Chapter 1.5, Problem 49HP

To determine

To find: the reason that the inequality symbol does need to be reversed when multiplying or dividing by a negative number.

Answer to Problem 49HP

Explanation of Solution

Given information:

Calculation:

Consider the numbered line that numbers are “climbing” from negative to positive. For example $1<2$ which is true but if multiply or divide it by $-1$ and do not change the inequality sign, and get that $-1<-2$ which is not true statement, what is easy to show on numbered line.

Sufficient proof would look like this:

Let’s take $x,y$ and say that,

  $x<y$ only if $-x>-y$ .

Given proposition is obviously true, now let’s prove it.

  $\begin{array}{l}x<y\\ -x<-y\end{array}$

Now, if add $x$ and $y$ on both sides, then

  $\begin{array}{c}-x+x+y<-y+y+x\\ y<x\end{array}$

So, it’s can’t be true statement, giving that on the beginning that $x<y$ . So, inequality sign has to be changed in order to fulfill this statement.

Therefore, proven the proposition.