I have a question, if $-2\le x \le 1$ and $-3.4\le x-y \le 3.4$ and $a\le y \le b$ find $a$ and $b$.
that's how I did it:
if $-2\le x \le 1$ then $-1\le -x \le 2$
$$-3.4-1\le x-y+(-x) \le 3.4+2$$
$$-4.4 \le -y \le 5.4$$
$$-5.4 \le y \le 4.4$$
but than if we subtract that $y$ to $x$ we should get $-3.4$ to $3.4$ but we get this:
$$-2 - 4.4 \le x-y \le 1 + 5.4$$
$$ -6.4\le x-y \le 6.4$$
that is not the same, and I wonder why.
also If I calculate it different way I think I get the correct answer:
$$a\le y \le b$$
$$-b\le -y \le -a$$
$$-2-b\le x-y \le 1-a$$
$$-2-b=-3.4$$
$$1-a = 3.4$$
$$a = -2.4$$
$$b = 1.4$$
$$-2.4 \le y \le 1.4$$
and this seems to be correct but I'm writing a math test and there is no answer like this but there is the one above which I think is wrong. So what I'm doing wrong there? I just want to know if there is an error in the test, so I can point this out to others or something.
P.S
Even ChatGPT thinks that I'm right(I know it's useless in math It makes mistakes 99% of the time but it got the same answer as me on the first try).
Thanks everyone in advance.
$y - 3.4 \leq x \leq 1 \implies y \leq 4.4.$
$-2 \leq x \leq 3.4 + y \implies -5.4 \leq y.$
Therefore, the candidate range is $~-5.4 \leq y \leq 4.4.$
$~x = -2, y = -5.4~$ satisfies both inequalities.
Further, having $~x~$ increase from $~-2~$ to $~1~$ indicates that $~y~$ can attain any value in the range $~-5.4 \leq y \leq -2.4.$
$~x = 1, ~y = 4.4 ~$ satisfies both inequalities.
Further, having $~x~$ decrease from $~1~$ to $~-2~$ indicates that $~y~$ can attain any value in the range $~1.4 \leq y \leq 4.4.$
Consideration of (for example) $~x = 0,~$ then shows that $~y~$ can also achieve any value between $~-2.4~$ and $~1.4.$
Therefore, the entire candidate range is validated, and the final answer is $~-5.4 \leq y \leq 4.4.$