Can we subtract an inequality from the same inequality, and if so, does that mean $0<0$ is a true statement?

Solution 1:

If we define what it means to subtract inequalities in terms of addition and multiplication, it becomes obvious why we cannot subtract two inequalities and expect the result to hold.

Suppose we have the following inequalities:

$$\begin{align} a&<b & &(1)\\ c&<d & &(2) \end{align}$$

To subtract these inequalities, what we actually do is to first negate the inequalities, and second add them.

Subtracting naively, we get:

$$\begin{align} a&<b & &(1)\\ -c&<-d & &(*)\\ \hline a-c&<b-d & &(3) \end{align}$$

But notice that when we negated $(2)$ to get $(*)$, we should have flipped the inequality. In fact, if $(2)$ is true, then $(*)$ cannot be true. Thus $(3)$ does not follow from $(1)$ and $(2)$, so inequality subtraction does not work, as you have defined it.

Solution 2:

You can add 2 inequalities, but you cannot subtract 2 inequalities.

1 < 2

0 < 3

If you subtract them, you get 1 < -1 which is definitely false.

Solution 3:

If you have

• $a < b$ and
• $c > d$

then you cannot say from this whether $a+c<b+d$ or $a+c>b+d$ or $a+c=b+d$, and any of these three could be the case.

If you then have

• $a < b$ and
• $e < f$

then, letting $c=-e$ and $d=-f$ and noting that this then implies implies $c>d$, so $a-e=a+c$ and $b-f=b+d$,

you therefore cannot say from this whether $a-e<b-f$ or $a-e>b-f$ or $a-e=b-f$ in exactly the same way as before, and any of these three could be the case.