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?
 A: You should start with expanding the innermost absolute value:
$$
    \vert \vert x-1 \vert -2 \vert = \begin{cases} \vert x-3 \vert & x \geqslant 1 \\ \vert -1-x \vert & x < 1 \end{cases} = \begin{cases} x-3 & x \geqslant 1, x > 3 \\ 3-x & x \geqslant 1, x \leqslant  3 \\  1+x  & x < 1, x \geqslant -1 \\ -1-x & x < 1, x < -1  \end{cases} = \begin{cases} x-3 & x > 3 \\ 3-x & 1 \leqslant x \leqslant 3\\  1+x  & -1 \leqslant x < 1 \\ -1-x & x < -1  \end{cases}
$$
You now seek solutions for each of these branches being equal to $3-x$ within the appropriate domain, giving you three integral solutions $x=1, x=2, x=3$.
A: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
 \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
 \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
 \newcommand{\dd}{{\rm d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\dsc}[1]{\displaystyle{\color{red}{#1}}}
 \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}
 \newcommand{\fermi}{\,{\rm f}}
 \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}
 \newcommand{\half}{{1 \over 2}}
 \newcommand{\ic}{{\rm i}}
 \newcommand{\iff}{\Longleftrightarrow}
 \newcommand{\imp}{\Longrightarrow}
 \newcommand{\Li}[1]{\,{\rm Li}_{#1}}
 \newcommand{\norm}[1]{\left\vert\left\vert\, #1\,\right\vert\right\vert}
 \newcommand{\pars}[1]{\left(\, #1 \,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\pp}{{\cal P}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,}
 \newcommand{\sech}{\,{\rm sech}}
 \newcommand{\sgn}{\,{\rm sgn}}
 \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}
 \newcommand{\ul}[1]{\underline{#1}}
 \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$
\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}
