qu.1.topic=THEORY@

qu.1.1.mode=Non Permuting Multiple Selection@
qu.1.1.name=AbsValTrueFalse@
qu.1.1.comment=@
qu.1.1.editing=useHTML@
qu.1.1.solution=@
qu.1.1.algorithm=@
qu.1.1.uid=e6d74512-1c62-4dfd-b580-26307d70ccda@
qu.1.1.info=  Author=Jack Weiner, Gord Clement;
  Difficulty=Medium;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Theory;
  Course=Introduction to Calculus I;
@
qu.1.1.question=<p>Which of the following are always true?</p>@
qu.1.1.answer=1, 2, 3, 4, 5, 6, 7@
qu.1.1.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='|' close='|' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo><mi>x</mi></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo></mrow></math>@
qu.1.1.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='|' close='|' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mo lspace='0.0em' rspace='0.0em'> </mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>x</mi></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow></math>@
qu.1.1.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo></mrow></math>@
qu.1.1.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mi>x</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo></mrow></math>@
qu.1.1.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mfenced open='|' close='|' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math> or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='|' close='|' separators=','><mrow><mi>x</mi></mrow></mfenced></mrow></math>@
qu.1.1.choice.6=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow></math> or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow></math>@
qu.1.1.choice.7=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow></math> or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo></mrow><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='|' close='|' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>x</mi></mrow></mfenced></mrow></math>@
qu.1.1.fixed=@

qu.1.2.mode=Inline@
qu.1.2.name=Abs Val Inequalities Method@
qu.1.2.comment=@
qu.1.2.editing=useHTML@
qu.1.2.solution=@
qu.1.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$c=rint(6,9);
$d=rint(-5,5);
condition:ne($a,0);
$f=rint(3);
$d1=rint(3);
$d2=rint(3);
condition:ne($f,$d1)*ne($f,$d2)*ne($d1,$d2);
$F=maple("
if $f=0 then 
printf(MathML[ExportPresentation](($a)*x+($b)<=abs(($c)*x+($d))))
elif $f=1 then
printf(MathML[ExportPresentation](abs(($a)*x+($b))<=abs(($c)*x+($d))))
else
printf(MathML[ExportPresentation](abs(($a)*x+($b))<=($c)*x+($d)))
end if
");
$ANS=switch($f, "You must use cases", "You should square both sides", "It is recommended that you use cases");
$D1=switch($f, "It is recommended that you use cases", "You must use cases", "You should square both sides");
$D2=switch($f,"You should square both sides", "It is recommended that you use cases", "You must use cases");@
qu.1.2.uid=b82a62f1-8c46-4abe-9593-da45f5e4c22a@
qu.1.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Theory;
  Difficulty=Medium;
@
qu.1.2.weighting=1@
qu.1.2.numbering=alpha@
qu.1.2.part.1.grader=exact@
qu.1.2.part.1.name=sro_id_1@
qu.1.2.part.1.editing=useHTML@
qu.1.2.part.1.display.permute=true@
qu.1.2.part.1.answer.3=$D2@
qu.1.2.part.1.question=(Unset)@
qu.1.2.part.1.answer.2=$D1@
qu.1.2.part.1.answer.1=$ANS@
qu.1.2.part.1.mode=List@
qu.1.2.part.1.display=menu@
qu.1.2.part.1.credit.3=0.0@
qu.1.2.part.1.credit.2=0.0@
qu.1.2.part.1.credit.1=1.0@
qu.1.2.question=<p>&nbsp;</p><p>What is the best strategy to solve $F ?</p><p><span>&nbsp;</span><1><span>&nbsp;</span></p><p>&nbsp;</p><p>&nbsp;</p>@

qu.1.3.mode=Multiple Selection@
qu.1.3.name=inequalityTrueFalse@
qu.1.3.comment=@
qu.1.3.editing=useHTML@
qu.1.3.solution=@
qu.1.3.algorithm=@
qu.1.3.uid=5fde516f-1d64-40cb-a3d9-8f25fa1af562@
qu.1.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Theory;
  Difficulty=Medium;
@
qu.1.3.question=<p>Check the statements that are ALWAYS true.</p>@
qu.1.3.answer=2, 5, 6@
qu.1.3.choice.1=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&rArr;</mo><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></math>@
qu.1.3.choice.2=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mrow><mfenced open='&LeftBracketingBar;' close='&RightBracketingBar;' separators=','><mrow><mi>a</mi></mrow></mfenced></mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>b</mi><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></math>@
qu.1.3.choice.3=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mfenced open='|' close='|' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></math>@
qu.1.3.choice.4=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo></mrow><mrow><mi>a</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&amp;verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><msup><mrow><mi>b</mi><mrow><mo lspace='0.0em' rspace='0.0em'>&hArr;</mo></mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></math>@
qu.1.3.choice.5=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo></mrow><mrow><mi>a</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mfenced open='|' close='|' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&hArr;</mo><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></math>@
qu.1.3.choice.6=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mfenced open='|' close='|' separators=','><mrow><mi>b</mi></mrow></mfenced></mrow><mrow><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' accent='true'>&rArr;</mo></mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>b</mi></mrow><mrow></mrow></math> or <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>b</mi></mrow></math>@
qu.1.3.choice.7=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup><mrow><mo lspace='0.0em' rspace='0.0em' stretchy='true' accent='true'>&rArr;</mo></mrow><mi>a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo><mi>b</mi></mrow></math>@
qu.1.3.fixed=@

