Let A=(x,y) R R:|x+y| 3 and B=(x,y) R R:|x|+|y| 3.

Question

Let $A={(x,y)\in R	imes R:|x+y|\geqslant 3}$ and $B={(x,y)\in R	imes R:|x|+|y|\le 3}$.
If $C={(x,y)\in A\cap B:x=0$ or $y=0}$, then $\sum _{(x,y)\in C}|x+y|$ is :

Accepted Answer

$B$ is the diamond $|x|+|y|\leq3$ and $A$ requires $|x+y|\geq3$. On the axes ($x=0$ or $y=0$), the only points of $B$ satisfying $|x+y|\geq3$ are the diamond's vertices, where equality $|x+y|=3$ holds exactly:
$$C = \{(3,0),(-3,0),(0,3),(0,-3)\}$$

$$\sum_{(x,y)\in C}|x+y| = 3+3+3+3 = 12$$

Solutions · 1