.
Cell Phone Plan Comparison
| Condition | Inequality |
|---|---|
| Plan A monthly cost ≤ $60 | 30 + 0.10t ≤ 60 |
| Plan A cheaper than Plan B | 30 + 0.10t < 50 |
| Simplified Plan A limit | t ≤ 300 |
| Plan A cheaper when | t < 200 |
Distance: Two friends walk in opposite directions from the same point. Friend A walks at 3 mph and Friend B at 4 mph. They want to be at least 14 miles apart after t hours. Write an inequality.
Define variable: t = time in hours.
Distance A: 3t miles. Distance B: 4t miles. Total apart: 7t miles.
Constraint: 7t ≥ 14.
Solve: t ≥ 2.
They must walk for at least 2 hours.
Distance–Rate–Time Model
| Distance (d) | Rate (r) | Time (t) | |
|---|---|---|---|
| Friend A | 3t | 3 mph | t hours |
| Friend B | 4t | 4 mph | t hours |
| Combined | 7t | — | t hours |
Constraint: 7t ≥ 14 → t ≥ 2 hours
Coins: A piggy bank contains only quarters and dimes. There are at most 40 coins total and the total value is at least $5.00. Write a system of inequalities.
Define variables: q = number of quarters; d = number of dimes.
Constraint 1 (count): q + d ≤ 40.
Constraint 2 (value): 0.25q + 0.10d ≥ 5.00.
Implied: q ≥ 0 and d ≥ 0.
Verify (20, 10): 30 ≤ 40 ✓; 0.25(20)+0.10(10)=$6 ≥ $5 ✓.
q quarters
$0.25 each
d dimes
$0.10 each
Coin System
q + d ≤ 400.25q + 0.10d ≥ 5.00q ≥ 0, d ≥ 0Count + value + non-negativity
Guided Practice
Guided Practice Video: Writing Systems of Inequalities
Watch the guided practice walkthrough for writing systems of inequalities, then complete the problems below.
Video by Sang Real Math
Watch on YouTube ↗Shopping: You have at most $45 to spend on shirts (x) at $12 each and socks (y) at $3 each. You need at least 2 shirts and at least 3 pairs of socks. Define variables and write a system of inequalities.
Hint: Constraint 1 (budget): 12x + 3y ≤ 45. Constraint 2: x ≥ 2. Constraint 3: y ≥ 3. Check: (2, 3) → 24+9=33 ≤ 45 ✓.
Tickets: A movie theater sells adult tickets for $12 and child tickets for $7. The theater holds at most 250 people and needs to earn at least $2,000. Define variables and write a system of inequalities.
Hint: Let a = adult tickets, c = child tickets. Constraint 1: a + c ≤ 250. Constraint 2: 12a + 7c ≥ 2000. Don't forget: a ≥ 0, c ≥ 0.
Time management: A student has at most 6 hours to study math (m) and science (s). They must study each subject for at least 1 hour. Write a system of inequalities.
Hint: Constraint 1: m + s ≤ 6. Constraint 2: m ≥ 1. Constraint 3: s ≥ 1. Verify (2, 3): 2+3=5 ≤ 6 ✓; 2 ≥ 1 ✓; 3 ≥ 1 ✓.
Geometry: A triangle has a perimeter of at most 30 cm. Two sides are equal (each of length x cm) and the third side (y cm) must be at least 4 cm. Write a system of inequalities.
Hint: Perimeter: 2x + y ≤ 30. Third side: y ≥ 4. Non-negativity: x ≥ 0. Verify (10, 5): 20+5=25 ≤ 30 ✓; 5 ≥ 4 ✓.
Fundraising: A club sells lemonade for $1.50 per cup and cookies for $2.00 each. They want to sell at most 150 items and earn at least $200. Write a system of inequalities.
Hint: Let l = cups of lemonade, c = cookies. Constraint 1: l + c ≤ 150. Constraint 2: 1.5l + 2c ≥ 200. Add l ≥ 0, c ≥ 0.
