no. of positive integral solutions of ||x - 1| - 2| + x = 3 What are the no. of positive integral solutions of ||x - 1| - 2| + x = 3 ?
My effort

||x - 1| - 2| = 3 - x
|x - 1| - 2 = 3 - x   OR   |x - 1| - 2 = x - 3
|x - 1| = 5 - x OR |x - 1| = x - 1
x - 1 = 5 - x OR x - 1 = x - 5 OR x - 1 $\geq$ 0
2x = 6 OR x $\geq$ 1
Therefore, x $\geq$ 1

But the any value of x greater than 1 except 3 does not satisfy the equation. Where have I gone wrong?

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\begin{align}&\color{#66f}{\large\sum_{x\ =\ 0}^{\infty}
\delta_{\verts{\vphantom{\large A}\verts{x\ -\ 1}\ -\ 2}\ +\ x,3}}
=\sum_{x\ =\ 0}^{\infty}\oint_{\verts{z}\ =\ 1}
{1 \over z^{-\verts{\verts{\vphantom{\Large A}x\ -\ 1}\ -\ 2}\ -\ x\ +\ 4}}
\,{\dd z \over 2\pi\ic}
\\[5mm]&=\oint_{\verts{z}\ =\ 1}{1 \over z^{4}}
\sum_{x\ =\ 0}^{\infty}z^{\verts{\verts{\vphantom{\Large A}x\ -\ 1}\ -\ 2}\ +\ x}
\,{\dd z \over 2\pi\ic}
=\oint_{\verts{z}\ =\ 1}{1 \over z^{4}}\pars{%
z + z^{3} + \sum_{x\ =\ 2}^{\infty}z^{\verts{x\ -\ 3}\ +\ x}}
\,{\dd z \over 2\pi\ic}
\\[5mm]&=\oint_{\verts{z}\ =\ 1}{1 \over z^{4}}\pars{%
z + z^{3} + z^{3} + z^{3} + \sum_{x\ =\ 4}^{\infty}z^{2x - 3}}
\,{\dd z \over 2\pi\ic}
\\[5mm]&=\oint_{\verts{z}\ =\ 1}\pars{%
{1 \over z^{3}} + {3 \over z} + \sum_{x\ =\ 4}^{\infty}{1 \over z^{7 - 2x}}}
\,{\dd z \over 2\pi\ic}
\\[5mm]&=\overbrace{\oint_{\verts{z}\ =\ 1}{1 \over z^{3}}\,{\dd z \over 2\pi\ic}}
^{\ds{=\ \dsc{0}}}\ +\
\overbrace{\oint_{\verts{z}\ =\ 1}{3 \over z}\,{\dd z \over 2\pi\ic}}
^{\ds{=\ \dsc{3}}}\ +\
\sum_{x\ =\ 4}^{\infty}\overbrace{%
\oint_{\verts{z}\ =\ 1}{1 \over z^{7 - 2x}}\,{\dd z \over 2\pi\ic}}
^{\ds{=\ \dsc{\delta_{x,3}}}}\ =\
\color{#66f}{\LARGE 3}
\end{align}
