Linear Inequalities And Shading
Inequalities in context: best profit and least cost
In this note
- 2.6Translate statements like 'at least' and 'no more than' into inequalities.
- 2.6Draw the region, then find the integer point giving the greatest profit or least cost.
- 2.6Say whether a combination is possible, with a reason.
Before you start: Regions and shading.
Formula bank
Words into inequalities
| Words (x small, y large) | Inequality |
|---|---|
| at least 3 small | \(x\ge3\) |
| no more than 10 altogether | \(x+y\le10\) |
| fewer than 9 large | \(y<9\) |
| more small than large | \(x>y\) |
| the number of large is at most twice the number of small | \(y\le2x\) |
| small holds 20, large 45; at least 180 people | \(20x+45y\ge180\) |
General methodHow to write the inequalities
- Say what x and y count.
- Take each sentence on its own and write ONE inequality for it.
- Check the direction with a sensible pair of numbers.
Try it yourself
Show answer
Answer \(x+y\le10\)
Show answer
Answer \(x\ge3\)
Finding the best point
Profit or cost (like \(20x+15y\)) keeps growing in one direction across the region, so the best value is on the edge of R - at or next to a corner. Counts must be whole numbers, so check the integer points near that corner.
General methodHow to find the greatest profit or least cost
- Draw the region R (shade the unwanted sides).
- Find the corner of R in the 'best' direction (more of the item that earns more, or less of the dearer one).
- List the integer points in R near that corner.
- Work out the profit or cost at each one and pick the best. Answer in words with the value.
Profit \(P=20x+15y\). Jackets earn more, so push \(x\) up along \(x+y=10\): (8, 2) gives $190, (7, 3) gives $185, (6, 4) gives $180.
Best: 8 jackets and 2 trousers, $190.
Try it yourself
Hint
Show answer
Answer no
Show answer
Answer yes
Ways they test it
Type 1 Write the inequalities
"Write down four inequalities ..." (4 marks, 0580 Paper 4)
Spot it a story with 'at least', 'no more than'.
- One inequality per sentence.
- Watch which letter is 'twice' the other.
Type 2 Show that
"Show that \(20x+45y\ge180\)."
Spot it one inequality is given to you.
- Explain where each number comes from (seats per bus × number of buses ≥ pupils).
Type 3 Is it possible?
"Can the school hire 4 small and 2 large buses?"
Spot it a specific pair.
- Test it in EVERY inequality.
- Say which one fails.
For example 4 small and 2 large
- seats \(80+90=170<180\)
Answer No: not enough seats
Type 4 Best profit / least cost
"Find the least cost." (2-3 marks)
Spot it after the region is drawn.
- Integer points near the best corner.
- Compare values.
For example Buses: (3, 3), (4, 3), (5, 2)
- costs $1380, $1540, $1400
Answer 3 small and 3 large, $1380
Mistakes to avoid
- ✗ 'at least 3' → x > 3✓ x ≥ 3'At least' includes 3.
- ✗ A best point that isn't a whole number✓ the nearest integer point inside RYou can't hire part of a bus.
- ✗ Checking one point only✓ compare the neighboursShow it is the best.
Summary
- One inequality per sentence: at least ≥, at most ≤.
- Draw R, then look at integer points near the best corner.
- Compare the profit/cost at each candidate.
- Answer in context with the value.
If you see …
- "greatest profit" / "least cost"test the integer points near a corner of R.
- "is it possible?"test the pair in every inequality.
Practice
Practice: easy, medium, hard
Easy
Show answer
Answer \(x>y\)
Show answer
Answer \(y\le25\)
Hint
Show answer
Answer no
Medium
Guide me
What is it asking? an inequality
What do we know? seats = 20x + 45y
Plan at least → ≥
Show working
- \(20x+45y\ge180\)
Answer \(20x+45y\ge180\)
Guide me
What is it asking? a profit
What do we know? \(20x+15y\)
Plan substitute
Show working
- \(140+45=185\)
Answer $185
Guide me
What is it asking? yes or no
What do we know? count the seats
Plan compare with 180
Show working
- 80 + 90 = 170 seats
- 170 < 180: no
Answer no
Hard
Guide me
What is it asking? the greatest profit
What do we know? jackets earn more: push x up
Plan (8, 2), (7, 3), (6, 4)
Show working
- (8, 2): 190
- (7, 3): 185
- (6, 4): 180
Answer $190
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| candidates tested | M1 | ||
| $190 | A1 | ||
| Total: 2 marks | |||
Guide me
What is it asking? the least cost
What do we know? integer points in R
Plan (3, 3), (4, 3), (5, 2)
Show working
- (3, 3): 1380
- (4, 3): 1540
- (5, 2): 1400
Answer $1380
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| integer points listed | M1 | ||
| costs compared | M1 | ||
| $1380 | A1 | ||
| Total: 3 marks | |||
Guide me
What is it asking? a count
What do we know? list integer points in R
Plan check every inequality
Show working
- (3, 3), (4, 3), (5, 2)
Answer 3
| Part | Answer / working | Marks | Guidance |
|---|---|---|---|
| 3 | B1 | ||
| Total: 1 mark | |||
Can I do this?
- I can turn context statements into inequalities.
- I can find the integer point with the greatest profit or least cost.
- I can explain whether a combination is possible.
Anything unticked: revise that section first. All ticked: take the quiz.
Quiz
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