inequality question in x and y

Question

1. Let $a,b\ge2$ Show that we have $a+b\le ab$. For this i just showed that the inequality $x+y\le xy$ has some of its solutions in the range $x,y\ge 2$ (i factorised $xy-x-y+1$) is this a valid way to show this?
2. What does it mean to find an example when equality holds given an inequality in $x$ and $y$, and when it does not hold. for example, does equality refer to when the inequality is true or speicifially when both sides are equal?

Thanks

Answer 1

Because $$ab-a-b=(a-1)(b-1)-1\geq0.$$ The equality occurs for $a-1=b-1=1$, id est, for $a=b=2.$