← Back to the Linear Inequalities worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: A-REI.B.3, A-CED.A.1, A-CED.A.3 · Level: Grade 9 Algebra 1
Problem 1 Add and compare
Subtract 7: x > 5.
Open circle at 5, shade right. Check: 6 + 7 = 13 > 12.
x > 5
Problem 2 Divide by a positive
Divide by 3 (positive, so the sign stays): x ≤ 7.
Check: 3(7) = 21 ≤ 21.
x ≤ 7
Problem 3 Two steps with a fraction
Add 2: x4 ≥ 3. Multiply by 4: x ≥ 12.
Check: 12/4 − 2 = 1 ≥ 1.
x ≥ 12
Problem 4 Reverse the sign
Divide by −2 and reverse the sign: x > −7.
Check: x = 0 gives 0 < 14, true; x = −8 gives 16 < 14, false.
x > −7
Problem 5 Write from a graph
The circle at −2 is closed, so −2 is included, and the shading goes left.
x ≤ −2
Problem 6 Variables on both sides
Subtract 2x: 4x − 5 > 11. Add 5: 4x > 16. Divide by 4: x > 4.
Check: x = 5 gives 25 > 21.
x > 4
Problem 7 Negative coefficient
Subtract 7: −3x ≥ 15. Divide by −3 and reverse: x ≤ −5.
Check: x = −6 gives 7 + 18 = 25 ≥ 22.
x ≤ −5
Problem 8 Distribute first
4x − 8 < 2x + 10. Subtract 2x: 2x − 8 < 10. Add 8: 2x < 18. So x < 9.
Check: x = 8 gives 24 < 26.
x < 9
Problem 9 Fraction inequality
Multiply by 3: 2x − 1 ≤ 15. Add 1: 2x ≤ 16. Divide: x ≤ 8.
Check: x = 8 gives 15/3 = 5 ≤ 5.
x ≤ 8
Problem 10 Compound inequality (and)
Add 1 to all parts: −6 < 2x ≤ 10. Divide all parts by 2: −3 < x ≤ 5.
−3 < x ≤ 5, which is (−3, 5]
Problem 11 Compound inequality (or)
First: 3x < −9, so x < −3. Second: 2x ≥ 14, so x ≥ 7.
x < −3 or x ≥ 7
Problem 12 Read a segment
Closed at −1 and open at 4.
−1 ≤ x < 4, which is [−1, 4)
Problem 13 Interval notation
(a) open at −1, going right: (−1, ∞). (b) closed at 9, going left: (−∞, 9]. (c) open at 1, closed at 8: (1, 8].
(a) (−1, ∞) (b) (−∞, 9] (c) (1, 8]
Problem 14 Test values
x = −3: 6 + 6 = 12 > 11, true. x = −2: 6 + 4 = 10, not greater than 11.
Solving: −2x > 5, so x < −2.5.
−3 is a solution, −2 is not; x < −2.5
Problem 15 Camp activities
18 + 7.5a ≤ 60, so 7.5a ≤ 42 and a ≤ 5.6.
Activities are whole numbers, so the greatest is 5 (cost $55.50).
5 activities
Problem 16 Test average
(78 + 85 + 91 + t)/4 ≥ 85. The first three add to 254, so 254 + t ≥ 340 and t ≥ 86.
Check: (254 + 86)/4 = 85.
At least 86
Problem 17 Food truck profit
Profit per wrap: 6.50 − 1.90 = 4.60. So 4.6w − 420 > 300, giving 4.6w > 720 and w > 156.52.
Check: 157 wraps give 722.20 > 720; 156 give 717.60, not enough.
At least 157 wraps
Problem 18 Greenhouse range
59 ≤ 1.8C + 32 ≤ 86. Subtract 32: 27 ≤ 1.8C ≤ 54. Divide by 1.8: 15 ≤ C ≤ 30.
15 ≤ C ≤ 30, or [15, 30]
Problem 19 Robotics budget
24(6) + 9s ≤ 300, so 144 + 9s ≤ 300 and 9s ≤ 156, giving s ≤ 17.33.
Whole sensors: 17 (cost 144 + 153 = 297).
At most 17 sensors
Problem 20 Phone plan comparison
15 + 4g < 35, so 4g < 20 and g < 5. The graph agrees: the blue line is below the orange line before 5 GB.
Since g ≥ 0, the answer is 0 ≤ g < 5.
Plan B is cheaper when g < 5