Equations and inequalities · Ages 14–15

Properties of 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.

1.6 > 2. After both sides are multiplied by −2, the bigger of the two products is
Answer:
2.−4 > −7. After both sides are multiplied by −2, the bigger of the two products is
Answer:
3.8 > 7. After both sides are multiplied by 3, the bigger of the two products is
Answer:
4.5 > −1. Take 5 from both sides: 0 > ; multiply the original sides by −3: −15 < .
Fill in each blank
Answer:
5.−2 > −7. Take 8 from both sides: −10 > ; multiply the original sides by −2: 4 < .
Fill in each blank
Answer:

13.x and y are two people's weights in kg, and x > y. Each picks up a 6 kg bag of rice. Which of these must be true?
- A. x + 6 < y + 6
- B. x − 6 < y − 6
- C. x + 6 > y + 6
- D. 6x < 6y
Choose the right answer
Working:
Answer:
14.The morning's 1°C is warmer than the evening's −3°C: 1 > −3. Multiply both temperatures by −1: the morning's becomes and the evening's , and the sign turns round.
Fill in each blank
Working:
Answer:
15.Tower A is 200 m tall, taller than tower B's 150 m: 200 > 150. Models of both are made at 1 : 50 (both sides divided by 50): A's model is m tall and B's m, A's still the taller.
Fill in each blank
Working:
Answer:
16.One job pays $82 an hour, more than another's $47. Both are worked for 4 hours, and each then loses $10 in fares. By how many dollars do the two earnings differ?
Working:
Answer:
17.One job pays $107 an hour, more than another's $88. Both are worked for 4 hours, and each then loses $14 in fares. By how many dollars do the two earnings differ?
Working:
Answer:
18.The morning's 0°C is warmer than the evening's −4°C: 0 > −4. Multiply both temperatures by −1: the morning's becomes and the evening's , and the sign turns round.
Fill in each blank
Working:
Answer:

Show the answers and working

Properties of inequalities worksheet: Answers and working

1. −4
6 × (−2) = −12 and 2 × (−2) = −4. Times a negative, the side that was smaller becomes the bigger.
2. 14
−4 × (−2) = 8 and −7 × (−2) = 14. Times a negative, the side that was smaller becomes the bigger.
3. 24
8 × (3) = 24 and 7 × (3) = 21. Times a negative, the side that was smaller becomes the bigger.
4. −6, 3
Taking the same number keeps the sign: −1 − 5 = −6; a negative turns it: −1 × (−3) = 3, and −15 < 3.
5. −15, 14
Taking the same number keeps the sign: −7 − 8 = −15; a negative turns it: −7 × (−2) = 14, and 4 < 14.

6. A. a + 7 < b + 7
Adding the same number, 7, to both sides keeps the sign as it is.
7. False
Times a positive number the sign stays; times a negative it turns round. −1 × (−3) = 3 and −3 × (−3) = 9: compare them.
8. True
Times a positive number the sign stays; times a negative it turns round. 4 × (5) = 20 and −2 × (5) = −10: compare them.
9. False
Both sides times a positive number: the sign stays; times a negative: it turns. Try x = 2, y = 0: compare −2 × 2 with 0.
10. C. −4a < −4b
Both sides times the negative −4: the sign turns, so a > b gives −4a < −4b.
11. A. x ≤ −2
14 ÷ (−7) = −2; dividing by a negative turns the sign: x ≤ −2.
12. B. x ≤ −6
30 ÷ (−5) = −6; dividing by a negative turns the sign: x ≤ −6.

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

13. C. x + 6 > y + 6
Both add 6 kg: the same number added to both sides keeps the sign, x + 6 > y + 6.
14. −1, 3
1 × (−1) = −1 and −3 × (−1) = 3, and −1 < 3: times a negative, the order flips.
15. 4, 3
200 ÷ 50 = 4 and 150 ÷ 50 = 3: dividing by a positive number keeps the sign, 4 > 3.
16. 140 dollars
82 > 47; times the positive 4, the sign stays; taking 10 from each changes neither the sign nor the gap: (82 − 47) × 4 = 140.
17. 76 dollars
107 > 88; times the positive 4, the sign stays; taking 14 from each changes neither the sign nor the gap: (107 − 88) × 4 = 76.
18. 0, 4
0 × (−1) = 0 and −4 × (−1) = 4, and 0 < 4: times a negative, the order flips.

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