qu.2.topic=LinInequalities@

qu.2.1.question=<p>Solve the inequality $El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;gt;</mo></mrow></math>$Er. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&lsqb;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo></mrow><mrow><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' fence='true' lspace='0.1666667em' rspace='0.1666667em' stretchy='true'>&rpar;</mo></mrow></math>enter [-3,infinity).</p>@
qu.2.1.maple=grade("$RESPONSE",$ANS);@
qu.2.1.allow2d=0@
qu.2.1.maple_answer=show($ANS);@
qu.2.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.2.1.type=maple@
qu.2.1.mode=Maple@
qu.2.1.name=ax+bGTcx+d@
qu.2.1.comment=<p>Solution:</p>
<p>$El > $Er</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo>&$symbol;</mo></mrow></math>$ans</p>
<p>Note:</p>
<p>The RED line is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math> $El and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the RED line is ABOVE the GREEN line. Look!</p>
<p>$plot</p>@
qu.2.1.editing=useHTML@
qu.2.1.solution=@
qu.2.1.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint(-5,5);
condition:ne($a,$c);
condition:ne($b,$d);
$m = maple("
if ($a) < ($c) then
MathML[ExportPresentation](($a)*x+($b)), MathML[ExportPresentation](($c)*x+($d)), 0
else
MathML[ExportPresentation](($a)*x+($b)), MathML[ExportPresentation](($c)*x+($d)), 1;
end if;
");
$El=switch(0, $m);
$Er=switch(1,$m);
$temp=switch(2,$m);
$ANS=switch($temp, '"(-infinity,(($d)-($b))/(($a)-($c)))"', '"((($d)-($b))/(($a)-($c)),infinity)"');
$diffac=($a)-($c);
$diffdb=($d)-($b);
$ans=mathml("$diffdb/$diffac");
$symbol=switch($temp, "lt", "gt");
$plot=plotmaple("plot([($a)*x+($b),($c)*x+($d)],x=($diffdb)/($diffac)-4..($diffdb)/($diffac)+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.1.uid=a2971498-4001-4359-b3a1-af98111aa6f9@
qu.2.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Linear Inequalities;
  Difficulty=Easy;
@

qu.2.2.question=<p>Solve the inequality $El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></math>$Er. Give your answer using interval notation and enter infinity for<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter [-3,infinity).</p>@
qu.2.2.maple=grade("$RESPONSE",$ANS);@
qu.2.2.allow2d=0@
qu.2.2.maple_answer=show($ANS);@
qu.2.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.2.2.type=maple@
qu.2.2.mode=Maple@
qu.2.2.name=ax+bLEcx+d@
qu.2.2.comment=<p>Solution:</p>
<p>$El <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo></mrow></math>$Er</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo>&$symbol;</mo></mrow></math>$ans</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>The RED line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the RED line is UNDER the GREEN line and also when they meet. Look!</p>
<p>$plot</p>@
qu.2.2.editing=useHTML@
qu.2.2.solution=@
qu.2.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint(-5,5);
condition:ne($a,$c);
condition:ne((($a)*($b)*($c)),0);
$m=maple("
if ($a) > ($c) then
MathML[ExportPresentation](($a)*x+($b)), MathML[ExportPresentation](($c)*x+($d)), 0
else
MathML[ExportPresentation](($a)*x+($b)), MathML[ExportPresentation](($c)*x+($d)), 1
end if;
");
$El=switch(0,$m);
$Er=switch(1,$m);
$temp=switch(2,$m);
$ANS=switch($temp,'"(-infinity,(($d)-($b))/(($a)-($c))]"','"[(($d)-($b))/(($a)-($c)),infinity)"');
$diffac=($a)-($c);
$diffdb=($d)-($b);
$ans=mathml("$diffdb/$diffac");
$symbol=if(lt($diffac,0),"ge","leq");
$plot=plotmaple("plot([($a)*x+($b),($c)*x+($d)],x=($diffdb)/($diffac)-4..($diffdb)/($diffac)+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.2.uid=6be57fa8-f576-42f3-8bb2-362c3b34ab69@
qu.2.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Sub-Topic=Linear Inequalities;
  Topic=Inequalities and Absolute Value;
  Difficulty=Easy;
@

qu.2.3.question=<p>Solve the inequality $El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$Er. Give your answer using interval notation and enter infinity for<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter [-3,infinity).</p>@
qu.2.3.maple=grade("$RESPONSE",$ANS);@
qu.2.3.allow2d=0@
qu.2.3.maple_answer=show($ANS);@
qu.2.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.2.3.type=maple@
qu.2.3.mode=Maple@
qu.2.3.name=ax+bLTcx+d@
qu.2.3.comment=<p>Solution:</p>
<p>$El < $Er</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo>&$symbol;</mo></mrow></math>$ans</p>
<p>Note:</p>
<p>The RED line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math> $Er. We want to know in this question when the RED line is UNDER the GREEN line. Look!</p>
<p>$plot</p>@
qu.2.3.editing=useHTML@
qu.2.3.solution=@
qu.2.3.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint(-5,5);
condition:ne($a*$b*$c,0);
condition:ne($a,$c);
$m = maple("
if ($a) > ($c) then
MathML[ExportPresentation](($a)*x+($b)),MathML[ExportPresentation](($c)*x+($d)), 0
else 
MathML[ExportPresentation](($a)*x+($b)),MathML[ExportPresentation](($c)*x+($d)), 1
end if;
");
$El=switch(0,$m);
$Er=switch(1, $m);
$temp=switch(2,$m);
$ANS=switch($temp, '"(-infinity,(($d)-($b))/(($a)-($c)))"','"((($d)-($b))/(($a)-($c)),infinity)"');
$diffac = $a - ($c);
$diffdb = $d - ($b);
$ans = mathml("$diffdb/$diffac");
$symbol=switch($temp,"lt", "gt");
$plot=plotmaple("plot([($a)*x+($b),($c)*x+($d)],x=($diffdb)/($diffac)-4..($diffdb)/($diffac)+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.3.uid=4306aca9-82f2-4624-9c9a-e1acf41d611f@
qu.2.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Linear Inequalities;
  Difficulty=Easy;
@

