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 
.
I would like to know where i went wrong in this method and is there any better method to draw the graph by hand.
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.
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 $y=2x+1$.