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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 4

Write a System of Inequalities

Create two linear inequalities with two variables to represent a real-world situation.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Two facts, two inequalities

Each restriction in the story becomes one inequality. Together they are a system.

And means overlap

A pair (x, y) must make both inequalities true. That is the overlap of the two half-planes.

Counts stay nonnegative

If x and y are counts or lengths, you may also need x ≥ 0 and y ≥ 0. Write the two story inequalities first, then add those if the problem asks for all restrictions.

Keep the wording

at least → ≥. at most → ≤. more than → >. less than → <.

Level 1

Easy

Two simple restrictions. Name the letters, then write both inequalities.

Example
You want at least 5 posters and frames combined, and you can spend at most \$40. Posters are \$4 and frames are \$7. Write a system for posters p and frames f.

Let p = posters and f = frames.

p + f ≥ 5

4p + 7f ≤ 40

Answer: p + f ≥ 54p + 7f ≤ 40

1
A student studies math m hours and history h hours. Combined time is at most 10 hours, and math is at least 3 hours. Write a system.
Show Answer

m + h ≤ 10

m ≥ 3

Answer: m + h ≤ 10m ≥ 3

2
A baker makes cakes c and pies p. There is at most 24 cups of flour (3 cups per cake, 2 per pie), and at least 4 desserts total. Write a system.
Show Answer

3c + 2p ≤ 24

c + p ≥ 4

Answer: 3c + 2p ≤ 24c + p ≥ 4

3
Cars c and buses b in a lot. Fewer than 30 vehicles, and at least 2 buses. Write a system.
Show Answer

c + b < 30

b ≥ 2

Answer: c + b < 30b ≥ 2

4
Adult tickets a are \$10 and child tickets k are \$4. Sales must be at least \$80, and no more than 12 tickets are sold. Write a system.
Show Answer

10a + 4k ≥ 80

a + k ≤ 12

Answer: 10a + 4k ≥ 80a + k ≤ 12

5
Miles walked w and miles run r. Total miles at least 6, and walking is no more than 4 miles. Write a system.
Show Answer

w + r ≥ 6

w ≤ 4

Answer: w + r ≥ 6w ≤ 4

Level 2

Medium

Rates plus a second comparison. Keep both inequalities.

Example
Weekday visits w cost \$8 and weekend visits e cost \$12. Kai can spend at most \$100 and wants at least 3 weekend visits. Write a system.

8w + 12e ≤ 100

e ≥ 3

Answer: 8w + 12e ≤ 100e ≥ 3

1
A truck carries boxes b (40 lb) and crates c (75 lb). At most 2,000 pounds, and at least 6 crates. Write a system.
Show Answer

40b + 75c ≤ 2000

c ≥ 6

Answer: 40b + 75c ≤ 2000c ≥ 6

2
A garden has tomato packs t (4 plants) and pepper packs p (6 plants). More than 20 plants, and no more than 8 packs total. Write a system.
Show Answer

4t + 6p > 20

t + p ≤ 8

Answer: 4t + 6p > 20t + p ≤ 8

3
A fundraiser sells adult tickets a at \$12 and student tickets s at \$5. At least \$250, and students must outnumber adults. Write a system.
Show Answer

12a + 5s ≥ 250

s > a

Answer: 12a + 5s ≥ 250s > a

4
Hours of tutoring x at \$20 and hours of group class y at \$12. Earnings at least \$160, and total hours at most 10. Write a system.
Show Answer

20x + 12y ≥ 160

x + y ≤ 10

Answer: 20x + 12y ≥ 160x + y ≤ 10

5
A phone plan allows at most 12 GB. Downloads d and streaming s use the data, and streaming must be at least twice the downloads. Write a system.
Show Answer

d + s ≤ 12

s ≥ 2d

Answer: d + s ≤ 12s ≥ 2d

Level 3

Hard

A starting amount, a comparison, and careful “at least twice.”

Example
A printer charges \$15 to set up plus \$2 per color page and \$1 per black-and-white page. The total must stay under \$60, and there must be at least as many black-and-white pages as color pages. Write a system for color pages c and black-and-white pages b.

15 + 2c + b < 60

2c + b < 45

b ≥ c

Answer: 2c + b < 45b ≥ c

1
A boat can hold at most 900 pounds. Adults 150 lb, children 60 lb, plus 40 lb of gear. At least 2 adults. Write a system for adults a and children k.
Show Answer

150a + 60k + 40 ≤ 900

150a + 60k ≤ 860

a ≥ 2

Answer: 150a + 60k ≤ 860a ≥ 2

2
You work h hours at \$14 and sell n items at \$3 each. You need more than \$200, and you work at most 12 hours. Write a system.
Show Answer

14h + 3n > 200

h ≤ 12

Answer: 14h + 3n > 200h ≤ 12

3
Miles walked w (20 min each) and miles biked b (6 min each). At most 90 minutes, and you walk at least 2 miles. Write a system.
Show Answer

20w + 6b ≤ 90

w ≥ 2

Answer: 20w + 6b ≤ 90w ≥ 2

4
A class needs at least twice as many pencils p as erasers e, and at most 40 supplies in all. Write a system.
Show Answer

p ≥ 2e

p + e ≤ 40

Answer: p ≥ 2ep + e ≤ 40

5
A tank starts with 12 gallons. Add x gallons and remove y gallons. At least 5 gallons must remain, and you cannot remove more than 8 gallons. Write a system.
Show Answer

12 + x − y ≥ 5

y ≤ 8

Answer: 12 + x − y ≥ 5y ≤ 8

Super Challenge

Super Challenge

1
Write the system for posters at \$4 and frames at \$7: spend at most \$40, buy at least 5 items, and you cannot buy a negative number of either. Then name one extra inequality students often forget.
Show Answer

4p + 7f ≤ 40

p + f ≥ 5

p ≥ 0

f ≥ 0

The extra pair is p ≥ 0 and f ≥ 0.

Answer: 4p + 7f ≤ 40p + f ≥ 5p ≥ 0f ≥ 0

2
A student writes x + y ≤ 10 and x > 3 for “at most 10 hours combined and more than 3 hours of math.” Which ordered pair is in the system: (3, 7) or (4, 6)? Why?
Show Answer

(3, 7): 3 + 7 = 10 ≤ 10 is true, but 3 > 3 is false.

(4, 6): 4 + 6 = 10 ≤ 10, and 4 > 3, both true.

(4, 6) is in the system. (3, 7) is on the open boundary x = 3, so it is not.

Answer: (4, 6)

3
Two inequalities can have no overlap. Write a system in x and y that has no solution, using one horizontal and one vertical comparison that cannot both be true.
Show Answer

Example: y ≥ 5 and y ≤ 2.

No y is both at least 5 and at most 2.

The half-planes do not overlap.

Answer: any pair of opposite half-planes with a gap

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