Linear inequalities in one unknown · Ages 14–15
Properties of inequalities
Times or divided by a negative: turn the sign round.
How to do it
Method
Times or divided by a negative: turn the sign round
4 > 1, times −2: −8 < −2
Another way
Move x to where it is positive
−2x > 6 ⇒ −6 > 2x ⇒ x < −3
In full
Add or take away the same number on both sides and the sign stays as it is: a > b, so a + 5 > b + 5. Multiply or divide both sides by the same positive number and it stays too: a > b, so 3a > 3b. But multiply or divide both sides by the same negative number and the sign turns round: a > b, so −2a < −2b. See it with numbers: 4 > 1; times −2 they are −8 and −2, and −8 < −2, so the sign has turned. Or never divide by a negative: −2x > 6; add 2x to both sides and take away 6: −6 > 2x, that is 2x < −6, x < −3, the same answer. Multiplying by 0 leaves no inequality at all (0 = 0).
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
4 > 0. After both sides are multiplied by −5, the bigger of the two products is
Show the answer and working
Answer: 0
4 × (−5) = −20 and 0 × (−5) = 0. Times a negative, the side that was smaller becomes the bigger.
Example 2
If x > y, then 2x > 2y.
True or false?
Show the answer and working
Answer: True
Both sides times a positive number: the sign stays; times a negative: it turns. Try x = 8, y = 0: compare 2 × 8 with 0.
Example 3
−3 > −8. Take 3 from both sides: −6 > ; multiply the original sides by −4: 12 < .
Fill in each blank
Show the answer and working
Answer: −11, 32
Taking the same number keeps the sign: −8 − 3 = −11; a negative turns it: −8 × (−4) = 32, and 12 < 32.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.8 > 2. Multiplying both sides by 3 gives 24 > 6.
True or false?
Answer:
2.If a < b, which of these must be true?
- A. a ÷ 9 > b ÷ 9
- B. −9a < −9b
- C. a − 9 > b − 9
- D. a + 9 < b + 9
Choose the right answer
Answer:
3.If x > y, then −2x > −2y.
True or false?
Answer:
4.−2 > −5. After both sides are multiplied by −5, the bigger of the two products is
Answer:
5.7 > 2. Take 3 from both sides: 4 > ; multiply the original sides by −3: −21 < .
Fill in each blank
Answer:
All the answers
- 1. True
- 2. D. a + 9 < b + 9
- 3. False
- 4. 25
- 5. −1, −6
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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