** Using WeBWorK **

• The context you're using already has "R" for "all real numbers" -- none of the answers actually require this string, but you also didn't have a string in the context for this response.
• All three of the methods used to display the correct answers in the SOLUTIONS block have (almost) identical behavior. The fourth answer in the SOLUTIONS block is just an example of how the answer object can be extended to give more details about why it is the correct answer.
• There is actually no issue with extending the scope of the problem out to the original use of four unique variables in absolute value equations. I'm not sure what problems you were running into, but they were not related to the number of variables.

DOCUMENT(); # This should be the first executable line in the problem.

loadMacros(
  "PGstandard.pl",
  "PGML.pl",
  "contextInequalitiesAllowStrings.pl",
  "PGchoicemacros.pl"
);

TEXT(beginproblem());

######################################
#  Setup

@letters = ("a", "b","g","p","q","t","w","x","y","z");
@pick = NchooseK(10,4);
$v0 = $letters[$pick[0]];
$v1 = $letters[$pick[1]];
$v2 = $letters[$pick[2]];
$v3 = $letters[$pick[3]]; 

$a = random(4,10)*random(-1,1,2);
$b = $a>0 ? random(-12,-2,1) : random(2,12,1);
do {$c = non_zero_random(-15,15);} until ($c != $a && $c != $b);
do {$d = non_zero_random(-15,15);} until ($d != $a && $d != $b && $d != $c);

$RequiredFormat = "Enter your answer in the form: ".
"'\(x=m\)'.$BR For more than one answer: $BR".
"'\(x=m\) or \(x=n\)' $BR".
"(include the word ${BBOLD}or$EBOLD between the solutions), ".
"or enter your answer as a set \(\{m, n\}\). $BR".
"For no solution, type ${BBOLD}No solution${EBOLD} and ".
"if all real numbers are solutions, ".
"type ${BBOLD}R${EBOLD}.";

######################################
#  Main text

BEGIN_PGML
Solve:

A) [`|[$v0]|=[$a]`]  
Answer: [__]
    
A) [`|[$v1]|=[$b]`]  
Answer: [__]
    
A) [`|[$v2]|=[$c]`]  
Answer: [__]
    
A) [`|[$v3]|=[$d]`]  
Answer: [__]

END_PGML

######################################
#  Answer

Context("Inequalities-AllowStrings");
Context()->variables->are(
    $v0=>'Real',
    $v1=>'Real',
    $v2=>'Real',
    $v3=>'Real'
);
Context()->strings->add(
    None=>{},
    "No solution"=>{None}
);

sub errMsg {
  my $ans = shift;
  $ans->{ans_message} = $RequiredFormat
    if $ans->{ans_message} =~ /it looks like/;
  return $ans;
}

$ans_a = ($a<0) ? Compute("No solution") : Compute("$v0=-$a or $v0=$a")->reduce;
ANS($ans_a->cmp->withPostFilter(~~&errMsg));

$ans_b = ($b<0) ? Compute("No solution") : Compute("$v1=-$b or $v1=$b")->reduce;
ANS($ans_b->cmp->withPostFilter(~~&errMsg));

$ans_c = ($c<0) ? Compute("No solution") : Compute("$v2=-$c or $v2=$c")->reduce;
ANS($ans_c->cmp->withPostFilter(~~&errMsg));

$ans_d = ($d<0) ? Compute("No solution") : Compute("$v3=-$d or $v3=$d")->reduce;
ANS($ans_d->cmp->withPostFilter(~~&errMsg));

$showPartialCorrectAnswers = 1;

###################################
# Solution

# the solution for part A is stored in a perl
# variable using \( ... \) delimiters for LaTeX
$display_a = ($a<0) ? "No solution" : " \($v0 = −$a
\) or \($v0 = $a\)";

# the solution for B uses [@ ... @] to execute perl
# inside of the PGML block, and the output of that 
# perl code includes PGML LaTeX delimiters [`...`], 
# so append "**" to tell PGML to parse the result

# the solution for C and D are simpler, using only
# the answer variables inside PGML LaTeX mode
BEGIN_PGML_SOLUTION
A) [$display_a]
A) [@($b<0) ? "No solution" 
    : " [`$v1 = -$b`] or [`$v1=$b`]"@]**
A) [`[$ans_c]`]
A) Because [$d] is [@ $d > 0 ? 'positive' : 'negative' @], the answer is [`[$ans_d]`]
END_PGML_SOLUTION

ENDDOCUMENT();