How to sketch graphs for sums of absolute value expressions?

Question

I know how to sketch graphs for expressions like $|x+a|+|x+b|=k$ and the ones involving only one variable but how would I go about sketching the same for expressions like $$|x+y|+|2y-x|=4$$

I tried to make cases and go about sketching them interval by interval but I can only imagine doing it in the first or the third quadrant as in those cases the mod opens up perfectly but in other quadrants there seem to be a lot of cases so can someone suggest a procedure as such functions really seem to be troublesome when trying to calculate the area by integration?

Answer 1

I would first draw the lines $x+y=0$ and $2x+y=0$ which divide the plane into four regions which are defined by the choice of sign of the two expressions you are adding.

For each choice of signs you get a straight line (take out the absolute values and allocate a sign instead and you get a linear expression), and the part of the line you want is the part (if any) which intersects the corresponding region.

So you end up with six lines in order to identify the relevant pieces.

Answer 2

For the case $|x+a|+|x+b|=k$, you have exactly four cases:

1. $x+a\geq 0$ and $x+b\geq 0$
2. $x+a< 0$ and $x+b\geq 0$
3. $x+a\geq 0$ and $x+b< 0$
4. $x+a< 0$ and $x+b< 0$.

Four cases aren't really that troublesome to do, are they? Besides, at least one of these four cases should be impossible