I4.2 Find the value of $b$ if $b$ and $y$ satisfy $|b-y|=b+y-a$ and $|b+y|=b+a$.
From the first equation:
$(b-y=b+y-2$ or $y-b=b+y-2)$ and $b+y-2 \geq 0$
$(y=1$ or $b=1)$ and $b+y-2 \geq 0$
When $y=1 \Rightarrow b \geq 1$
When $b=1 \Rightarrow y \geq 1$
From the second equation:
$(b+y=b+2$ or $b+y=-b-2)$ and $b+2 \geq 0$
$(y=2$ or $2 b+y=-2)$ and $b \geq-2$
When $y=2$ and $b \geq-2$
When $2 b+y=-2$ and $b \geq-2 \Rightarrow(y \leq 2$ and $b \geq-2$ and $2 b+y=-2)$
(3) and (5): $y=1, b \geq 1$ and $y=2$ and $b \geq-2 \Rightarrow$ contradiction
(3) and (6): $y=1, b \geq 1$ and $(y \leq 2, b \geq-2,2 b+y=-2) \Rightarrow y=1$ and $b=-1.5$ and $b \geq 1$ !!!
(4) and (6): $(y \geq 1, b=1)$ and $(y \leq 2, b \geq-2,2 b+y=-2) \Rightarrow y=-4, b=1$ and $y \geq 1$ !!!
(4) and (5): $(b=1, y \geq 1)$ and $(y=2, b \geq-2) \Rightarrow b=1$ and $y=2$
$$
\therefore b=1
$$