qu.2.4.question=<p>Solve the inequality $El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&ge;</mo></mrow></math>$Er. Give your answer using interval notation and enter infinity for<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter [-3,infinity).</p>@
qu.2.4.maple=grade("$RESPONSE",$ANS);@
qu.2.4.allow2d=0@
qu.2.4.maple_answer=show($ANS);@
qu.2.4.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.2.4.type=maple@
qu.2.4.mode=Maple@
qu.2.4.name=ax+bGEcx+d@
qu.2.4.comment=<p>Solution:</p>
<p>$El <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo></mrow></math>$Er</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo>&$symbol;</mo></mrow></math>$ans</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>The RED line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the RED line is ABOVE the GREEN line and when they meet. Look!</p>
<p>$plot</p>@
qu.2.4.editing=useHTML@
qu.2.4.solution=@
qu.2.4.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint(-5,5);
condition:ne($a,$c);
condition:ne($b,$d);
condition:ne(($a)*($b)*($c),0);
$m=maple("
if ($a)<($c) then
MathML[ExportPresentation](($a)*x+($b)),MathML[ExportPresentation](($c)*x+($d)), 0
else
MathML[ExportPresentation](($a)*x+($b)),MathML[ExportPresentation](($c)*x+($d)), 1
end if;
");
$El=switch(0,$m);
$Er=switch(1,$m);
$temp=switch(2,$m);
$ANS=switch($temp,'"(-infinity,(($d)-($b))/(($a)-($c))]"', '"[(($d)-($b))/(($a)-($c)),infinity)"');
$diffac=($a)-($c);
$diffdb=($d)-($b);
$ans=mathml("$diffdb/$diffac");
$symbol=if(lt($diffac,0),"leq","ge");
$plot=plotmaple("plot([($a)*x+($b),($c)*x+($d)],x=($diffdb)/($diffac)-4..($diffdb)/($diffac)+4,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.2.4.uid=db74b14f-51a7-4d4d-8093-309ed6e5e8b2@
qu.2.4.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Linear Inequalities;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
@

qu.3.topic=aLTbx+cLTd@

qu.3.1.question=<p>Solve the inequality <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$E<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'></mi></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow><mrow><mi mathvariant='normal'>$d</mi></mrow></math>. Give your answer using interval notation.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&infin;</mi></mrow></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.3.1.maple=grade("$RESPONSE",$ANS);@
qu.3.1.allow2d=0@
qu.3.1.maple_answer=show($ANS);@
qu.3.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.3.1.type=maple@
qu.3.1.mode=Maple@
qu.3.1.name=a LT bx+c LE d@
qu.3.1.comment=<p>Solution:</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$b</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$c</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diff1</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$b</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi mathvariant='normal'>$diff2</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$sol1</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.0em' rspace='0.0em'>&ge;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$sol2</mi></mrow></math></p>
<p>&nbsp;</p>
<p>In interval notation, the answer is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='[' close=')' separators=','><mrow><mi mathvariant='normal'>$sol2</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$sol1</mi></mrow></mfenced></mrow></math>.</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>&nbsp;</p>
<p>The RED line is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$a</mi></mrow></math>. The GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$E. The BLUE line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$d</mi></mrow></math>. Look at where the RED line is UNDER the GREEN line <strong>AND </strong>the GREEN line is UNDER or ON the BLUE line.</p>
<p>$plot</p>@
qu.3.1.editing=useHTML@
qu.3.1.solution=@
qu.3.1.algorithm=$a=rint(-5,1);
$b=rint(-5,5);
$c=rint(-5,5);
$d=rint($a+1,5);
condition:ne($b,0);
condition:ne($b,1);
$m=maple("
MathML[ExportPresentation](($b)*x+($c)), convert((($a)-($c))/($b),string), convert((($d)-($c))/($b),string) 
");
$E=switch(0,$m);
$sol1=switch(1,$m);
$sol2=switch(2,$m);
$ANS=switch(lt($b,0),'"((($a)-($c))/($b),(($d)-($c))/($b)]"','"[(($d)-($c))/($b),(($a)-($c))/($b))"');
$plot=plotmaple("plot([($a),($b)*x+($c),$d],x=-5..5,thickness=2,
color=[red,green,blue]),plotdevice='gif', plotoptions='height=250,width=250'");
$symbol1=switch(lt($b,0), "lt", "gt");
$symbol2=switch(lt($b,0), "?", "?");
$diff1= ($a)-($c);
$diff2= ($d)-($c);@
qu.3.1.uid=29ade35c-a04f-4b72-ae20-e3376cb2f04d@
qu.3.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Basic Inequalities;
  Difficulty=Easy;
@

