3

Consider the following

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}


\begin{document}
\begin{align}
X_{n}=\begin{cases}
               X_{n}^{I}= t_{1}\nu_{1}^{n}+r_{1}\nu_{2}^{n}, \quad -N\leq n<-d, \\
               X_{n}^{II}= t_{2}\nu_{1}^{n}+r_{2}\nu_{2}^{n}, \quad \ -d<n<-1, \\
               X_{n}^{III}=t_{3}\nu_{1}^{n}+r_{3}\nu_{2}^{n}, \quad \quad 1<n<d, \\
               X_{n}^{IV}=t_{4}\nu_{1}^{n}+r_{4}\nu_{2}^{n}, \quad \quad d<n\leq N, \\
\end{cases}
\end{align}

\end{document}

how can I make sure that the = sign in the left column is aligned as well as the inequalities on the right?

2

One option is to try the array environment instead of the cases one.

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}    
\begin{document}

\begin{equation}
a_{n}=\left\{%
       \begin{array}{@{}r@{{}={}}l@{\quad}r@{{\,n\,}}l@{}}
       X_{n}^{I}  & t_{1}\nu_{1}^{n}+r_{1}\nu_{2}^{n}, & -N \leq& <-d, \\
       X_{n}^{II} & t_{2}\nu_{1}^{n}+r_{2}\nu_{2}^{n}, & -d    <& <-1, \\
       X_{n}^{III}& t_{3}\nu_{1}^{n}+r_{3}\nu_{2}^{n}, &  1    <& <d, \\
       X_{n}^{IV} & t_{4}\nu_{1}^{n}+r_{4}\nu_{2}^{n}, &  d    <& \leq N, \\
\end{array}\right .
\end{equation}

\end{document}

enter image description here

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5

You can nest aligned inside cases:

\documentclass[12pt]{article}
\usepackage{amsmath}
\usepackage{amssymb}

\begin{document}

\begin{equation}
X_{n}=
\begin{cases}
  \begin{aligned}
  X_{n}^{I}   &= t_{1}\nu_{1}^{n}+r_{1}\nu_{2}^{n}, & -N &\leq n < -d, \\
  X_{n}^{II}  &= t_{2}\nu_{1}^{n}+r_{2}\nu_{2}^{n}, & -d &<    n < -1, \\
  X_{n}^{III} &= t_{3}\nu_{1}^{n}+r_{3}\nu_{2}^{n}, &  1 &<    n < d, \\
  X_{n}^{IV}  &= t_{4}\nu_{1}^{n}+r_{4}\nu_{2}^{n}, &  d &<    n \leq N,
  \end{aligned}
\end{cases}
\end{equation}

\end{document}

This takes advantage on the fact that the middle term in the inequalities is always the same. Otherwise a slightly more complicated approach with alignedat would be needed.

The advantage over array is in the slightly larger spacing between lines and, most important, that display style is automatically used, which allows for less cramped placement of subscripts and superscripts.

enter image description here

Note. Don't use align as a replacement for equation.

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