Property of a system of two inequalities

Question

I have this system $$\begin{cases} a+b>1 \\ a-b>1 \end{cases}$$ can I sum the second inequality to the first getting $a>1$? Or this property can be used only equations?

Answer 1

You can. If $\alpha > 0$ and $\beta > 0$, $\alpha+\beta > 0$ and $\frac{\alpha+\beta}{2} > 0$.

In your case, you can even get more: as all terms are positive and the inequalities are in the same direction, there is nothing to be worried about here and you can multiply the two inequalities term-by-term. You then get $a^2-b^2 > 1^2=1$, so that $a^2 > 1+b^2$.

Answer 2

You can sum them, yes. But you can't subtract them. $$a+b \gt 1 \implies a+b= 1 + \alpha^2 \\a-b \gt 1 \implies a-b = 1+\beta^2 \\[12pt]2b = \alpha^2 - \beta^2$$but we don't know the values of $\alpha$ and $\beta$, so $\alpha^2 - \beta^2$ could take any value.

You can prove that multiplication works (and probably division too), although in these cases you have to be careful about whether the inequality will change direction or not.