Given a formula $2|x|+3|y|\leq 12$, how many integer pairs of (x, y) are there?

Question

I just had a seventh grade math Olympiad test, and I had this question that confused me.

Given a formula $2|x|+3|y|\leq 12$, how many integer pairs of (x, y) are there?

I just checked in Desmos, and the answer was 15.

In the test, I didn't really have much time to ponder this. I don't know why I did this, but somehow I managed to just remove the absolute value signs and wrote this: $$\begin{align}&2|x|+3|y|\leq 12\\ \Rightarrow &2x+3y\leq 12\end{align}$$

I really don't know how I did that. But what I know is that I did it wrong during the test. So how DO I do this question?

Answer 1

$F:2|x|+3|y|\leq 12$ is the figure formed by the four lines $\pm 2x\pm 3y=12$. Count the number of integer points on these four lines. The number of points inside $F$ will then be anyways easy to count.

Looking at the graph of $2|x|+3|y|\leq 12$ (or in general $m|x|+n|y|\leq r$) may help.