Forming Linear Inequalities from Real Contexts

S4 Mathematics — Term 1, Topic 3

In this lesson:

• Translate verbal constraints into linear inequalities
• Write non-negativity restrictions for every model
• Validate inequalities by checking against the context

What You Will Learn Today

By the end of this lesson, you should be able to:

1. Translate verbal constraints into linear inequalities (, , , )
2. Add non-negativity restrictions (, ) to every model
3. Check each inequality against the original context to validate it

Quick Review from Lesson One

A tuck-shop sells chapatis () and mandazis ().

• Decision variables: = chapatis/day, = mandazis/day
• Objective: Maximize

Today: turn the shop's limits into inequalities.

Translating Words to Inequality Symbols

Whenever you see these phrases in a constraint, use the symbol on the right.

At Most vs At Least: Key Contrast

"At least 10" = floor (). "At most 50" = ceiling ().

Quick Check: Match Phrase to Symbol

Match each phrase to the correct inequality for :

1. "Total items cannot exceed 40"
2. "At least 15 items must be made"
3. "Exactly 30 items produced"

Which symbol — , , or — for each?

Answers: Match Phrase to Symbol

1. "Cannot exceed 40" → (ceiling)
2. "At least 15 items" → (floor)
3. "Exactly 30 items" → (fixed)

"Cannot exceed" = at most = . "At least" = minimum = .

Worked Example: Translating a Budget Constraint

Context: A tuck-shop has a daily ingredient budget of 12,000 UGX.
Each chapati costs 500 UGX; each mandazi costs 300 UGX.

Step 1: Total spending = UGX

Step 2: "Cannot exceed 12,000" →

Worked Example: Translating a Time Constraint

Context: The tuck-shop has 8 hours of preparation time per day.
Each chapati takes 2 hours to prepare; each mandazi takes 1 hour.

Step 1: Total time = hours

Step 2: "Available time" sets a ceiling →

Worked Example: Minimum Quantity Requirements

Context: The owner must stock at least 5 chapatis and at least 3 mandazis each day to meet customer demand.

Minimum requirements produce constraints.

Quick Check: Write the Inequality

Context: A farmer can plant at most 20 acres in total.
Let = acres of maize, = acres of beans.

Write the constraint inequality for total acreage.

What is the combined expression? What is the limit? What is the direction?

Answer: Writing the Total Acreage Constraint

Total acreage: acres

"At most 20 acres" →

Check: if and , the total is 22 — that violates the constraint. ✓

Non-Negativity Constraints Are Always Required

• — you cannot produce a negative number of chapatis
• — you cannot stock a negative number of mandazis

Add these to every LP model, even when the problem does not mention them.

Why Non-Negativity Constraints Always Matter

Without , :

• A mathematical solution might say "produce −8 chapatis"
• This satisfies other constraints but is physically impossible
• The feasible region would extend into negative quadrants

These two constraints confine solutions to the real world.

Complete Constraint System for Tuck-Shop

All five constraints together form the complete system.

Validating Your Inequalities: Three-Step Check

To verify each inequality is correct, apply three checks:

1. Units match — left side units = right side units
2. Direction correct — test a value that should be valid; it must satisfy the inequality
3. Context sensible — the constraint captures the real-world limit described

Worked Example: Validating a Constraint

Constraint:

Check 1 (units): shs/chapati chapatis shs ✓

Check 2 (direction): Try , : ✓

Check 3 (context): "Cannot exceed budget" — constraint prevents overspending ✓

Guided Practice: Forming Farm Constraints

A farmer plants maize ( acres) and beans ( acres):

• Total land: at most 20 acres
• Maize requires at least 6 acres
• Budget 200,000 UGX; maize 15,000/acre, beans 8,000/acre

Write all constraints and validate each one.

Quick Check: Spot the Error

A student wrote these constraints for the farm scenario:

Which inequality is wrong? Why?

Check each one against the context before advancing.

Guided Practice: School Supplies Constraint System

A school buys chairs () and desks ():

• Total furniture: at most 60 pieces
• Desks: at least 20
• Budget 500,000 UGX; chairs 5,000, desks 18,000

Write the complete inequality system (include non-negativity).

Exit Task: Write All Inequalities

A vendor makes samosas () and mandazis ():

• Total items: at most 80 per day
• Samosas: at least 20 per day
• Budget 14,000 UGX; samosas 200 UGX, mandazis 150 UGX each

Write all inequalities (include non-negativity).

Exit Task: Check Your Answers

Five inequalities. Non-negativity always included.

Key Takeaways From Today's Lesson

✓ "At most" → ; "at least" →

✓ Left side = coefficient × variable (units must match)

✓ Always include ,

✓ Validate: check units, direction, and context

Use or , not ; "at least 10" =

Next Lesson: Graphing the Constraints

In Lesson 3, we will:

• Convert each inequality into a boundary line equation
• Graph the line and shade the correct half-plane
• Identify the feasible region where all constraints are satisfied

Today's inequalities become tomorrow's graph.