Inequalities- QA Formulae

By BYJU'S CAT|Updated : June 7th, 2023

In inequalities, we do have LHS and RHS which are not equal as any of the following cases are possible:

LHS RHS; LHS > RHS; LHS < RHS; LHS RHS.

Basic rules of operations on inequality:

  • We can add or subtract the same number from both sides with no change in the truth of the inequality. If a > b, then a + k > b + k.

For example: 

If 9 > 7, then 9 + 2 > 7 + 2; if 4 < 5, then 4 – 3 < 5 – 3.

 

  • We can multiply or divide both sides with the same number; however, the sign will depend as follows:

 

 

 

Table of Content

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
    • ac > bc 

 

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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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