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Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 4

Write a Linear Inequality

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

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Name the variables

Two quantities, two letters. Include units (hours, dollars, tickets).

Translate the comparison

at least / no less than / no fewer than → ≥   at most / no more than → ≤   more than → >   less than → <.

One inequality, two variables

The graph will be a half-plane, not a single line. Counts and amounts usually cannot be negative, even if you do not write x ≥ 0 yet.

Boundary vs region

The equation is the boundary. The inequality says which side of that boundary is allowed.

Level 1

Easy

One sentence. Write one linear inequality in two variables.

Example
Notebooks cost \$3 and pens cost \$2. A student can spend at most \$40. Write an inequality for notebooks n and pens p.

Let n = the number of notebooks.

Let p = the number of pens.

3n + 2p ≤ 40

Answer: 3n + 2p ≤ 40

1
A baker uses at most 30 cups of flour. Cakes use 3 cups and muffins use 1 cup. Write an inequality for cakes c and muffins m.
Show Answer

Let c = the number of cakes.

Let m = the number of muffins.

3c + m ≤ 30

Answer: 3c + m ≤ 30

2
Jada wants at least 8 hours of math and science combined this week. Write an inequality for math hours x and science hours y.
Show Answer

Let x = hours of math.

Let y = hours of science.

x + y ≥ 8

Answer: x + y ≥ 8

3
A store sells posters for \$4 and frames for \$7. You can spend no more than \$40. Write an inequality for posters p and frames f.
Show Answer

Let p = the number of posters.

Let f = the number of frames.

4p + 7f ≤ 40

Answer: 4p + 7f ≤ 40

4
A parking lot has space for fewer than 50 cars and trucks. Write an inequality for cars c and trucks t.
Show Answer

Let c = the number of cars.

Let t = the number of trucks.

c + t < 50

Answer: c + t < 50

5
Smoothies need at least 5 cups of fruit. Strawberries add 1 cup each and bananas add 2 cups each. Write an inequality for strawberries s and bananas b.
Show Answer

Let s = the number of strawberries.

Let b = the number of bananas.

s + 2b ≥ 5

Answer: s + 2b ≥ 5

Level 2

Medium

Rates, starting amounts, and careful wording of at least / at most.

Example
A gym charges \$8 per weekday visit and \$12 per weekend visit. Kai can spend at most \$100 this month. Write an inequality for weekday visits w and weekend visits e.

Let w = weekday visits.

Let e = weekend visits.

8w + 12e ≤ 100

Answer: 8w + 12e ≤ 100

1
A truck can carry no more than 2,000 pounds. Boxes weigh 40 pounds and crates weigh 75 pounds. Write an inequality for boxes b and crates c.
Show Answer

Let b = the number of boxes.

Let c = the number of crates.

40b + 75c ≤ 2000

Answer: 40b + 75c ≤ 2000

2
A garden needs more than 20 plants. Tomato plants come in 4-packs and pepper plants come in 6-packs. Write an inequality for tomato packs t and pepper packs p.
Show Answer

Let t = tomato packs.

Let p = pepper packs.

4t + 6p > 20

Answer: 4t + 6p > 20

3
A phone plan includes 2 GB already used, plus d extra gigabytes of downloads and s extra gigabytes of streaming. The plan allows at most 12 GB total. Write an inequality for d and s.
Show Answer

Total data is 2 + d + s.

2 + d + s ≤ 12

d + s ≤ 10

Answer: d + s ≤ 10

4
A fundraiser must raise at least \$250. Adult tickets are \$12 and student tickets are \$5. Write an inequality for adults a and students s.
Show Answer

Let a = adult tickets.

Let s = student tickets.

12a + 5s ≥ 250

Answer: 12a + 5s ≥ 250

5
A recipe allows no fewer than 9 cups of mix. Flour is measured in cups f and sugar in cups g, and you always add 1 cup of oil. Write an inequality for f and g.
Show Answer

Total mix is f + g + 1.

f + g + 1 ≥ 9

f + g ≥ 8

Answer: f + g ≥ 8

Level 3

Hard

Rewrite a sentence so the inequality is in two variables, and watch extra constants.

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. Write an inequality for color pages c and black-and-white pages b.

Let c = color pages.

Let b = black-and-white pages.

15 + 2c + b < 60

2c + b < 45

Answer: 2c + b < 45

1
A boat can hold at most 900 pounds. Each adult weighs 150 pounds and each child weighs 60 pounds, and 40 pounds of gear is already on board. Write an inequality for adults a and children k.
Show Answer

150a + 60k + 40 ≤ 900

150a + 60k ≤ 860

Answer: 150a + 60k ≤ 860

2
You work h hours at \$14 per hour and sell n items with a \$3 commission each. You need to earn more than \$200. Write an inequality for h and n.
Show Answer

14h + 3n > 200

Answer: 14h + 3n > 200

3
A tank starts with 12 gallons. You add x gallons of rainwater and remove y gallons for the garden. After that, the tank must still have at least 5 gallons. Write an inequality for x and y.
Show Answer

12 + x − y ≥ 5

x − y ≥−7

Or x + 12 − y ≥ 5

Answer: 12 + x − y ≥ 5

4
A class needs at least twice as many pencils as erasers, and there are p pencils and e erasers. Write an inequality.
Show Answer

“At least twice as many pencils as erasers” means p is at least 2e.

p ≥ 2e

Answer: p ≥ 2e

5
Miles walked w and miles biked b. Walking takes 20 minutes per mile and biking takes 6 minutes per mile. You have no more than 90 minutes. Write an inequality, then write it with hours on the left if each minute is 160 of an hour.
Show Answer

20w + 6b ≤ 90

Divide by 60 if you want hours.

2060w + 660b ≤ 9060

13w + 110b ≤ 32

Answer: 20w + 6b ≤ 90

Super Challenge

Super Challenge

1
A student writes 3x + 2y < 40 for “at most \$40 on \$3 notebooks and \$2 pens.” What is wrong, and what inequality should they write?
Show Answer

“At most” includes spending exactly \$40, so the boundary counts.

Use ≤, not <.

3x + 2y ≤ 40

Answer: 3x + 2y ≤ 40

2
Is (0, 0) allowed in 4p + 7f ≤ 40 for posters and frames? Is it a useful point in the story?
Show Answer

4(0) + 7(0) = 0, and 0 ≤ 40 is true, so (0, 0) is in the half-plane.

In the story it means buying nothing. That is allowed by the inequality, but it may not be the purchase you care about.

Counts still cannot be negative, even though the inequality alone does not say that.

Answer: yes algebraically; buying nothing

3
Write an inequality for “the number of adult tickets a is more than three times the number of child tickets c,” then name one ordered pair that works and one that does not.
Show Answer

a > 3c

(10, 3): 10 > 9, true.

(9, 3): 9 > 9 is false.

The boundary a = 3c is not included.

Answer: a > 3c

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