ACT Strange Math Question "Check my way of solving please"

Question

I got my answer as J. This is my way of solving it.

$1 < x + y < 2$ *MINUS $1$ FROM EVERYTHING"

\begin{align} x + y - 1 &< 1 \\ +1 &\,+1 \end{align} "PLUS 1 TO EVERYTHING"

Result: $y = 2 - x$ When I graphed it, the graph looked exactly like J.

The only possible answers left would be J and K. Is the better answer J because plugging in $(0, 0)$ would make K wrong?

Someone check this with me please.

Answer 1

The boundaries for the region in question are $x+y=1$ and $x+y=2$. Rewriting in standard form, we get $y=-x+1$ and $y=-x+2$. So both boundaries should have a slope of $-1$, one having a $y$-intercept of $1$ and the other $2$. The answer is indeed J.

Answer 2

You have the right idea. Given $1 < x+y < 2$, just solve for $y$.

Subtract $x$ on both sides: $$1-x < y < 2-x$$

So $1-x$ is the lower dashed line below the shaded region and $2-x$ is the upper dashed line above the shaded region. The answer is $J$.

Answer 3

Yes. Though your working is confusing.

You want the region between the lines: $y=1-x$ and $y=2-x$

These both have slopes of $-1$, and further the $y$ intercepts are $1$ and $2$. (To double check, the $x$ intercepts are $1$ and $2$.) This matches only $J$.