qu.3.2.question=<p>Solve the inequality $E1<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></math>$E2<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&lt;</mo></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$e</mi></mrow></math>. Give your answer using interval notation.<o:p></o:p></p>@
qu.3.2.maple=grade("$RESPONSE",$ANS);@
qu.3.2.allow2d=0@
qu.3.2.maple_answer=show($ANS);@
qu.3.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.3.2.type=maple@
qu.3.2.mode=Maple@
qu.3.2.name=ax + b LE cx + d LT d@
qu.3.2.comment=<p>Solution:<br />
Solve the inequalities seperately, then see which <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math></em> values make both inequalities true.</p>
<p>&nbsp;</p>
<p>$E1 <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo></mrow></math>$E2</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$ans1</mi></mrow></math></p>
<p>AND</p>
<p>$E2 <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$e</mi></mrow><mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$e</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$d</mi></mrow></mfenced></mrow><mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$ans2</mi></mrow></math></p>
<p>&nbsp;</p>
<p>We need the <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math></em> values that make <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$ans1</mi></mrow></math>AND <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$ans2</mi></mrow></math>.</p>
<p>Therefore, the interval is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=']' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='normal' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mi mathvariant='normal'>$ans1</mi></mrow></mfenced></mrow></math>.</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>The RED line is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math> $E1. The GREEN line is&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math> $E2. The BLUE line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$e</mi></mrow></math>. Look at where the RED line is UNDER (or on) the GREEN line <strong>AND </strong>the GREEN line is UNDER the BLUE line.</p>
<p>$plot</p>@
qu.3.2.editing=useHTML@
qu.3.2.hint.1=Solve the two inequalities seperately, then find the values of <em>x</em> that solve both inequalities at the same time.@
qu.3.2.solution=@
qu.3.2.algorithm=$a=rint(1,10);
$b=rint(-5,5);
$c=rint(1,5);
$d=rint(-5,5);
$e=rint(-5,10);
condition:ne($a,$c);
condition:lt(($d-$b)/($a-$c),($e-$d)/$c);
$m=maple("
if $c<($a) then 
MathML[ExportPresentation](($a)*x+($b)), MathML[ExportPresentation](($c)*x+($d)), 0, convert((($d)-($b))/(($a)-$c),string), convert((($e)-($d))/$c,string);
else
MathML[ExportPresentation](($a)*x+($b)), MathML[ExportPresentation](($c)*x+($d)), 1, convert((($d)-($b))/(($a)-$c),string), convert((($e)-($d))/$c,string);
end if;
");
$E1=switch(0, $m);
$E2=switch(1, $m);
$temp=switch(2, $m);
$symbol=switch($temp, "?", "?");
$ANS=switch($temp,'"(-infinity,(($d)-($b))/(($a)-($c))]"', '"[(($d)-($b))/(($a)-($c)),(($e)-($d))/($c))"');
$plot=plotmaple("plot([($a)*x+($b),($c)*x+($d),$e],x=-8..8,thickness=2,
color=[red,green,blue]),plotdevice='gif', plotoptions='height=250,width=250'");
$diffac=$a-$c;
$diffdb=$d-$b;
$diffed=$e-$d;
$ans1=switch(3,$m);
$ans2=switch(4,$m);
$sol=switch($temp,"x?$ans1",  "$ans1?x<$ans2");@
qu.3.2.uid=c3a44cb5-22f5-400a-b4a3-7f2a5f0fc8f3@
qu.3.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Solving Multiple Inequalities;
  Difficulty=Medium;
@

qu.4.topic=(x-a)^m*(x-b)^n*(x-c)@

qu.4.1.question=<p>Solve the inequality $Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.4.1.maple=grade("$RESPONSE",$ANS);@
qu.4.1.allow2d=0@
qu.4.1.maple_answer=show($ANS);@
qu.4.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.1.type=maple@
qu.4.1.mode=Maple@
qu.4.1.name=(x+a)^(1/n) * (x-b) * (x-c) GT 0@
qu.4.1.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Q. Look at where the graph is ABOVE the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>-axis! Note that the <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math> intercepts are<strong> NEVER </strong>part of the solution because the expression is greater than <strong>but not equal to</strong> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn></mrow></math>.</p>
<p>$plot</p>@
qu.4.1.editing=useHTML@
qu.4.1.hint.1=Set up a number line and consider whether each factor will be positive or negative on each interval.@
qu.4.1.solution=@
qu.4.1.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,3);
$m=maple("
f:=surd((x-($a))^$n,3)*(x-($b))*(x-$c):
if ($n=1 or $n=3) then 
convert(f,string),MathML[ExportPresentation](f), 0
else 
convert(f,string),MathML[ExportPresentation](f), 1
end if;
");
$f=switch(0,$m);
$Q=switch(1,$m);
$A=switch(2,$m);
$ANS=switch($A, '"($a,$b) U ($c,infinity)"','"(-infinity,$a) U ($a,$b) U ($c,infinity)"');
$plot=plotmaple("plot($f,x=$a-1..$c+1,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.1.uid=65c498ca-dda5-4a27-89e0-28cba8db57f2@
qu.4.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Difficulty=Easy;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Number Line Analysis;
  Feature=Interval Answer;
@

qu.4.2.question=<p>Solve the inequality $Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mn>0</mn></mrow></math>. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.4.2.maple=grade("$RESPONSE",$ANS);@
qu.4.2.allow2d=0@
qu.4.2.maple_answer=show($ANS);@
qu.4.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.2.type=maple@
qu.4.2.mode=Maple@
qu.4.2.name=(x+a)^(n/3) * (x-b) * (x-c) LE 0@
qu.4.2.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Q. Look at where the graph is BELOW OR ON the x axis.</p>
<p>$plot</p>@
qu.4.2.editing=useHTML@
qu.4.2.hint.1=Set up a number line and consider whether each factor will be positive or negative on each interval.@
qu.4.2.solution=@
qu.4.2.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,9,4);
$m=maple("
f:= surd((x-($a))^$n,3)*(x-($b))*(x-$c):
if ($n=1 or $n=5) then 
convert(f,string),MathML[ExportPresentation](f), 0
else 
convert(f,string),MathML[ExportPresentation](f), 1
end if;
");
$f=switch(0,$m);
$Q=switch(1,$m);
$A=switch(2,$m);
$ANS=switch($A, '"(-infinity,$a] U [$b,$c]"
','"[$b,$c]"');

