How to plot the 2-D graph of $|x|+|x+y|=1$?

Question

Problem Description:

Today，I am countered with a integral problem that I need visualize a 2D graph of the equation $|x|+|x+y|=1$

I plots it step-by-step by hand:

• When $x \geq 0, x+|x+y|=1$ $$\begin{cases} 2x+y=1 & y\geq -x \\ y=-1 & y<-x \end{cases} $$
• When $x<0, -x+|x+y|=1$

$$\begin{cases} y=1 & y \geq -x \\ 2x+y=-1 & y<-x \end{cases} $$

So I ploted this graph by four lines.

However,now I want to use Mathematica to solve it.

Have a try:

Plot[y /. {Solve[Abs@x + Abs@(x + y) == 1, y]}, {x, -1, 1}, 
AspectRatio -> Automatic]

but the Mathematica give the following warning information:

“Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information. >>”

So I think my solution has flaws.My question is how to revise it or give another method to solve it?

Answer 1

No warning information:

With[{sol = y /. Solve[Abs@x + Abs@(x + y) == 1, y, Reals]},
 Plot[sol, {x, -2, 2}, PlotRange -> 2, AspectRatio -> Automatic]
 ]

Plot[y /. Solve[Abs@x + Abs@(x + y) == 1, y, Reals] //  Evaluate,
 {x, -2, 2}, PlotRange -> 2, AspectRatio -> Automatic, PlotStyle -> Blue]

Answer 2

ContourPlot[Abs[x] + Abs[x + y] == 1, {x, -2, 2}, {y, -2, 2}]