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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.

Tornike Kacadze
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  • note that when you say "$-5.4\le y\le 4.4$", this is essentially talking about a function $y(x)$, i.e., with respect to $x$. For instance, when you say $y = 4.4$ the implicit assumption is that $x = 1$. Therefore, when you do the subtraction $x-y$ again, the values you attain are not freely variable. – SummerAtlas Jul 12 '23 at 15:22
  • so of course the new range you obtain is still reasonable, i.e., $-6.4\le x-y\le 6.4$ contains $-3.4\le x-y\le 3.4$, it is just that many of those values are not feasible. – SummerAtlas Jul 12 '23 at 15:23
  • correction: okay that is not really a function, but the point is having a certain value of $y$ in $-5.4\le y\le 4.4$ already says a lot about what $x$ can be – SummerAtlas Jul 12 '23 at 15:29
  • "Even ChatGPT" - that sounds as chatgpt is a good tool which it is not. – Peter Jul 24 '23 at 12:22

1 Answers1

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$y - 3.4 \leq x \leq 1 \implies y \leq 4.4.$

$-2 \leq x \leq 3.4 + y \implies -5.4 \leq y.$

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 $~-5.4 \leq y \leq 4.4.$

user2661923
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