$plot=plotmaple("plot($f,x=$a-1..$c+1,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.2.uid=f91c4bf2-ee02-4cfb-b303-22ea3ece6d60@
qu.4.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Number Line Analysis;
  Difficulty=Easy;
  Feature=Interval Answer;
@

qu.4.3.question=<p>Solve the inequality $Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo mathvariant='italic' separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math> you would enter (-infinity,-3) U [4,infinity).</p>@
qu.4.3.maple=grade("$RESPONSE",$ANS);@
qu.4.3.allow2d=0@
qu.4.3.maple_answer=show($ANS);@
qu.4.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.3.type=maple@
qu.4.3.mode=Maple@
qu.4.3.name=(x+a)^n * (x-b) * (x-c) GT 0@
qu.4.3.comment=<p>Here is the plot of y=$Q. Look at where the graph is ABOVE the x axis!</p>
<p>$plot</p>@
qu.4.3.editing=useHTML@
qu.4.3.hint.1=Set up a number line and consider whether each factor will be positive or negative on each interval.@
qu.4.3.solution=@
qu.4.3.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,4);
$m=maple("
f:=(x-($a))^$n*(x-($b))*(x-$c):
if ($n=1 or $n=3) then 
convert(f,string),MathML[ExportPresentation](f), 0
else 
convert(f,string),MathML[ExportPresentation](f), 1
end if;
");
$f=switch(0,$m);
$Q=switch(1,$m);
$A=switch(2,$m);
$ANS=switch($A, '"($a,$b) U ($c,infinity)"','"(-infinity,$a) U ($a,$b) U ($c,infinity)"');
$plot=plotmaple("plot($f,x=$a-1..$c+1,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.3.uid=da789ecf-1109-4f61-b203-75e6d07127ad@
qu.4.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Number line analysis;
  Difficulty=Easy;
  Feature=Interval Answer;
@

qu.4.4.question=<p>Solve the inequality $Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mn>0</mn></mrow></math>. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.4.4.maple=grade("$RESPONSE",$ANS);@
qu.4.4.allow2d=0@
qu.4.4.maple_answer=show($ANS);@
qu.4.4.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.4.4.type=maple@
qu.4.4.mode=Maple@
qu.4.4.name=(x+a)^n * (x-b) * (x-c) LE 0@
qu.4.4.comment=<p>Here is the plot of y=$Q. Look at where the graph is BELOW OR ON the x axis.</p>
<p>$plot</p>@
qu.4.4.editing=useHTML@
qu.4.4.hint.1=Set up a number line and consider whether each factor will be positive or negative on each interval.@
qu.4.4.solution=@
qu.4.4.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,5,2);
$m=maple("
f:= (x-($a))^$n*(x-($b))*(x-$c):
if ($n=1 or $n=3) then 
convert(f,string),MathML[ExportPresentation](f), 0
else 
convert(f,string),MathML[ExportPresentation](f), 1
end if;
");
$f=switch(0,$m);
$Q=switch(1,$m);
$A=switch(2,$m);
$ANS=switch($A, '"(-infinity,$a] U [$b,$c]"','"[$b,$c]"');
$plot=plotmaple("plot($f,x=$a-1..$c+1,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.4.4.uid=c6c5daff-3095-4e79-8f4e-17f540ea8ea1@
qu.4.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Number Line Analysis;
  Difficulty=Easy;
  Feautre=Interval Answer;
@

qu.5.topic=(x+a)^n * (x-c) / (x-b)@

qu.5.1.question=<p>Solve the inequality $Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mn>0</mn></mrow></math>. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.5.1.maple=grade("$RESPONSE",$ANS);@
qu.5.1.allow2d=0@
qu.5.1.maple_answer=show($ANS);@
qu.5.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.5.1.type=maple@
qu.5.1.mode=Maple@
qu.5.1.name=(x+a)^n/(x-b)*(x-c) LE 0@
qu.5.1.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow></math>$Q. Look at where the graph is BELOW OR ON the x axis.</p>
<p>$plot</p>@
qu.5.1.editing=useHTML@
qu.5.1.hint.1=Set up a number line and consider whether each factor will be positive or negative on each interval.@
qu.5.1.solution=@
qu.5.1.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$m=maple("
f:=(x-($a))/(x-($b))*(x-($c)):
convert(f,string),MathML[ExportPresentation](f)
");
$f=switch(0,$m);
$Q=switch(1,$m);
$ANS='"(-infinity,$a] U ($b,$c]"';
$plot=plotmaple("plot($f,x=$a-2..$c+2,y=-25..25,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.1.uid=c06b9579-17e9-4b6a-91bc-474b3c0159ea@
qu.5.1.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Number Line Analysis;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Feature=Interval Answer;
@

qu.5.2.question=<p>Solve the inequality $Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>. Give your answer using interval notaition. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&infin;</mi></mrow></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.5.2.maple=grade("$RESPONSE",$ANS);@
qu.5.2.allow2d=0@
qu.5.2.maple_answer=show($ANS);@
qu.5.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.5.2.type=maple@
qu.5.2.mode=Maple@
qu.5.2.name=(x+a)^(n/3)  / (x-b) * (x-c)  GT 0@
qu.5.2.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Q. Look at where the graph is ABOVE the x axis!</p>
<p>$plot</p>@
qu.5.2.editing=useHTML@
qu.5.2.hint.1=Set up a number line and consider whether each factor will be positive or negative on each interval.@
qu.5.2.solution=@
qu.5.2.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,3);
$m=maple("
f:=surd((x-($a))^($n),3)/(x-($b))*(x-($c)):
if ($n=1) then 
convert(f,string),MathML[ExportPresentation](f), 0
else 
convert(f,string),MathML[ExportPresentation](f), 1
end if;
");
$f=switch(0,$m);
$Q=switch(1,$m);
$A=switch(2,$m);
$ANS=switch($A, '"($a,$b) U ($c,infinity)"',' "(-infinity,$a) U ($a,$b) U ($c,infinity)"');
$plot=plotmaple("plot($f,x=$a-2..$c+2,y=-25..25,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.2.uid=1b2018bd-5a6a-41d7-a071-7c22463b86a2@
qu.5.2.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Number Line Analysis;
  Course=Introduction to Calculus I;
  Feature=Interval Answer;
  Difficulty=Easy;
@

