A constraint is one limit from the question, written as an inequality in x and y. Your first task is to define x and y in words, then turn each sentence into a separate inequality.
This lesson opens the skills listed on the linear programming guide. Next comes drawing the feasible region.
How do words map to symbols?
Look for a small set of phrases. Each one has a fixed meaning.
| Phrase in the question | Symbol | Example |
|---|---|---|
| at most, not more than, up to | ≤ | x + y ≤ 40 |
| at least, not less than, a minimum of | ≥ | y ≥ 3 |
| at least twice as many x as y | x ≥ 2y | x ≥ 2y |
| x exceeds y by at most 4 | x − y ≤ 4 | x − y ≤ 4 |
| the number cannot be negative | x ≥ 0, y ≥ 0 | x ≥ 0 |
Worked example: a school club
A club makes x greeting cards and y posters. A card takes 10 minutes and a poster takes 25 minutes. The club has at most 300 minutes in total. It makes at least twice as many cards as posters, and at least 3 posters.
Define first: x = number of cards, y = number of posters.
- Time: 10x + 25y ≤ 300, which simplifies to 2x + 5y ≤ 60.
- Cards and posters: x ≥ 2y.
- Posters: y ≥ 3.
- Counting: x ≥ 0 (y ≥ 0 is already covered by y ≥ 3).
Check with a trial point, x = 12 and y = 4.
Time: 24 + 20 = 44 ≤ 60, which works. Ratio: 12 ≥ 8, and posters: 4 ≥ 3, which both work. The point is allowed.
The mistake that costs marks
The common error is the ratio sentence. “At least twice as many cards as posters” is easy to write backwards as y ≥ 2x.
Test it with numbers. If there are 12 cards and 4 posters, cards are three times posters, so the sentence is true.
Check y ≥ 2x: 4 ≥ 24 is false. Check x ≥ 2y: 12 ≥ 8 is true. A single trial pair catches a reversed inequality.
Check yourself
A shop sells x pens and y notebooks, at most 40 items in total.
It sells at most three times as many notebooks as pens. Pens are RM2 each, notebooks are RM5 each, and the shop buys at most RM150 of stock. Write all the constraints.
Answer
Total items: x + y ≤ 40.
Notebooks at most three times pens: y ≤ 3x.
Stock cost: 2x + 5y ≤ 150.
Counting: x ≥ 0 and y ≥ 0.
Trial check with x = 10 and y = 20: 30 ≤ 40, 20 ≤ 30 and 20 + 100 = 120 ≤ 150. All three hold.
What to study next
Continue with drawing the feasible region, which uses these inequalities as its input. The word-problem structure worksheet helps separate a long question into labelled parts.
If sentence-to-symbol translation is where you lose marks, see online one-to-one Additional Mathematics tuition.