Chapter 5.2, Problem 45HP

To determine

Explain the statement: if a > b, then $\frac{1}{a}>\frac{1}{b}$ is sometimes, always or never true.

Answer to Problem 45HP

The statement is only true when a > 0 and b < 0.

Explanation of Solution

Given:

If a > b, then $\frac{1}{a}>\frac{1}{b}$

Concept Used:

Sometimes: Consider three cases.

Case 1: a > 0 and b > 0:

Choose a = 3 and b = 2, then 3 > 2 but $\frac{1}{3}<\frac{1}{2}$

Case 2: a < 0 and b < 0:

Choose a = − 2 and b = − 3. Then − 2 > − 3 , but $-\frac{1}{2}<-\frac{1}{3}$

Case 3: a > 0 and b < 0:

Choose a = 2 and b = − 3, then 2 > − 3 but $\frac{1}{2}>-\frac{1}{3}$

The statement is only true when a > 0 and b < 0.

Thus, the statement is only true when a > 0 and b < 0.