qu.5.3.question=<p>Solve the inequality $Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo><mn>0</mn></mrow></math>. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&infin;</mi></mrow></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.5.3.maple=grade("$RESPONSE",$ANS);@
qu.5.3.allow2d=0@
qu.5.3.maple_answer=show($ANS);@
qu.5.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.5.3.type=maple@
qu.5.3.mode=Maple@
qu.5.3.name=(x+a)^n/(x-b)*(x-c) GT 0@
qu.5.3.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Q. Look at where the graph is ABOVE the x axis!</p>
<p>$plot</p>@
qu.5.3.editing=useHTML@
qu.5.3.hint.1=Set up a number line and consider whether each factor will be positive or negative on each interval.@
qu.5.3.solution=@
qu.5.3.algorithm=$a=rint(-8,-1);
$b=rint(0,5);
$c=rint(6,10);
$n=rint(1,3);
$m=maple("
f:=(x-($a))^$n/(x-($b))*(x-($c)):
if ($n=1) then 
convert(f,string),MathML[ExportPresentation](f), 0
else 
convert(f,string),MathML[ExportPresentation](f), 1
end if;
");
$f=switch(0,$m);
$Q=switch(1,$m);
$A=switch(2,$m);
$ANS=switch($A, '"($a,$b) U ($c,infinity)"
','"(-infinity,$a) U ($a,$b) U ($c,infinity)"');
$plot=plotmaple("plot($f,x=$a-2..$c+1,y=-25..25,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.3.uid=9b4ff3f4-7f7d-4ead-b663-016c8544226d@
qu.5.3.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Number Line Analysis;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
  Feature=Interval Answer;
@

qu.5.4.question=<p>Solve the inequality $Q<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&le;</mo><mn>0</mn></mrow></math>. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>&infin;</mi><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&infin;</mi></mrow></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.5.4.maple=grade("$RESPONSE",$ANS);@
qu.5.4.allow2d=0@
qu.5.4.maple_answer=show($ANS);@
qu.5.4.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.5.4.type=maple@
qu.5.4.mode=Maple@
qu.5.4.name=(x+a)^(n/3) / (x-b) * (x-c) LE 0@
qu.5.4.comment=<p>Here is the plot of <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math>$Q. Look at where the graph is BELOW OR ON the x axis.</p>
<p>$plot</p>@
qu.5.4.editing=useHTML@
qu.5.4.hint.1=Set up a number line and consider whether each factor will be positive or negative on each interval.@
qu.5.4.solution=@
qu.5.4.algorithm=$a=rint(-5,-1);
$b=rint(0,5);
$c=rint(6,10);
$m=maple("
f:=surd(x-($a),3)/(x-($b))*(x-($c)):
convert(f,string),MathML[ExportPresentation](f)
");
$f=switch(0,$m);
$Q=switch(1,$m);
$ANS='"(-infinity,$a] U ($b,$c]"';
$plot=plotmaple("plot($f,x=$a-2..$c+2,y=-25..25,thickness=2,discont=true),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.5.4.uid=bc1e8e57-12d4-4555-85a8-9a6acd700013@
qu.5.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Number Line Analysis;
  Difficulty=Easy;
  Feature=Interval Answer;
@

qu.6.topic=1/(x+a)LE1/(2x-b)@

qu.6.1.question=<p>Solve the inequality $Q. Give your answer using interval notation. Use infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&infin;</mi></mrow></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.6.1.maple=grade("$RESPONSE",$ANS);@
qu.6.1.allow2d=0@
qu.6.1.maple_answer=show($ANS);@
qu.6.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.6.1.type=maple@
qu.6.1.mode=Maple@
qu.6.1.name=1/(x+a) LE 1/(2*x-b)@
qu.6.1.comment=<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfrac></mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mn>1</mn><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfrac><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>0</mn></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfrac><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$sum</mi></mrow><mrow><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfrac><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>0</mn></mrow></math></p>
<p>&nbsp;</p>
<p>From here perform number line analysis to find the answer.</p>@
qu.6.1.editing=useHTML@
qu.6.1.hint.1=Do not cross multiply.@
qu.6.1.hint.2=Bring to both terms to one side and find a common denominator.@
qu.6.1.solution=@
qu.6.1.algorithm=$a=rint(1,5);
$b=rint(2,10,2);
$sum=$b+$a;
$f="1/(x+($a))<=1/(2*x-($b))";
$Q=maple("printf(MathML[ExportPresentation]($f))");
$ANS='"(-infinity,-($a)) U (($b)/2,($a)+($b)]"';@
qu.6.1.uid=72880ea4-9f74-4211-8870-2fd6f9e8538b@
qu.6.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Solving inequalities with rational expressions on both sides;
  Difficulty=Medium;
@

qu.7.topic=BasicAbsoluteValue@

qu.7.1.question=<p>Solve the inequality $E <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></math> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.7.1.maple=grade("$RESPONSE",$ANS);@
qu.7.1.allow2d=0@
qu.7.1.maple_answer=show($ANS);@
qu.7.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.7.1.type=maple@
qu.7.1.mode=Maple@
qu.7.1.name=|x - a| LE b@
qu.7.1.comment=<p>$E measures the distance between <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math></em> and <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>.</em></p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$E and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. We want to know in this question when the absolute value is UNDER or ON the line. Look!</p>
<p>$plot</p>@
qu.7.1.editing=useHTML@
qu.7.1.hint.1=Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfenced></mrow></math>&nbsp;measures the distance between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math> and <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>.</em>@
qu.7.1.solution=@
qu.7.1.algorithm=$a=rint(-5,5);
$b=rint(1,6);
condition: ne($a,0);
$l=$a-$b;
$r=$a+($b);
$E=mathml("abs(x-$a)");
$ANS='"[$l,$r]"';
$plot=plotmaple("plot([abs(x-($a)),$b],x=-8..8,y=-8..8,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.7.1.uid=05731ca0-d3e5-462b-a80c-ce3d1ff73d00@
qu.7.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Basic absolute value inequalities;
  Difficulty=Easy;
@

