Question

На координатной плоскости изобразить фигуру и найти площадь
Ix-1I+Iy+1I+Ix+yI≤2

Answer 1

$|x-1|+|y+1|+|x+y| leq 2, \ x-1=0, x=1, \ y+1=0, y=-1, \ x+y=0, y=-x; \ 1) x-1 textless 0, y-1 textless 0, x+y textless 0, \ 1-x-y-1-x-y leq 2, \ -2x-2y-2 leq 0, \ x+y+1 geq 0, \ x+y+1 = 0, y=-x-1; \ 2)x-1 textless 0, y+1 textgreater 0, x+y textless 0, \ 1-x+y+1-x-y leq 2, \ -2x leq 0, \ x geq 0, \ x=0  (Oy); \ 3) x-1 textless 0, y+1 textgreater 0, x+y textgreater 0, \ 1-x+y+1+x+y leq 2, \ 2y leq 0, \ y leq 0, \ y=0  (Ox); \ 4) x-1 textgreater 0, y+1 textgreater 0, x+y textgreater 0, \ x-1+y+1+x+y leq 2, \ 2x+2y-2 leq 0, \ x+y-1 leq 0, \ x+y-1=0, y=1-x;$
$5)x-1 textgreater 0, y+1 textless 0, x+y textgreater 0, \ x-1-y-1+x+y leq 2, \ 2x-4 leq 0, \ x-2 leq 0, \ x-2=0, x=2, \ 6) x-1 textgreater 0, y+1 textless 0, x+y textless 0, \ x-1-y-1-x-y leq 2, \ -2y-4 leq 0, \ y+2 geq 0, \ y+2=0, y=-2; \ S=6S_triangle=6cdotfrac{1}{2}cdot1cdot1=3.$