Chapter 3.7, Problem 81PPE

To determine

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

Answer to Problem 81PPE

Given statement $\left|\frac{a}{b}\right|=\frac{\left|a\right|}{\left|b\right|}$ is always true $\forall a,b\ne 0,\in R$ .

Explanation of Solution

Given information :

Given statement $\left|\frac{a}{b}\right|=\frac{\left|a\right|}{\left|b\right|},\forall a,b\ne 0,\in R$

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

So, the left side part $\left|\frac{a}{b}\right|$ is positive ratio of $a\text{and}b.$

Similar way, the right side part $\frac{\left|a\right|}{\left|b\right|}$ is also a positive ratio of $a\text{and}b,$ as $\left|a\right|$ and $\left|b\right|$ are positive values.

Hence, the given statement $\left|\frac{a}{b}\right|=\frac{\left|a\right|}{\left|b\right|}$ is always true $\forall a,b\ne 0,\in R$ .