qu.7.2.question=<p>Solve the inequality $E <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&gt;</mo></mrow></math> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.7.2.maple=grade("$RESPONSE",$ANS);@
qu.7.2.allow2d=0@
qu.7.2.maple_answer=show($ANS);@
qu.7.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.7.2.type=maple@
qu.7.2.mode=Maple@
qu.7.2.name=|x - a| GT b@
qu.7.2.comment=<p>$E measure the distance between <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math></em> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>.</p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$E and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. We want to know in this question when the absolute value is ABOVE the line. Look!</p>
<p>$plot</p>@
qu.7.2.editing=useHTML@
qu.7.2.hint.1=Remember&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>measures the distance between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math> .@
qu.7.2.solution=@
qu.7.2.algorithm=$a=rint(-5,5);
$b=rint(1,6);
condition: ne($a,0);
$l=$a-$b;
$r=$a+$b;
$E=mathml("abs(x-$a)");
$ANS='"(-infinity,$l) U ($r,infinity)"';
$plot=plotmaple("plot([abs(x-($a)),$b],x=-8..8,y=-8..8,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.7.2.uid=0f4ba691-5801-488a-8271-c42aac45acaf@
qu.7.2.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Basic absolute value inequalities;
  Difficulty=Easy;
  Course=Introduction to Calculus I;
@

qu.7.3.question=<p>Solve the inequality $E<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&infin;</mi></mrow></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.7.3.maple=grade("$RESPONSE",$ANS);@
qu.7.3.allow2d=0@
qu.7.3.maple_answer=show($ANS);@
qu.7.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.7.3.type=maple@
qu.7.3.mode=Maple@
qu.7.3.name=|x - a| LT b@
qu.7.3.comment=<p>$E measures the distance between <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math></em> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>.</p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$E and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. We want to know in this question when the absolute value (RED) is UNDER the line (GREEN). Look!</p>
<p>$plot</p>@
qu.7.3.editing=useHTML@
qu.7.3.hint.1=Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfenced></mrow></math>&nbsp;measures the distance between <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math></em>&nbsp;and <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math>.</em>@
qu.7.3.solution=@
qu.7.3.algorithm=$a=rint(-5,5);
$b=rint(1,6);
condition: ne($a,0);
$l=($a)-($b);
$r=($a)+($b);
$E=mathml(abs(x-$a));
$ANS='"($l,$r)"';
$plot=plotmaple("plot([abs(x-($a)),$b],x=-8..8,y=-8..8,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.7.3.uid=a8223bf7-6116-4306-9803-1cb66818aa7e@
qu.7.3.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Absolute Value and Inequalities;
  Sub-Topic=Basic absolute value inequalities;
  Difficulty=Easy;
@

qu.7.4.question=<p>Solve the inequality $E <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&ge;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.7.4.maple=grade("$RESPONSE",$ANS);@
qu.7.4.allow2d=0@
qu.7.4.maple_answer=show($ANS);@
qu.7.4.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.7.4.type=maple@
qu.7.4.mode=Maple@
qu.7.4.name=|x - a| GE b@
qu.7.4.comment=<p>$E measures the distance between <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math> </em>and <em><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow></math>.</em></p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$E and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo></mrow><mrow><mi mathvariant='normal'>$b</mi></mrow></math>. We want to know in this question when the absolute value is ABOVE or ON the line. Look!</p>
<p>$plot</p>@
qu.7.4.editing=useHTML@
qu.7.4.hint.1=Remember <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi>a</mi></mrow></mfenced></mrow></math>&nbsp;measures the distance between <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi></mrow></math><em> </em>and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>a</mi></mrow></math><em>.</em>@
qu.7.4.solution=@
qu.7.4.algorithm=$a=rint(-5,5);
$b=rint(1,6);
condition: ne($a,0);
$l=$a-$b;
$r=$a+$b;
$E=mathml("abs(x-$a)");
$ANS='"(-infinity,$l] U [$r,infinity)"';
$plot=plotmaple("plot([abs(x-($a)),$b],x=-8..8,y=-8..8,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.7.4.uid=8c473b45-28d0-492d-8a98-5e97a65a5f67@
qu.7.4.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Basic absolute value inequalties;
  Difficulty=Easy;
@

qu.8.topic=LinAbsValIneqONESIDE@

qu.8.1.question=<p>Solve the absolute value inequality $El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$Er. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union if you need it. Enter N if the solution is the null set.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mrow><mi>&infin;</mi></mrow></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.8.1.maple=evalb(($ANSWER)-($RESPONSE)=0);@
qu.8.1.allow2d=1@
qu.8.1.maple_answer=$ANS;@
qu.8.1.type=formula@
qu.8.1.mode=Maple@
qu.8.1.name=LT Parallel Null@
qu.8.1.comment=<p>Solution:</p>
<p><strong>Case 1:</strong> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$ax</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$condition.</p>
<p>With the assumption of this case $El=<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a2</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$ans1</p>
<p>Therefore for this case we need <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$condition and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$ans1, which yields no solution.</p>
<p>&nbsp;</p>
<p><strong>Case 2:</strong> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$condition</p>
<p>With the assumption of this case, $El = <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$d</mi></mrow><mrow><mi mathvariant='normal'></mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$sumdb</mi></mrow></math></p>
<p>Therefore this case yields no solution.</p>
<p>Since both cases gave no solution, the solution is the null set.</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the absolute value is UNDER the line.</p>
<p>$plot</p>@
qu.8.1.editing=useHTML@
qu.8.1.hint.1=Use&nbsp;Cases.@
qu.8.1.solution=@
qu.8.1.algorithm=$a=rint(1,5);
$b=rint(-5,-1);
$c=-($a);
$d=rint(-5,-1);
$m = maple("
MathML[ExportPresentation](abs(($a)*x+($b))), MathML[ExportPresentation](($c)*x+($d))
");
$El = switch(0, $m);
$Er = switch(1, $m);
$ANS=N;
$plot=plotmaple("plot([abs(($a)*x+($b)),($c)*x+($d)],x=-8..8,y=-8..8,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");
$condition= mathml("-($b)/$a");
$a2 = $a * 2;
$diffdb= $d-$b;
$ans1 = mathml("($d-($b))/$a");
$sumdb=$d + ($b);@
qu.8.1.uid=a795d114-fa5f-4e81-8f2a-6607d6bf33a4@
qu.8.1.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Linear absolute value inequalities;
  Difficulty=Medium;
@

