sketching the graph

Question

how to sketch the graph $|x+y|= m$, when $ m $ is some real number?

I personally can not see any efficient or known method to do so...

Answer 1

Case 1: Both $x$ and $y$ are positive(1st quadrant).

Then, the equality changes to $x+y=m$

So draw a line in the first quadrant joining $(0,m)$ and $(m,0)$.

Case 2: Both are negative (third quadrant)

Proceed as in 1.

$x+y=-m$.

Case 3: 2nd quadrant $(x<0,y>0)$

Note: if $|x|<|y|$, then $x+y$ is positive. On the other hand, negative.

Case 4: proceed as in 3

Answer 2

We can get a better idea of what the graph might look like by using the fact that $|x| = x$ if $x\geq 0$, and $|x| =-x$ if $x\lt 0$.

$$ |x-y| = m \implies \begin{cases} x-y = m & \text{when}\;x-y\geq 0 \iff x\geq y\\ x-y = -m & \text{when}\;x-y \lt 0\iff x < y\end{cases}$$

Answer 3

A very naive approach to draw this ends up like this: