Chapter 2.7, Problem 14WE

To determine

To prove:

If $a<0$ and $b>0$ then $\frac{a}{b}<0$

Explanation of Solution

Given information:

  $\begin{array}{l}a<0\\ b>0\end{array}$

Proof:

Here, the given proof statement is:

If $a<0$ and $b>0$ then $\frac{a}{b}<0$

1. $a<0$ and $b>0$

This is the given statement as per the problem.

2. $-a>0,-b<0$

Here, use the second multiplication property for order. On the both sides of the given inequality equation need to multiply by $\left(-1\right)$

3. Assume that:

If $a<0\Rightarrow \frac{1}{a}<0$

So, get that:

  $-a>0\Rightarrow -\frac{1}{b}<0$

4. Use the multiplication property of order to get that:

  $\left(-a\right)\cdot \left(-\frac{1}{b}\right)<0$

5. Use the multiplication property as:

  $\therefore \frac{a}{b}<0$

Hence proved.