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I tried to find out a way to calculate the graph for multivariable modulus functions(eg. $|x|+|y|=1+x$)

in this case made a table to find out all possible cases like:

$y$/$x$ $x > 0$ $x < 0$
$y > 0$ $y = x$ $y = 0$
$y < 0$ $x = 0$ $R$

On graphing this equation it gives this graph.
I would like to know where i went wrong in this method and is there any better method to draw the graph by hand.

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You're table is wrong, it should be

y/x x $\geq$ 0 x < 0
y $\geq$ 0 y = 1 y = 1 + 2x
y < 0 y = -1 y = -1 - 2x

You can find the formulas for each cell by using the fact that for $x<0, |x|=-x$ and for $x\geq 0, |x|=x$. Substitute this relation for $x,y$ for each cell.

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Since LHS is nonnegative, we know that $x \ge -1$.

Also, we know from $|y|$ that the function must be symmetric about the $x$-axis, hence we can focus on $y \ge 0$ and then reflect it.

If $y \ge 0$ and $x \ge 0$, then the equation becomes $|x|=x$, and hence our function becomes $y=1$.

If $y \ge 0$ and $x < 0$, then $|x|=-x$, and hence $y=2x+1$.

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