Answer Keys: Inequalities

Student Concept Worksheet Answers

1

x > 7

"Greater than" translates to >

2

y ≤ 15

"At most" means less than or equal to (the number IS included)

3

x < 8

x + 6 < 14 → x < 14 - 6 → x < 8

4

n ≥ 5

4n ≥ 20 → n ≥ 20/4 → n ≥ 5

5

Open circle at 4, shaded to the right

x > 4: open circle (4 NOT included), shade right toward bigger numbers

6

Closed circle at 2, shaded to the left

y ≤ 2: closed circle (2 IS included), shade left toward smaller numbers

Practice Worksheet Answers

Part A: Translating Words to Inequalities

#	Answer	Explanation
1	x < 12	"Less than" = <
2	x ≥ 5	"Greater than or equal to" = ≥
3	x ≤ 20	"At most" means ≤
4	x ≥ 8	"At least" means ≥
5	x ≤ 15	"No more than" means ≤
6	x > -3	"Greater than" = >

Part B: Solving One-Step Inequalities

#	Answer	Work
7	x > 7	x + 5 > 12 → x > 12 - 5 → x > 7
8	y < 11	y - 3 < 8 → y < 8 + 3 → y < 11
9	n ≥ 6	4n ≥ 24 → n ≥ 24/4 → n ≥ 6
10	m ≤ 15	m/3 ≤ 5 → m ≤ 5 × 3 → m ≤ 15
11	a ≤ 6	a + 9 ≤ 15 → a ≤ 15 - 9 → a ≤ 6
12	b > 7	5b > 35 → b > 35/5 → b > 7

Part C: Graphing Inequalities

#	Answer
13	Open circle at 2, shaded right (x > 2)
14	Closed circle at 4, shaded left (x ≤ 4)
15	Closed circle at -1, shaded right (x ≥ -1)
16	Open circle at 0, shaded left (x < 0)

Part D: Writing Inequalities from Graphs

17

x ≤ 3

Closed circle means ≤ or ≥; shaded left means less than

18

x > 2

Open circle means < or >; shaded right means greater than

Part E: Identifying Solutions

19

4.5, 5, 10

x > 4: Solutions must be greater than 4. 4.5 > 4, 5 > 4, 10 > 4. NOT: 3, 4, or 0

20

6, 5.9, -2, 0

y ≤ 6: Solutions must be 6 or less. NOT: 7, 100

Part F: Word Problems

21

a ≥ 13

"At least 13" means 13 or older (13 IS included)

22

s ≤ 65

"At most 65" means 65 or less

23

18 + m ≤ 50 OR m ≤ 32

Total spent must be at most $50. 18 + m ≤ 50 → m ≤ 32. Maya can spend up to $32 more.

24

42 + h > 48 OR h > 6

Height + growth must be MORE than 48. 42 + h > 48 → h > 6. Tomas needs to grow more than 6 inches.

Challenge

25

5 is a solution to x ≥ 5 but NOT to x > 5

x > 5 has an open circle (5 NOT included). x ≥ 5 has a closed circle (5 IS included). The only number that is a solution to x ≥ 5 but not x > 5 is 5 itself.

26

8t ≤ 50; Maximum = 6 tickets

8t ≤ 50 → t ≤ 6.25. Since you can't buy partial tickets, max is 6 tickets ($48).

FAST Practice Quiz Answers

1

A) x > 8

x + 7 > 15 → x > 15 - 7 → x > 8. B adds instead of subtracts. C flips the inequality. D gives a single value, not an inequality.

2

B) Closed circle at 3, shaded to the left

x ≤ 3: The ≤ symbol means CLOSED circle (3 IS included). "Less than" means shade LEFT.

3

D) a ≥ 25

"At least $25" means $25 or MORE. The customer must spend ≥ $25. A/C are wrong direction. B excludes $25 exactly.

4

Select: 5.1, 6, 100

5 - NOT a solution (5 is NOT greater than 5)
5.1 - Solution (5.1 > 5)
6 - Solution (6 > 5)
4.9 - NOT a solution (4.9 < 5)
100 - Solution (100 > 5)

5

A) n ≤ 6

3n ≤ 18 → n ≤ 18/3 → n ≤ 6. B divides wrong. C flips inequality. D changes ≤ to <.

6

B) x > 3

Open circle means < or > (NOT ≤ or ≥). Shaded right means greater than. Answer: x > 3.

FAST Quiz Scoring Guide

Score	Interpretation	Recommended Action
6/6	Mastery	Ready for two-step inequalities and compound inequalities
4-5/6	Approaching Mastery	Review specific areas - graphing conventions or word problem translation
2-3/6	Developing	Reteach with emphasis on open vs closed circles and key vocabulary
0-1/6	Needs Intervention	Small group instruction starting with inequality symbols and their meanings

Common Error Patterns to Watch For:

Confusing open/closed circles: Review that < and > use OPEN circles; ≤ and ≥ use CLOSED circles
"At least" vs "at most": Both INCLUDE the number (use ≤ or ≥). "At least" = ≥, "At most" = ≤
Shading wrong direction: Less than shades LEFT (toward smaller numbers), greater than shades RIGHT
Including boundary in strict inequalities: x > 5 does NOT include 5 as a solution

ANSWER KEYS - Teacher Use Only