Trying to understand idea behind absolute value?

Question

I came up with this question when trying to understand how to process the equation $|x| + |y| = 4$, and how its graph should look. It only started to make sense when I broke $|x|$ down into two separate statements for $x \geq 0$ and $x <0$. Then I took those two new statements and broke each one into two more statements involving $y\geq 0$ and $y <0$. I ended up with four statements, one for each quadrant of the $xy$-plane, each with a separate functions.

Is it necessary to look at the absolute value sign like this in higher level math? Focusing on the idea that it breaks a single statement into two separate statements that describe separate subsets of whatever $U$ you're working with? And that the union of those subsets equals the universal set?

Answer 1

Yes, this is the proper way because the absolute value of a real number is defined exactly this way:

$|x|= x $ if $x\ge 0$

$|x|=-x$ if $x<0$

Answer 2

In some cases you need to deal with absolute value as a piecewise linear function, just like you did. But other times, you can use geometry and symmetry to create a picture of the solution set.

Remember that $|x|$ is the distance from $x$ to $0$ on the number line. So in the plane, $|x|+|y|$ is the sum of the distances from $(x,y)$ to the $y$-axis and $x$-axis, respectively (yes, in that order—think about that for a minute).

Another way to think about it uses that fact that $|x|+|y|$ stays the same if $x$ is replaced by $-x$, or $y$ is replaced by $-y$. So the solution set $|x|+|y|=4$ has vertical and horizontal symmetry to it. So you can draw the portion in the first quadrant and reflect in both axes. If $(x,y)$ is in the first quadrant, then $|x|+|y| = x+y$. So take the segment of the line $x+y=4$ in the first quadrant, which connects $(0,4)$ to $(4,0)$, and reflect it around.