Multiplying inequalities.

Question

Suppose, I have two inequalities as follows:

$$x^2>1\tag{1}$$ $$y<-1\tag{2}$$

Is it possible to form a new inequality by combining $(1)$ and $(2)$ where the left-hand side would contain $x^2y$ and the right-hand side would contain only numbers?

Answer 1

You can multiply $$x^2>1\tag{1}$$ by a positive number without changing the direction of inequality.

In this case since $$ y<-1\tag{2}$$ we have $$-y>1$$ so you may multiply $$x^2>1$$ by $-y$ to get $$-x^2y>-y>1\implies-x^2y>1 $$ which is the same as multiplying the two inequalities $$-y>1$$ and $$x^2>1$$

Answer 2

We have $y<-1\iff -y>1>0$. So $-y>1$ and $x^2>1$. Notice that both sides of these two inequalities are non-negative, thus we can multiply them. So we have $-y \cdot x^2>1 \cdot 1=1$.

Answer 3

The two basic axioms are:

If $a < b$ then $a + c < b + c$ for all $c$

If $a < b$ and $c > 0$ then $ac < bc$.

From there there are some basic propositions that can be proven with those two axioms:

If $a > 0$ then $-a < 0$.

If $a < b$ and $c < 0$ then $ac > bc$.

And $1 > 0$. and for $c\ne 0$ then $c^2 > 0$.

....

So if $x^2 > 1$ and $y < - 1<0$ then $x^2\cdot y < 1\cdot y = y < -1$.

That's it.