What are the properties of the positive real numbers pair $(a,b)$ for which $a b \geq a + b$?

Question

Working set: $\left\{ a,b\in(0,\infty)\right\} $

For example I'm considering these as including/excluding sets:

Excluding 1. For any pair of numbers that, $a\in (0,1)$ and $b\in (0,\infty)$, the product always makes the result smaller than the pair of numbers.

$\left\{ a\in(0,1),b\in(0,\infty)\mid(a\times b\leq a)\land(a\times b\leq b)\right\} $

Including 2. For any pair of numbers that, $a\in [2,\infty)$ and $b\in [2,\infty)$, the product of both is always bigger than the sum of both.

$\left\{ a\in[2,\infty),b\in[2,\infty)\mid(a\times b\geq a + b)\right\} $

Excluding 3. For any pair of numbers that, $a\in [1,2)$ and $a = b$, the product always makes the result smaller than the sum of both.

$\left\{ a\in[1,2)\mid a^{2}\leq2a\right\} $

So I then can exclude pairs like $\left\{ (0.2,100),(1.2,1.2)\right\} $ and including $\left\{ (2,3),(2,4.3)\right\} $ just by checking if it belongs to any of those sets.

Question A: For any given pair where $a\in [1,2)$ and $b\in [2,\infty)$, which properties could I find about $[1,2)$, so I can reduce the set instead of evaluating it by $ab - a - b \geq 0$.

Question B: Could you tell me the proper name of all proposed properties, I mean if can be expressed by an specific theorem, conjecture, etc.

Answer 1

You can rearrange $ab \ge a + b$ to get $$\begin{eqnarray} ab - a - b & \ge & 0 \\ (a-1)(b-1) - 1 & \ge & 0 \\ (a-1)(b-1) & \ge & 1 \end{eqnarray}$$ Since $xy = 1$ is the simplest hyperbola, you have the points which lie "outside" a rectangular hyperbola centred on $(1, 1)$.

Answer 2

So you can rearrange to get $a(b-1) \geq b$. You can divide by b-1, and get two cases:

$b>1: a \geq \frac{b}{b-1}$

$b<1: a \leq \frac{b}{b-1}$

It's symmetrical, so the same basic algebra works for a and b swapped around. I hope that helps get you going.