Forming Linear Inequalities

S4 Mathematics — Term 1, Topic 3

In this lesson:

• Translate a worded constraint into a linear inequality
• Write the implicit constraints the problem leaves unsaid
• Assemble a complete system for the carpenter problem

What You Will Learn Today

By the end of this lesson:

1. Translate a worded resource limit into a linear inequality in two variables
2. Identify and write all implicit constraints, including non-negativity
3. Assemble a complete system of inequalities for a multi-constraint LP scenario

The Carpenter Problem from Lesson 1

With = tables, = chairs:

• Table: 2 h cutting, 1 h finishing
• Chair: 3 h cutting, 1 h finishing
• Limits: 12 h cutting, 5 h finishing
• Profit: $20/table, $10/chair

Fill In the Cutting-Hours Translation

The Five Steps of a Translation

Worked: Cutting Hours, Five Steps

• 1. Variables: tables, chairs
• 2. Contributions: h, h
• 3. Sum:
• 4. "Available" →
• 5.

Your Turn: Finishing Hours Constraint

Each table: 1 h finishing. Each chair: 1 h. Available: 5 h.

• Contributions: ,
• Sum:
• Symbol:
• Inequality:

Pick the Symbol from the Wording

Worked: Match Wording to Symbol

Translate, with as the left-hand side:

• "No more than 30 sacks" →
• "At least 5 hectares" →
• "Does not exceed 100" →
• "Exceeds 50,000 shillings" →

Quick Check: Pick the Symbol

Write the symbol; flag the strict ones:

1. "No more than 200 kg of maize"
2. "At least 5 workers on site"
3. "Fewer than 20 vehicles per hour"
4. "Available: 18 m of wire"
5. "Profit must exceed 100,000"

Predict First: "At Least 30 Sacks"

The wording: "The truck carries at least 30 sacks of maize per trip."

Let be the number of sacks. Which is correct?

• A.
• B.

Commit to A or B before advancing. This direction is the one most students reverse under pressure.

Quick Drill: Match Four Phrases

Match each phrase to its symbol:

• "Reaches a minimum of 8" →
• "Carries no more than 50" →
• "Strictly less than 100" →
• "Greater than 25, never equal" →

Could the Carpenter Make −3 Tables?

So far: and .

• Try : ✓
• And ✓
• The algebra accepts negative tables.

The Rule the Problem Never Says

When a variable counts a physical quantity — tables, chairs, hectares, kilograms — it cannot be negative.

Write these constraints every time, even when the problem statement is silent:

The situation says it, not the wording.

With and Without the Non-Negativity Lines

A Note on Whole Numbers

In real life, the carpenter cannot make 3.7 tables.

• The algebra here treats and as continuous — they may take any non-negative real value
• Integrality is also implicit, but we defer it to Lesson 8
• For now: continuous variables, non-negativity, and the explicit limits

Predict: Which Reads More Cleanly?

Two students wrote the carpenter system. Both are correct.

Student A:

Student B:

Which version makes the system easier to read and graph in Lesson 3?

Template for a Complete System

• One line per inequality
• — first
• Non-negativity last
• Objective separate

The Full Carpenter System on Paper

Constraints:

Objective (set up only):

Predict: Does Profit Belong Here?

A student writes:

• Does the last line belong with the others?
• If not, where does go?

Constraint vs. Objective: the Difference

Validate the System with

Pick a plan that should be feasible — substitute into every constraint:

• ✓
• ✓
• ✓ and ✓

Your Turn: The Tailor Problem

Shirts (), trousers (). Shirt: 2 m cloth, 1 h sewing. Trouser: 3 m, 2 h. Have 24 m, 14 h.

In pairs, produce: definitions, both constraints, non-negativity, validation, objective.

Exit Task: The Baker, On Your Own

Loaves (), cakes (). Loaf: 0.5 kg flour, 10 min. Cake: 0.3 kg, 20 min. Have 5 kg, 240 min. Profit: 2,000 UGX/loaf, 5,000/cake.

Produce the complete setup — no scaffolding. 3 minutes.

Four Mistakes to Catch and Fix

M1. Missing non-negativity → always write
M2. Flipped symbol → "at least" ; read aloud
M3. Mixed vs → always
M4. Profit as constraint → objective has no RHS number

Today's System Is Tomorrow's Graph

A worded LP problem becomes:

✓ A system of inequalities — with non-negativity
✓ A separate objective to maximise or minimise

Next: graph the carpenter system. Corners appear.