Here are three ways of achieving your goal, depending on what the rest of your table looks like:
\documentclass{article}
\usepackage{mathtools,array}
\begin{document}
\[
\begin{array}{r@{}>{{}}l}
\multicolumn{2}{c}{\text{Some length heading for the LHS and RHS}} \\
\hline
v_2 & \geq 0 \\
(k-1)v_2 - (k+1)v_3 & \geq 0 \\
kv_3 & \geq 0 \\
kv_2 - kv_3 & \geq 0 \\
%\hspace{3em}% Possible horizontal alignment required
(k+1)v_1 + (-2k+1)v_2 & \geq 1
\end{array}
\]
\newcommand{\LHS}{\phantom{(k+1)v_1 + (-2k+1)v_2}}%
\[
\begin{array}{c}
\text{Some length heading for the LHS and RHS} \\
\hline
\LHS\mathllap{v_2} \geq 0 \\
\LHS\mathllap{(k-1)v_2 - (k+1)v_3} \geq 0 \\
\LHS\mathllap{kv_3} \geq 0 \\
\LHS\mathllap{kv_2 - kv_3} \geq 0 \\
(k+1)v_1 + (-2k+1)v_2 \geq 1
\end{array}
\]
\[
\begin{tabular}{>{\centering\arraybackslash}p{20em}}
Some length heading for the LHS and RHS \\
\hline
$\begin{aligned}
v_2 &\geq 0 \\
(k-1)v_2 - (k+1)v_3 &\geq 0 \\
kv_3 &\geq 0 \\
kv_2 - kv_3 &\geq 0 \\
(k+1)v_1 + (-2k+1)v_2 &\geq 1
\end{aligned}$
\end{tabular}
\]
\end{document}
The first construction may require some additional horizontal adjustment in order to make the alignment fit properly across the column:

The second uses the technique listed in Alignment of equals sign in multiple align environments with a combination of \phantoms and overlaps:

The third inserts an aligned in order to achieve the alignment. It requires a pre-specified column width (which can be measured, if needed):

As a side-note: The right-hand sides are all of similar size, so using an r-column would suffice. However, I assume the array construction is far larger than depicted and an r-column might affect other (currently invisible) layout.
lcolumn qualifier. – Bernard Dec 08 '15 at 16:39r:-) – David Carlisle Dec 08 '15 at 16:39