How Do Absolute Value Inequalities Apply to Different Sign Scenarios?

[wonnabewith]

Homework Statement

I was trying to show that
1) |a+b|≤|a|+|b|
2) |a+b|≥|a|-|b|
and find out how they were true when a,b>0, a,b<0, and a>0,b<0

Homework Equations

1) |a+b|≤|a|+|b|
2) |a+b|≥|a|-|b|

The Attempt at a Solution

For |a+b|≤|a|+|b|
a,b>0
I got that |a+b|=a+b
|a|+|b|=a+b
so |a+b|=|a|+|b|
but I was confused when I was trying to solve when a,b<0, and a>0,b<0
please help!

 

[Hurkyl]

What were your problems? What did you try?

 

[HallsofIvy]

Homework Statement

I was trying to show that
1) |a+b|≤|a|+|b|
2) |a+b|≥|a|-|b|
and find out how they were true when a,b>0, a,b<0, and a>0,b<0

Homework Equations

1) |a+b|≤|a|+|b|
2) |a+b|≥|a|-|b|

The Attempt at a Solution

For |a+b|≤|a|+|b|
a,b>0
I got that |a+b|=a+b
|a|+|b|=a+b
so |a+b|=|a|+|b|
but I was confused when I was trying to solve a,b<0, and a>0,b<0
please help!

If a and b are both less than 0, then so is a+ b.
|a|= -a, |b|= -b, |a+ b|= -(a+ b)= -a- b.

If one of a> 0 and b< 0, there are still two cases: |a|> |b| or |a|< |b|.
If |a|> |b| then a+ b= a-(-b)> 0 so |a+ b|= a+ b= |a|- |b|