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Amy Ferguson Moncure

7th Grade Math · Unit 4

Inequalities in
Real Life

Write an inequality for the situation, then solve it.

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Key Ideas Easy Medium Hard
What to Remember

Words and symbols

At most means ≤. At least means ≥. Fewer than and less than both mean <. More than means >.

Choose the symbol that matches the words, then solve.

A positive rate

If the amount goes up, the coefficient is positive and the symbol stays.

Undo the starting amount, then divide by the rate.

A rate that goes down

If the amount goes down, the coefficient is negative.

Dividing by that negative rate reverses the symbol.

Level 1

Easy

The starting amount is 0, and the rate is positive.

Example
The score starts at 0 and goes up by 5 for each level. When is the result more than 20?

Let n be the number of levels.

5n > 20

Divide both sides by 5.

n > 4

Answer: n > 4

1
The distance starts at 0 and goes up by 4 for each hour. When is the result less than 24?
Show Answer

Let n be the number of hours.

4n < 24

Divide both sides by 4.

n < 6

Answer: n < 6

2
The savings starts at 0 and goes up by 3 for each week. When is the result at least 18?
Show Answer

Let n be the number of weeks.

3n ≥ 18

Divide both sides by 3.

n ≥ 6

Answer: n ≥ 6

3
The cost starts at 0 and goes up by 6 for each day. When is the result at most 30?
Show Answer

Let n be the number of days.

6n ≤ 30

Divide both sides by 6.

n ≤ 5

Answer: n ≤ 5

4
The point total starts at 0 and goes up by 8 for each game. When is the result more than 40?
Show Answer

Let n be the number of games.

8n > 40

Divide both sides by 8.

n > 5

Answer: n > 5

5
The height starts at 0 and goes up by 2 for each hour. When is the result less than 16?
Show Answer

Let n be the number of hours.

2n < 16

Divide both sides by 2.

n < 8

Answer: n < 8

6
The fund starts at 0 and goes up by 7 for each student. When is the result at least 28?
Show Answer

Let n be the number of students.

7n ≥ 28

Divide both sides by 7.

n ≥ 4

Answer: n ≥ 4

7
The total starts at 0 and goes up by 5 for each mile. When is the result at most 35?
Show Answer

Let n be the number of miles.

5n ≤ 35

Divide both sides by 5.

n ≤ 7

Answer: n ≤ 7

8
The reading starts at 0 and goes up by 9 for each day. When is the result more than 36?
Show Answer

Let n be the number of days.

9n > 36

Divide both sides by 9.

n > 4

Answer: n > 4

9
The temperature starts at 0 and goes up by 4 for each hour. When is the result less than 12?
Show Answer

Let n be the number of hours.

4n < 12

Divide both sides by 4.

n < 3

Answer: n < 3

10
The score starts at 0 and goes up by 10 for each level. When is the result at least 50?
Show Answer

Let n be the number of levels.

10n ≥ 50

Divide both sides by 10.

n ≥ 5

Answer: n ≥ 5

Level 2

Medium

A positive rate and a starting amount. The limit is at least or at most.

Example
The savings starts at 6 and goes up by 4 for each week. When is the result at least 30?

Let n be the number of weeks.

4n + 6 ≥ 30

4n + 6 − 6 ≥ 30 − 6

4n ≥ 24

Divide both sides by 4.

n ≥ 6

Answer: n ≥ 6

1
The cost starts at 5 and goes up by 3 for each day. When is the result at most 26?
Show Answer

Let n be the number of days.

3n + 5 ≤ 26

3n + 5 − 5 ≤ 26 − 5

3n ≤ 21

Divide both sides by 3.

n ≤ 7

Answer: n ≤ 7

2
The score starts at 4 and goes up by 5 for each level. When is the result at least 34?
Show Answer

Let n be the number of levels.

5n + 4 ≥ 34

5n + 4 − 4 ≥ 34 − 4

5n ≥ 30

Divide both sides by 5.

n ≥ 6

Answer: n ≥ 6

3
The distance starts at 8 and goes up by 2 for each hour. When is the result at most 24?
Show Answer

Let n be the number of hours.

2n + 8 ≤ 24

2n + 8 − 8 ≤ 24 − 8

2n ≤ 16

Divide both sides by 2.

n ≤ 8

Answer: n ≤ 8

4
The point total starts at 3 and goes up by 6 for each game. When is the result at least 33?
Show Answer

Let n be the number of games.

6n + 3 ≥ 33

6n + 3 − 3 ≥ 33 − 3

6n ≥ 30

Divide both sides by 6.

n ≥ 5

Answer: n ≥ 5

5
The fund starts at 2 and goes up by 4 for each student. When is the result at most 22?
Show Answer

Let n be the number of students.

4n + 2 ≤ 22

4n + 2 − 2 ≤ 22 − 2

4n ≤ 20

Divide both sides by 4.

n ≤ 5

Answer: n ≤ 5

6
The height starts at 9 and goes up by 3 for each hour. When is the result at least 30?
Show Answer

Let n be the number of hours.

