Proving Inequality: Using a Hint to Show (a+b)>(c+d)

[azizlwl]

Homework Statement

Prove: If a>b and c>d, then a+c>b+d
Hint: (a-b)+(c-d)=(a+c)-(b+d)>0

Homework Equations

The Attempt at a Solution

How to use the hint to prove the inequality?
My method, not sure it's right.
Given c>d, c-d>0
Given a>b => a+(c-d)>b
Thus a+c>b+d

 

[Mark44]

Homework Statement

Prove: If a>b and c>d, then a+c>b+d
Hint: (a-b)+(c-d)=(a+c)-(b+d)>0

Homework Equations

The Attempt at a Solution

How to use the hint to prove the inequality?
My method, not sure it's right.
Given c>d, c-d>0

right.

Given a>b => a+(c-d)>b

Why not continue with the same line of thinking as above?

a > b implies what about a - b?

Thus a+c>b+d

 

[azizlwl]

ok i see it now

Given a>b =>a-b>0
Given c>d =>c-d>0

(a-b)+(c-d)>0
(a+c)-(b+d)>0
thus a+c>b+d

thank you

 

[Mark44]

You're welcome!

 

[HallsofIvy]

Another way to do this, and, in my opinion, simpler, is this:
Since a> b, a+ c> b+ c.
Since c> d, b+ c> b+ d.
Since ">" is transitive a+ c> b+ d.