Chapter 3.7, Problem 82PPE

To determine

Whether the statement $\left|a+b\right|=\left|a\right|+\left|b\right|$ is always, sometimes, or never true for real numbers $a\text{and}b.$

Answer to Problem 82PPE

The given statement $\left|a+b\right|=\left|a\right|+\left|b\right|$ is true sometimes for all real numbers.

Explanation of Solution

Given information :

Given statement $\left|a+b\right|=\left|a\right|+\left|b\right|,\forall a,b\in R$

Since absolute value of a real numbers is a positive number.

Given statement is need not be trays always as if $a=5,b=-3$

  $\begin{array}{l}\text{Consider}\left|a+b\right|=\left|5+\left(-3\right)\right|\\ =\left|5-3\right|\\ =\left|2\right|\\ \therefore \left|a+b\right|=2\\ Now,\left|a\right|+\left|b\right|=\left|5\right|+\left|-3\right|\\ =5+3\\ \therefore \left|a\right|+\left|b\right|=8\\ Hence,\left|a+b\right|\ne \left|a\right|+\left|b\right|.\end{array}$

However, the given statement $\left|a+b\right|=\left|a\right|+\left|b\right|$ is true for all non-negative real numbers.

Hence, the given statement $\left|a+b\right|=\left|a\right|+\left|b\right|$ is true sometimes for all real numbers.