3n + 9 ≥ 30

3n + 9 − 9 ≥ 30 − 9

3n ≥ 21

Divide both sides by 3.

n ≥ 7

Answer: n ≥ 7

7
The total starts at 10 and goes up by 5 for each mile. When is the result at most 40?
Show Answer

Let n be the number of miles.

5n + 10 ≤ 40

5n + 10 − 10 ≤ 40 − 10

5n ≤ 30

Divide both sides by 5.

n ≤ 6

Answer: n ≤ 6

8
The reading starts at 4 and goes up by 8 for each day. When is the result at least 44?
Show Answer

Let n be the number of days.

8n + 4 ≥ 44

8n + 4 − 4 ≥ 44 − 4

8n ≥ 40

Divide both sides by 8.

n ≥ 5

Answer: n ≥ 5

9
The temperature starts at 7 and goes up by 2 for each hour. When is the result at most 21?
Show Answer

Let n be the number of hours.

2n + 7 ≤ 21

2n + 7 − 7 ≤ 21 − 7

2n ≤ 14

Divide both sides by 2.

n ≤ 7

Answer: n ≤ 7

10
The savings starts at 5 and goes up by 7 for each week. When is the result at least 40?
Show Answer

Let n be the number of weeks.

7n + 5 ≥ 40

7n + 5 − 5 ≥ 40 − 5

7n ≥ 35

Divide both sides by 7.

n ≥ 5

Answer: n ≥ 5

Level 3

Hard

The amount goes down, so the inequality symbol reverses.

Example
The temperature starts at 30 and goes down by 3 for each hour. When is the result more than 6?

Let n be the number of hours.

−3n + 30 > 6

−3n + 30 − 30 > 6 − 30

−3n > −24

Divide both sides by −3. The inequality symbol reverses.

n < 8

Answer: n < 8

1
The water starts at 40 and goes down by 2 for each hour. When is the result at least 10?
Show Answer

Let n be the number of hours.

−2n + 40 ≥ 10

−2n + 40 − 40 ≥ 10 − 40

−2n ≥ −30

Divide both sides by −2. The inequality symbol reverses.

n ≤ 15

Answer: n ≤ 15

2
The charge starts at 50 and goes down by 4 for each hour. When is the result at most 10?
Show Answer

Let n be the number of hours.

−4n + 50 ≤ 10

−4n + 50 − 50 ≤ 10 − 50

−4n ≤ −40

Divide both sides by −4. The inequality symbol reverses.

n ≥ 10

Answer: n ≥ 10

3
The budget starts at 60 and goes down by 5 for each day. When is the result less than 20?
Show Answer

Let n be the number of days.

−5n + 60 < 20

−5n + 60 − 60 < 20 − 60

−5n < −40

Divide both sides by −5. The inequality symbol reverses.

n > 8

Answer: n > 8

4
The point total starts at 24 and goes down by 3 for each level. When is the result more than 6?
Show Answer

Let n be the number of levels.

−3n + 24 > 6

−3n + 24 − 24 > 6 − 24

−3n > −18

Divide both sides by −3. The inequality symbol reverses.

n < 6

Answer: n < 6

5
The savings starts at 28 and goes down by 2 for each week. When is the result at least 8?
Show Answer

Let n be the number of weeks.

−2n + 28 ≥ 8

−2n + 28 − 28 ≥ 8 − 28

−2n ≥ −20

Divide both sides by −2. The inequality symbol reverses.

n ≤ 10

Answer: n ≤ 10

6
The distance starts at 48 and goes down by 6 for each hour. When is the result at most 12?
Show Answer

Let n be the number of hours.

−6n + 48 ≤ 12

−6n + 48 − 48 ≤ 12 − 48

−6n ≤ −36

Divide both sides by −6. The inequality symbol reverses.

n ≥ 6

Answer: n ≥ 6

7
The score starts at 36 and goes down by 4 for each game. When is the result less than 8?
Show Answer

Let n be the number of games.

−4n + 36 < 8

−4n + 36 − 36 < 8 − 36

−4n < −28

Divide both sides by −4. The inequality symbol reverses.

n > 7

Answer: n > 7

8
The height starts at 20 and goes down by 1 for each minute. When is the result more than 5?
Show Answer

Let n be the number of minutes.

−n + 20 > 5

−n + 20 − 20 > 5 − 20

−n > −15

Divide both sides by −1. The inequality symbol reverses.

n < 15

Answer: n < 15

9
The fund starts at 45 and goes down by 3 for each day. When is the result at least 9?
Show Answer

Let n be the number of days.

−3n + 45 ≥ 9

−3n + 45 − 45 ≥ 9 − 45

−3n ≥ −36

Divide both sides by −3. The inequality symbol reverses.

n ≤ 12

Answer: n ≤ 12

10
The total starts at 40 and goes down by 5 for each hour. When is the result at most 15?
Show Answer

Let n be the number of hours.

−5n + 40 ≤ 15

−5n + 40 − 40 ≤ 15 − 40

−5n ≤ −25

Divide both sides by −5. The inequality symbol reverses.

n ≥ 5

Answer: n ≥ 5

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