Show that a b lessthanorequalto a b Show that a b lesstha

Show that |a| - |b| lessthanorequalto |a - b|. Show that |a| - |b| lessthanorequalto |a - b|.

Case 1. a and b are of the same sign

|a|-|b|=(a-b) or -(a-b)

(a-b)<=|a-b|

-(a-b)<=|a-b|

HEnce, |a|-|b|<=|a-b|

Case 2. a and b are of the different sign

|a|-|b|=(a+b) or -(a+b)

(a+b)<=|a-b| ,because a and b are of different sign so subtracting will give a larger number eg. 1,-2

1+(-2)=-1 ,|1-(-2)|=3

-(a+b)<=|a-b|

HEnce, |a|-|b|<=|a-b|

(b)

Case 1. |a|<=|b|

using part a.

|a|-|b|<=|a-b|

Case 2. |a|>=|b|

Using part a.

|b|-|a|=-(|a|-|b|)<=|a-b|

Hence, ||a|-|b||<=|a-b|