Key Vocabulary
Mathematical Model
A mathematical representation (equation, inequality, or system) that describes a real-world situation.
Constraint
A condition or restriction expressed as an inequality that limits the possible values of the variables.
Variable
A letter representing an unknown quantity. Always define variables with units (e.g., x = number of adult tickets).
Non-negativity Constraint
An implied constraint that a quantity cannot be negative (e.g., x ≥ 0 for number of items).
Feasible Region
The set of all ordered pairs that satisfy every inequality in the system — the solution to the model.
Objective
The goal of a problem — what you are trying to maximize or minimize within the constraints.
Implied Constraint
A restriction not stated explicitly but understood from context (e.g., you cannot buy a negative number of items).
Interpretation
Stating the answer to a problem in the context of the original situation, including units and meaning.
Interactive Practice — 5 Questions
Which inequality correctly translates "at most 80 items"?
A problem says "at least $500 in revenue." Which inequality is correct (R = revenue)?
A student writes "at least 10 notebooks" as n < 10. What is the error?
Which ordered pair is a solution to the system: x + y ≤ 50 and 3x + 5y ≥ 150?
Why should you always include non-negativity constraints (x ≥ 0, y ≥ 0) in real-world problems?
Independent Practice
Independent Practice
Budget: You have at most $80 to spend on books ($15 each, x books) and magazines ($5 each, y magazines). You need at least 2 books and at least 3 magazines. Define variables and write the complete system of inequalities.
Tickets: A concert sells floor tickets for $75 and balcony tickets for $40. The venue holds at most 500 people and the promoter needs at least $25,000 in revenue. Define variables and write the system.
Coins: A jar has only nickels (n) and quarters (q). There are at most 60 coins and the total value is at least $8.00. Write the system and verify one solution.
Time: A worker can work at most 40 hours per week at job A (x hours) and job B (y hours). They must work at least 10 hours at each job. Write the system.
Geometry: A rectangle has a perimeter of at most 60 cm. The width must be at least 5 cm and the length must be at least twice the width. Define variables and write the system.
Common Mistakes
Using a solid line for < or > — strict inequalities require a dashed boundary line.
Use a dashed line for < or >. Points on a dashed boundary are not part of the solution.
Using a dashed line for ≤ or ≥ — inequalities that include equality require a solid boundary line.
Use a solid line for ≤ or ≥. Points on a solid boundary are included in the solution.
Shading the wrong side of the boundary line.
For y > mx + b or y ≥ mx + b, shade above. For y < mx + b or y ≤ mx + b, shade below. When not solved for y, use a test point to confirm the correct side.
Forgetting to reverse the inequality sign when multiplying or dividing by a negative number.
Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign. Example: −2y > 6 → y < −3.
Writing only one inequality when the problem or graph has multiple conditions.
A system must include one inequality for every condition. Count the constraints and make sure your system has the same number of inequalities.
Forgetting that the solution is the overlapping region — not just any shaded area.
The solution to a system of inequalities is the region where all shaded areas overlap simultaneously. A point must satisfy every inequality to be a solution.
Math Tips
"At most" → ≤ (maximum); "at least" → ≥ (minimum); "more than" → >; "fewer than" → <. Memorize these four keyword-to-symbol mappings.
Always write "Let x = ..." and "Let y = ..." with units before writing any inequality — vague variables lead to wrong inequalities.
Count your constraints: if the problem has 3 conditions, your system should have at least 3 inequalities (plus non-negativity).
Non-negativity constraints (x ≥ 0, y ≥ 0) are almost always implied in real-world problems — include them even if not stated.
Verify your system: substitute a test point that should be feasible and confirm it satisfies every inequality in the system.
Before You Finish
Student Checklist
Did I write one inequality for every condition?
Did I choose the correct solid or dashed boundary line?
Did I shade the correct side?
Did I include nonnegative restrictions when needed?
Did I identify the overlapping solution region?