It will not change the sign of the inequality if the number is positive.
If a > b, then ak > bk; k > 0
For example:. 7 > 5 ⟹ 7 × 4 > 5 × 4
It will change the sign of the inequality if the number is negative.
If a > b, then ak < bk; k < 0
For example: 8 > 4 ⟹ 8 × (−3) < 4 × (−3)
It will change the sign to equal if the number is zero.
If a > b, then ak = bk; k = 0
For example: 8 > 4 ⟹ 8 × (0) = 4 × (0).
Linear inequalities:
If a > b and c > 0, then we get the following:
- a + c > b + c
- a − c > b − c
- ac > bc
Modular inequalities:
|x − y| = |y − x|
|x . y| = |x|.|y|
|x + y| |x| + |y|
|x + y| |x| − |y|
Quadratic inequalities:
(x – a) (x – b) > 0 {since a < b}
⟹ (x < a) U (x > b)
(x – a) (x – b) < 0 {since a > b}
⟹ a < x < b
Important points:
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