Define a binary operation * on the real numbers as $x * y=xy+x+y$ for all real numbers x and y.

Question

Define a binary operation * on the real numbers as $x * y=xy+x+y$ for all real numbers x and y. Let $a, b \in \mathbb{R}, a\geq b$. Prove if $x \ge -1$ in the real numbers, then $x * a \le x * b$.

I have absolutely no clue how to prove this. I have tried starting with the assumption but I generally end up with inequities that don't form a chain. It seems like it should be easy to prove with just knowing $a\leq b$. Any hints for how to start when assuming $x \geq -1$ ?

Answer 1

It's $$xa+x+a\leq xb+x+b$$ or $$(a-b)(x+1)\leq0,$$ which is true for $x\geq-1$ and $a\leq b$.

Answer 2

Hint:   use that $x * y = (x+1)(y+1) - 1\,$, so $\,x*a - x*b = (x+1)(a-b)\,$.