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Amy Ferguson Moncure

Algebra 1 · Unit 1

Write Inequalities

Turn a real-life sentence into a linear inequality in one variable.

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Key Ideas Easy Medium Hard Super Challenge
What to Remember

Name the variable first

Say what the letter stands for before you write the math. Include units when the story has them (hours, dollars, tickets).

Translate the words

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

“More than” depends on the sentence

“n is more than 5” is n > 5.

“5 more than n” is n + 5. That is an expression, not an inequality.

Starting amount plus a rate, with a limit

Many stories look like (start) + (rate)(quantity), then compare to a bound with ≥, ≤, >, or <.

Level 1

Easy

One sentence. Name the variable, then write the inequality.

Example
Write an inequality. A number n is at most 14.

“At most” means ≤.

n ≤ 14

Answer: n ≤ 14

1
Write an inequality. A number n is at least 9.
Show Answer

“At least” means ≥.

n ≥ 9

Answer: n ≥ 9

2
Write an inequality. A number n is more than 6.
Show Answer

“More than” means >.

n > 6

Answer: n > 6

3
Write an inequality. A number n is less than 20.
Show Answer

“Less than” means <.

n < 20

Answer: n < 20

4
A club needs no fewer than 12 volunteers. Write an inequality for the number of volunteers n.
Show Answer

“No fewer than” means ≥.

n ≥ 12

Answer: n ≥ 12

5
A parking lot holds no more than 40 cars. Write an inequality for the number of cars n.
Show Answer

“No more than” means ≤.

n ≤ 40

Answer: n ≤ 40

Level 2

Medium

Longer stories and “at least / at most.”

Example
Maya has $52. Smoothies cost $4 each. She wants at least $20 left. Write an inequality for the number of smoothies n she can buy.

Let n = the number of smoothies.

Money left: 52 − 4n

At least $20 left.

52 − 4n ≥ 20

Answer: 52 − 4n ≥ 20

1
A ride share costs $5 plus $2 per mile. Write an inequality for the number of miles m if the trip should cost no more than $21.
Show Answer

Let m = the number of miles.

Cost = 5 + 2m

No more than $21.

5 + 2m ≤ 21

Answer: 5 + 2m ≤ 21

2
Write an inequality. Twice a number n is at most 30.
Show Answer

Twice n is 2n.

“At most” means ≤.

2n ≤ 30

Answer: 2n ≤ 30

3
Write an inequality. 4 more than a number n is at least 19.
Show Answer

4 more than n is n + 4.

“At least” means ≥.

n + 4 ≥ 19

Answer: n + 4 ≥ 19

4
Luis has $40. Snacks cost $3 each. He wants at least $10 left. Write an inequality for the number of snacks s he can buy.
Show Answer

Let s = the number of snacks.

Money left: 40 − 3s

At least $10 left.

40 − 3s ≥ 10

Answer: 40 − 3s ≥ 10

5
A gym charges $25 plus $8 per class. Write an inequality for the number of classes c if the total should be less than $65.
Show Answer

Let c = the number of classes.

Cost = 25 + 8c

Less than $65.

25 + 8c < 65

Answer: 25 + 8c < 65

Level 3

Hard

Two-part stories and careful wording.

Example
Jae has already biked 12 miles and then rides at 15 miles per hour for h hours. The whole trip should be at most 42 miles. Write an inequality for h.

Distance = 12 + 15h

At most 42 miles.

12 + 15h ≤ 42

Answer: 12 + 15h ≤ 42

1
A banquet has 8 tables. There should be at least twice as many chairs as tables. Write an inequality for the number of chairs n.
Show Answer

Twice the tables is 16.

At least twice as many chairs.

n ≥ 16

Answer: n ≥ 16

2
Write an inequality. 5 less than 3 times a number n is greater than 19.
Show Answer

3 times n is 3n.

5 less than that is 3n − 5.

3n − 5 > 19

Answer: 3n − 5 > 19

3
A tank starts with 20 gallons and fills at 4 gallons per minute. Write an inequality for the number of minutes t if the tank may hold no more than 50 gallons.
Show Answer

Amount = 20 + 4t

No more than 50.

20 + 4t ≤ 50

Answer: 20 + 4t ≤ 50

4
Adult tickets cost $9 and child tickets cost $6. A group buys 3 adult tickets and c child tickets and can spend at most $45. Write an inequality for c.
Show Answer

3 adult tickets cost 3(9) = 27.

At most $45.

27 + 6c ≤ 45

Answer: 27 + 6c ≤ 45

5
The length of a rectangle is 2 cm more than the width w. The perimeter is at least 40 cm. Write an inequality for w.
Show Answer

Length = w + 2

Perimeter formula.

P = 2ℓ + 2w

At least 40.

2(w + 2) + 2w ≥ 40

Answer: 2(w + 2) + 2w ≥ 40

Super Challenge

Super Challenge

1
A garden needs 48 feet of fence if it is fenced exactly, or at most 48 feet if leftover fencing is allowed. Write both the equation and the inequality for the length ℓ of one side of a square garden. How do the solutions differ?
Show Answer

A square has 4 equal sides, so the perimeter is 4ℓ.

Exact: 4ℓ = 48, so ℓ = 12.

At most: 4ℓ ≤ 48, so ℓ ≤ 12.

The equation has one side length. The inequality allows any side length 12 feet or shorter (and ℓ > 0 in the garden story).

Answer: 4ℓ = 48 has ℓ = 12; 4ℓ ≤ 48 has ℓ ≤ 12

2
Maya has $52. Smoothies cost $4 each. She wants at least $20 left. Student A writes 52 − 4n ≥ 20. Student B writes 4n ≤ 32. Show that both describe the same story, and say when each form is useful.
Show Answer

Student A starts with money left.

52 − 4n ≥ 20

Add 4n to both sides.

52 ≥ 20 + 4n

Subtract 20 from both sides.

32 ≥ 4n

4n ≤ 32

Student B’s form asks “how much can she spend on smoothies?” Student A’s form asks “how much will she have left?” Same story, different first step.

Answer: 52 − 4n ≥ 20 matches 4n ≤ 32

3
A student mixes up “n is more than 5” and “5 more than n.” Write the inequality and the expression. Then write an inequality for “5 more than n is at least 12.”
Show Answer

“n is more than 5” compares n to 5.

n > 5

“5 more than n” is an expression, not a comparison.

n + 5

“5 more than n is at least 12.”

n + 5 ≥ 12

Answer: n > 5; n + 5; n + 5 ≥ 12

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