If a and b have opposite signs then |a + b| < |a| + |b|

[Physicaa]

Homework Statement

If a and b have opposite signs then |a + b| < |a| + |b|

Homework Equations

No equations.

The Attempt at a Solution

Well first start with "a" positive and "-b" negative.

We have :

|a|=a

|-b|=b

|a-b|=a-b

We begin with 0 < |a| + |-b|

Then : 0 < |a| + |-b| -a

a < |a| + |-b|

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

which gives us : |a-b| < |a| + |-b|

We see that the result stays the same when we have -a and b.

|-a|=a

|b|=b

|b-a|=b-a

We begin with 0 < |b| + |-a|

Then : 0 < |b| + |-a| -b

b < |b| + |-a|

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

which gives us : |b-a| < |b| + |-a|

Thus wee see that when the signs are different the following inequality holds :

|a + b| < |a| + |b|

Is it any good ?

 

[Mark44]

Homework Statement

If a and b have opposite signs then |a + b| < |a| + |b|

Homework Equations

No equations.

The Attempt at a Solution

Well first start with "a" positive and "-b" negative.

No, this is wrong. In the problem statement, you're given that a and b have opposite signs.
For your first case, a > 0, and b < 0.

To say, as you did, ' "-b" negative ', means that b > 0, if -b < 0.

We have :

|a|=a

|-b|=b

|a-b|=a-b

No.
The problem asks you to show that |a + b| < |a| + |b|. It has nothing to do with |a - b|.

We begin with 0 < |a| + |-b|

Then : 0 < |a| + |-b| -a

a < |a| + |-b|

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

which gives us : |a-b| < |a| + |-b|

Again, the problem has nothing to do with |a - b|.

We see that the result stays the same when we have -a and b.

|-a|=a

|b|=b

|b-a|=b-a

We begin with 0 < |b| + |-a|

Then : 0 < |b| + |-a| -b

b < |b| + |-a|

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

which gives us : |b-a| < |b| + |-a|

Thus wee see that when the signs are different the following inequality holds :

|a + b| < |a| + |b|

Is it any good ?