Answer to Question #122829 in Real Analysis for NILAM JYOTI DAS

Question

Under what condition the equality holds in |a+b|≤|a|+|b|,a,b∈R ?

Answer 1

Concept- Modulus always gives positive quantity i.e. it can take positive or negative values as input but it will always give positive value as output.

We will take four possible cases here and our cases will be true if L.H.S is equal to R.H.S:

Case-1

a>0,b>0

Therefore summation of two positive values value will also be positive so it's modulus values will also be positive

L.H.S=|a+b| = a+b

R.H.S= |a| + |b| = a+b

Here, L.H.S = R.H.S

Hence this case is true

Case-2

a>0,b<0

L.H.S = |a+(-b)| = |a-b|

R.H.S = |a| + |(-b)| = a+b

L.H.S < R.H.S

Since, a-b will always be less than a+b so here L.H.S will always be less than R.H.S

Therefore this case is not true.

Case-3

a<0,b>0

L.H.S = |(-a)+b| = |b-a|

R.H.S = |(-a)| + |b| = a+b

Since, b-a will always be less than a+b so here L.H.S will always be less than R.H.S

Therefore this case is also not true.

Case-4

a<0,b<0

L.H.S=|(-a)+(-b)| = |-(a+b)| = a+b

R.H.S= |(-a)| + |(-b)| = a+b

Here also L.H.S = R.H.S

So this case is true.

Now we can conclude that for all cases possible equality holds for only cases (1) & (4)

Conclusion - For equation  |a+b|≤|a|+|b|,a,b∈R the equality holds only if both a & b are positive or both a & b are negative.