# Graphing absolute value inside absolute value equation

How would I graph an equation with an absolute value inside an absolute value?

For example, $$\left|-\left|x\right|+1\right|+\left|y-2\right|=3$$

I tried graphing this on Desmos and it gave me some weird closed polygon. https://www.desmos.com/calculator/ysnqhkiuuk

In this case, you should know what $$y=|x|$$ looks like, which tells you that your equation is going to consist of line segments with vertices at places where the interior of the absolute values is $$0$$.
It is helpful to solve it for $$y$$: $$|y - 2| = 3 - | 1 - |x|\,|\\y = 2\pm(3 - | 1 - |x|\,|)$$ Which also tells you that the graph will be symmetric about the line $$y = 2$$, and since $$x$$ only appears inside the absolute value, the graph will also be symmetric about $$x = 0$$.
If we examine just the quadrant where $$x \ge 0$$ and $$y \ge 2$$, we have $$y = 5 - |1 - x|$$ For $$x \le 1, y = 5 - (1 -x) = 4 + x$$. For $$x \ge 1, y = 5 + (1 - x) = 6 - x$$ Note that since $$y \ge 2$$ in this quadrant, this only holds when $$6 - x \ge 2$$, thus $$x \le 4$$. So in this quadrant, this is the graph of $$y = \begin{cases} 4 + x& 0\le x \le 1\\6 -x & 1 \le x \le 4\end{cases}$$
Reflect that graph through $$x = 0$$, and then reflect again though $$y = 2$$ to get the full graph.