Operations on Inequalities

 
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When we add or subtract the same number from both sides of an inequality, the inequality sign remains unchanged.

Example:

Find the new inequality when:
a) 3 is added to both sides of 4 < 10
b) 4 is subtracted from 7 > 2

Solution:

a) 4 < 10
4 + 3 < 10 + 3
7 < 13
b) 7 > 2
7 – 4 > 2 – 4
3 > – 2

When we multiply or divide the same positive number from both sides of an inequality, the inequality sign remains unchanged.

Example:

Find the new inequality when:
a) 3 < 5 is multiplied both sides by 2
b) 18 > 9 is divided both sides by 3

Solution:

a) 3 < 5
3 × 2 < 5 × 2
6 < 10
b) 18 > 9
18 ÷ 3 > 9 ÷ 3
6 > 3
When we multiply or divide the same negative number from both sides of an inequality, the inequality sign must be reversed . (change < to > and > to <).

Example:

Find the new inequality when:
a) 4 < 11 is multiplied both sides by – 2
b) 30 > –9 is divided both sides by –3

Solution:

a) 4 < 11
4 × (– 2) < 11 × (– 2)
–8 > –22 (reverse the inequality sign)
b) 30 > –9
30 ÷ (–3) > (–9) ÷ (–3)
–10 < 3 (reverse the inequality sign)

The following videos show more examples of solving inequalities: