# Aligning equations across the rows of a table

I would like to align all the "for all x > something" expressions in the following tabular environment.

My code for the above is as follows:

\begin{tabular}{rl|l}
1. & $f(x) \le g(x + y) \quad \forall x > 1$ & $x$ blah blah $f(x)$ and blah\\
2. & $g(x) \le g(x + y) + \log(x) \quad \forall x > 1$ & $\log(x)$ and $e^x$ blah blah\\
3. & $x \le \frac{(2\log(y))}{\sin(x)} \quad \forall x > 10$ & Blah blah $\cos(x)$ and $y$\\
4. & $f(x, y) = \left(e^{h(x)} + g(y)\right)^{e^x}$ & Using 3. and blah blah $x + y$\\
\hline
Hence & $x$ is $\mathcal{O}(x^x)$ & Because blah blah
\end{tabular}


I tried to perform the alignment using the aligned environment, but I have been unsuccessful. The following are my attempts:

## Attempt 1

\begin{tabular}{rl|l}
1. & \begin{aligned} &f(x) \le g(x + y) &&\quad \forall x > 1\\ &g(x) \le g(x + y) + \log(x) &&\quad \forall x > 1\\ &x \le \frac{(2\log(y))}{\sin(x)} &&\quad \forall x > 10\\ &f(x, y) = \left(e^{h(x)} + g(y)\right)^{e^x} \end{aligned} & $x$ blah blah $f(x)$ and blah\\
2. & & $\log(x)$ and $e^x$ blah blah\\
3. & & Blah blah $\cos(x)$ and $y$\\
4. & & Using 3. and blah blah $x + y$\\
\hline
Hence & $x$ is $\mathcal{O}(x^x)$ & Because blah blah
\end{tabular}


Output:

## Attempt 2

\begin{tabular}{rl|l}
\begin{aligned} 1. \\ 2. \\ 3. \\ 4. \end{aligned} &
\begin{aligned} &f(x) \le g(x + y) &&\quad \forall x > 1\\ &g(x) \le g(x + y) + \log(x) &&\quad \forall x > 1\\ &x \le \frac{(2\log(y))}{\sin(x)} &&\quad \forall x > 10\\ &f(x, y) = \left(e^{h(x)} + g(y)\right)^{e^x} \end{aligned} &
\begin{aligned} &\text{x$blah blah$f(x)$and blah}\\ &\text{$\log(x)$and$e^x$blah blah}\\ &\text{Blah blah$\cos(x)$and$y$}\\ &\text{Using 3. and blah blah$x + y} \end{aligned}\\
\hline
Hence & $x$ is $\mathcal{O}(x^x)$ & Because blah blah
\end{tabular}


Output:

In my second attempt, I managed to achieve the desired equation alignment, but the alignment across the rows of the table is still off.

Use \hphantom and zero-width box:

\documentclass{article}

\begin{document}

\begin{tabular}{rl|l}
\gdef\tmpbox{$g(x)\leq g(x+y)+\log(x)$}%
1. & \hbox to0pt {$f(x) \le g(x + y)$}\hphantom{\tmpbox}\quad $\forall x > 1$ & $x$ blah blah $f(x)$ and blah\\
2. & $g(x) \le g(x + y) + \log(x) \quad \forall x > 1$ & $\log(x)$ and $e^x$ blah blah\\
3. & \hbox to0pt{$x \le \frac{(2\log(y))}{\sin(x)}$}\hphantom{\tmpbox}\quad $\forall x > 10$ & Blah blah $\cos(x)$ and $y$\\
4. & $f(x, y) = \left(e^{h(x)} + g(y)\right)^{e^x}$ & Using 3. and blah blah $x + y$\\
\hline
Hence & $x$ is $\mathcal{O}(x^x)$ & Because blah blah
\end{tabular}

\end{document}