A word problem gives you constraints in sentences, and your job is to turn each sentence into one inequality. Do that one sentence at a time, and the shading that follows is the easy part.
This lesson belongs to linear inequalities in two variables. If the shading step is still shaky, revisit choosing the shaded side using a test point first.
How do I turn a sentence into an inequality?
Name the two variables first, with units. Then read one sentence, find the quantity it limits, and pick the symbol from the wording.
| Wording in the question | Symbol |
|---|---|
| at most, not more than, maximum of | ≤ |
| at least, not less than, minimum of | ≥ |
| more than, exceeds | > |
| fewer than, less than | < |
| twice, three times as many | multiplier goes on the quantity it is compared with |
The table only gives the symbol. You still have to decide which side each quantity goes on, and that needs a check.
Worked example: the school club shopping list
Aina buys x packs of pens and y packs of notebooks for a school club. A pack of pens costs RM4 and a pack of notebooks costs RM6.
She spends at most RM60 and buys at least 2 packs of pens. The number of notebook packs is not more than three times the number of pen packs.
Sentence 1 (the budget): total cost is 4x + 6y, and it is at most 60, so 4x + 6y ≤ 60. Divide every term by 2 to get 2x + 3y ≤ 30.
Sentence 2 (the pens): at least 2 packs means x ≥ 2.
Sentence 3 (the comparison): notebooks are the quantity that is limited, so y sits alone on the left. Not more than three times the pens gives y ≤ 3x.
Sentence 4 (the hidden constraints): packs cannot be negative, so x ≥ 0 and y ≥ 0. The line x ≥ 2 already covers x ≥ 0, so you only need to add y ≥ 0.
The full set is 2x + 3y ≤ 30, x ≥ 2, y ≤ 3x and y ≥ 0.
Check the translation with a test point
Pick one pair that the story says is allowed, and put it in every inequality. Try 5 packs of pens and 5 packs of notebooks: cost is 4(5) + 6(5) = RM50, which is within RM60.
Now check each line. 2(5) + 3(5) = 25, which is at most 30.
Then x = 5 is at least 2, and y = 5 is at most 3(5) = 15. All four hold, so the translation matches the story.
Try a pair the story forbids, such as 2 pen packs and 8 notebook packs. The cost is 8 + 48 = RM56, which is within budget, but 8 is more than 3(2) = 6. The inequality y ≤ 3x rejects it, as it should.
The mistake that flips a region
The common slip is writing the comparison the wrong way round, as x ≤ 3y. The wording “three times” attaches to the pens, and the slip attaches it to the notebooks.
| Step | Wrong | Right |
|---|---|---|
| Which quantity is limited? | not checked | notebooks, so y is alone |
| Inequality | x ≤ 3y | y ≤ 3x |
| Test with (2, 8) | 2 ≤ 24, so it is accepted | 8 ≤ 6 is false, so it is rejected |
The wrong version accepts a pair the story forbids. One test point exposes it in ten seconds.
Check yourself
A bakery makes x cakes and y cupcakes in a day. The total number of items is at most 40.
The number of cupcakes is at least three times the number of cakes, and at least 5 cakes are made. Write the inequalities, then test the pair (6, 20).
Answer
Total items at most 40: x + y ≤ 40.
Cupcakes at least three times the cakes: y ≥ 3x.
At least 5 cakes: x ≥ 5. Adding the hidden constraint y ≥ 0 completes the set.
Test (6, 20): 26 ≤ 40, 20 ≥ 18 and 6 ≥ 5 are all true, so the pair is allowed.
Test (10, 25) for contrast: 35 ≤ 40 holds, but 25 ≥ 30 is false, so this pair is not allowed.
What to study next
Once the inequalities are right, the next step is to find the region that satisfies several inequalities. Then revisit drawing an inequality boundary correctly if solid and dashed lines still confuse you.
The algebra step repair trainer is useful if the rearranging inside each inequality is where you slip. For a teacher to work through your translations with you, see online one-to-one Mathematics tuition.