Size of fractions and spaces in arrays

I have the following code:

documentclass[12pt]{article}
usepackage{amsmath}
begin{document}
    $$hat{x}_n(t)=left{ begin{array}{cc}
    1 & -1leq t leq -frac{1}{n+1}  
    -(n+1) t & -frac{1}{n+1} leq t leq frac{1}{n+1} 
    -1 & frac{1}{n+1} leq t leq 1. 
    end{array}right.$$
end{document}

This looks like this:

The problem is that the size of the fractions is too small, if I try to change it using dfrac, the size is too big and they end up overlapping, because the spaces between each line are the same. Is there an easy way to change this? Thanks.

documentclass{article}
usepackage{mathtools}
begin{document}
[
hat{x}_n(t)=
begin{dcases}
    1 & -1leq t leq -frac{1}{n+1} ,, 
    -(n+1) t & -frac{1}{n+1} leq t leq frac{1}{n+1} ,, 
    -1 & frac{1}{n+1} leq t leq 1 , . 
end{dcases}
]
end{document}