prove | a + b | ≤ | a | + | b | when a and b be any real number
remember there is a positive and negative case for each of the absolute values. There is only one rule to remember is you multiply or divide by a negative number the sign changes.
Start with the positive case of | a + b |:
(a + b) ≤ |a | + | b |
Positive of | a + b | and | a |: Positive of | a + b | and Negative of | a |:
(a + b) ≤ a + | b | (a + b) ≥ -a + | b |
Positive of |a + b | and | a | and | b |: Positive of | a + b | and negative of | a | and | b|
1. (a + b) ≤ a + b true 2. (a + b) ≤ -a – b false
Positive of | a + b | and | a | and negative of | b |: pos | a + b| and neg | a | and | b |
3. (a + b) ≥ a – b true 4. (a + b) ≥ -a – b true
I started the negative cases you find the 4 answers.
Negative of | a + b |
-a – b ≥ | a | + | b |