Perhaps this will be suitable (for your class exercise):

#!/usr/bin/env perl -l

use strict;
use warnings;

my @initial = qw{q w e r t y q w e r t y};
print "Initial: @initial";

my $last = '';
my @unique = map { $last eq $_ ? () : ($last = $_) } sort @initial;
print "Unique:  @unique";
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Output:

Initial: q w e r t y q w e r t y
Unique:  e q r t w y
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Update (alternative solution): You could even skip the creation of the unsorted array (your @array_q3):

#!/usr/bin/env perl -l

use strict;
use warnings;

my $last = '';
my @unique = map { $last eq $_ ? () : ($last = $_) } sort map { split 
+} <DATA>;
print "Unique:  @unique";

__DATA__
I am supposed to create a unique array
I am not supposed to use hashes
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Output:

Unique:  I a am array create hashes not supposed to unique use
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