Im trying to find all solutions satisfying $|x|+|y|<10 000$ given the Diophantine equation
$$1619x + 460y = 100 $$
I've already found $x$ and $y$ using Euclides algoritm that
$$x=17900+460k$$ $$y=-63000-1619k$$
Now to the part im stuck on. I realized I want to keep the inequality from $$|x|+|y|< 10000$$ to be able to find some interval for which $k$ it is satisfied.
Given $x$ and $y$:
$$|17900+460k|+|-63000-1619k|<10 000$$
The part I need help understanding is how to handle the abs signs to solve this mathematically.
I do know I can simply first find for what $x$ and $y$ that minimizes the Diophantine equation, and simply try different $k$ that satisfies it, but I want to solve it mathematically without guessing game.
Would anyone like to give me hint on how to handle the abs. signs and how to solve it? :)