Equations and inequalities · Ages 14–15

Solving linear inequalities worksheet

Part of: Linear inequalities in one unknown

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Equations and inequalities · Ages 14–15

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Write each answer in the space. For word problems, show your working and write a sentence answer.

1.Solve 3x − 1 ≥ −13: x ≥
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2.The least whole-number solution of 3x − 5 > 14 is
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3.Solve 5(x + 5) ≤ x + 37: x ≤
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4.The least whole-number solution of 5x − 3 > 14 is
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5.Solve 51 − 9x > −3 without dividing by a negative: x terms to the right, numbers to the left, giving 54 > x, so x <
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6.The greatest whole-number solution of x + 54 < −2 is
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7.Solve 18 − 2x > 6 without dividing by a negative: x terms to the right, numbers to the left, giving 12 > x, so x <
Fill in each blank
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13.A rectangular flower bed is 15 m long, and there is fencing for only 48 m, so its perimeter can't be more than 48 m. With a width of x m: 2(15 + x) ≤ 48, 15 + x ≤ , x ≤ .
Fill in each blank
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14.A phone plan costs $70 a month plus $15 for each GB of data. Fen spends no more than $175 a month. With x GB: 70 + 15x ≤ 175, 15x ≤ , x ≤ .
Fill in each blank
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15.Bike hire is $31 for the first hour and $14 for each hour after. Dong has $123. At most how many whole hours can he hire a bike for?
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16.A hilltop is 9°C this morning, and the temperature then falls 4°C an hour. After x hours it is below −7°C: 9 − 4x < −7. With x on the right and the numbers on the left: < 4x, so x > .
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17.Yan has $470 and spends $13 a day; Kit has $50 and saves $12 a day. At least how many days until Kit has more than Yan?
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18.A phone plan costs $48 a month plus $9 for each GB of data. Fen spends no more than $66 a month. With x GB: 48 + 9x ≤ 66, 9x ≤ , x ≤ .
Fill in each blank
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Show the answers and working

Solving linear inequalities worksheet: Answers and working

1. −4
Take −1 from both sides: 3x ≥ −13 − (−1) = −12; divide both sides by the positive 3, the sign staying: x ≥ −4.
2. 7
3x > 19, so x > 193. That isn't whole, and the least whole number above it is 7.
3. 3
Open the bracket and bring x to the left: 4x ≤ 37 − 5 × (5); divide by the positive 4: x ≤ 3.
4. 4
5x > 17, so x > 175. That isn't whole, and the least whole number above it is 4.
5. 9, 6
Add 9x and take away −3 on both sides: 51 − −3 > 9x, that is 54 > 9x; divide by the positive 9: 6 > x, so x < 6.
6. −14
Multiply both sides by 4: x + 5 < 4 × (−2), so x < −13. −13 is left out, so the greatest whole number is −14.
7. 2, 6
Add 2x and take away 6 on both sides: 18 − 6 > 2x, that is 12 > 2x; divide by the positive 2: 6 > x, so x < 6.

8. B. x < 4
6x < 21 − (−3), so 6x < 6 × 4; divide by 6: x < 4.
9. C. x < −6
3x < −23 − (−5), so 3x < 3 × −6; divide by 3: x < −6.
10. B. x > −6
−3x < 6 − −12; dividing by −3 turns it: x > −6. Or move x to the right: −12 − 6 < 3x, so x > −6.
11. False
Put in x = 3: 7 − 4 × (3) = −5. Is −5 ≥ −2?
12. True
Put in x = 6: 3 − 6 × (6) = −33. Is −33 ≥ −33?

Marking: 2 for the working, 1 for the answer, 1 for the sentence.

13. 24, 9
Halve both sides: 15 + x ≤ 24; take away 15: x ≤ 9. The bed is at most 9 m wide.
14. 105, 7
Take 70 from both sides: 15x ≤ 175 − 70 = 105; divide by the positive 15: x ≤ 7. She uses at most 7 GB a month.
15. 7 hours
With x hours: 31 + 14(x − 1) ≤ 123, 14(x − 1) ≤ 123 − 31, x − 1 ≤ 123 − 3114. In whole hours: at most 7.
16. 16, 4
Add 4x and take away −7 on both sides: 9 − (−7) < 4x, 16 < 4x; divide by the positive 4: x > 4, with no negative to divide by.
17. 17 days
After x days: 50 + 12x > 470 − 13x; x terms to the left: (12 + 13)x > 470 − 50, x > 470 − 5012 + 13. The least whole number above that is 17.
18. 18, 2
Take 48 from both sides: 9x ≤ 66 − 48 = 18; divide by the positive 9: x ≤ 2. She uses at most 2 GB a month.

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