qu.8.2.question=<p>Solve the absolute value inequality $El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$Er. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union if you need it. Enter N if the solution is the null set.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo mathvariant='italic' lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.8.2.maple=grade("$RESPONSE",$ANS);@
qu.8.2.allow2d=0@
qu.8.2.maple_answer=show($ANS);@
qu.8.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.8.2.type=maple@
qu.8.2.mode=Maple@
qu.8.2.name=LT (A,infinity)@
qu.8.2.comment=<p>Solution:</p>
<p><strong>Case 1: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo></mrow></math></strong>$condition</p>
<p>With the assumption of this case we have $El = <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi></mrow><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow><mrow></mrow></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>$ans1</p>
<p>Therefore we need<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$condition and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>$ans1, so this case contributes <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>$ans1 to the solution.</p>
<p>&nbsp;</p>
<p><strong>Case 2:&nbsp; <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math></strong>$condition</p>
<p>With the assumption of this case we have $El = <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$d</mi></mrow><mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac2</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$sumdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>$ans2</p>
<p>Therefore we need <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$condition and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>$ans2, so this case contributes nothing to the solution.</p>
<p>From Case 1 and Case 2 we see the answer is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>$ans1</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the absolute value is UNDER the line.</p>
<p>$plot</p>@
qu.8.2.editing=useHTML@
qu.8.2.hint.1=Use Cases.@
qu.8.2.solution=@
qu.8.2.algorithm=$a=rint(1,3);
$b=rint(1,5);
$c=rint($a+1,6);
$d=rint(-5,-1);
condition:gt(-$d/$c,-$b/$a);
$m=maple("
MathML[ExportPresentation](abs(($a)*x+($b))), MathML[ExportPresentation](($c)*x+($d))
");
$El=switch(0,$m);
$Er=switch(1,$m);
$ANS='"((($d)-($b))/(($a)-($c)),infinity)"';
$plot=plotmaple("plot([abs(($a)*x+($b)),($c)*x+($d)],x=-8..8,y=-8..8,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");
$condition=mathml("-$b/$a");
$diffac=$a-$c;
$diffdb=$d-$b;
$ans1=mathml("$diffdb/$diffac");
$diffac2=-$a-$c;
$sumdb=$d+$b;
$ans2=mathml("$sumdb/$diffac2");@
qu.8.2.uid=dc0a01ac-63bc-4775-ba53-ee8b00cd646d@
qu.8.2.info=  Author=Jack Weiner, Gord Clement;
  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Linear bsolute value inequalites;
  Difficulty=Medium;
@

qu.8.3.question=<p>Solve the absolute value inequality $El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$Er. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>if you need it. Enter N if the solution is the null set.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.8.3.maple=grade("$RESPONSE",$ANS);@
qu.8.3.allow2d=0@
qu.8.3.maple_answer=show($ANS);@
qu.8.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.8.3.type=maple@
qu.8.3.mode=Maple@
qu.8.3.name=LT (A,B)@
qu.8.3.comment=<p>Solution:</p>
<p><strong>Case 1:</strong> <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$ax</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo></mrow></math>$condition</p>
<p>&nbsp;</p>
<p>In this case $El <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$ax</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$ans1</p>
<p>&nbsp;</p>
<p>Therefore we need <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$condition AND <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$ans1, meaning this case contributes $condition <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$ans1 to the solution.</p>
<p>&nbsp;</p>
<p><strong>Case 2: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math></strong>$condition</p>
<p>&nbsp;</p>
<p>In this case $El <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ax</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$ax</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$diffac2</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$sumdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>$ans2</p>
<p>Therefore we need <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$condition and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&gt;</mo></mrow></math>$ans2, meaning this case contributes $ans2 <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$condition to the solution.</p>
<p>Putting both cases together we get the solution $ans2 <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$ans1.</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>The RED graph is<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the absolute value is UNDER the line.</p>
<p>$plot</p>@
qu.8.3.editing=useHTML@
qu.8.3.hint.1=Use&nbsp;cases.@
qu.8.3.solution=@
qu.8.3.algorithm=$a=rint(4,10);
$b=rint(-5,5);
$c=rint(1,3);
$d=rint(1,5);
condition:gt($a,abs($c));
condition: gt(($d-$b)/($a-$c),-$b/$a);
condition: lt((-$d-$b)/($a+$c),-$b/$a);
$El=mathml("abs($a*x+$b)");
$Er=mathml("$c*x+$d");
$ANS=maple '"((-($d)-($b))/(($a)+($c)),(($d)-($b))/(($a)-($c)))"';
$plot=plotmaple("plot([abs(($a)*x+($b)),($c)*x+($d)],x=-8..8,y=-8..8,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");
$diffac=$a-$c;
$diffac2=-($a)-$c;
$diffdb= $d-($b);
$condition=mathml("-($b)/$a");
$sumdb=$d+($b);
$ans1=mathml("(($d)-($b))/(($a)-($c))");
$ans2=mathml("(-($d)-($b))/(($a)+($c))");@
qu.8.3.uid=51276a08-824b-4b4f-815c-e054667e9312@
qu.8.3.info=  Course=Introduction to Calculus I;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Linear absolute value inequalities;
  Difficulty=Medium;
  Author=Jack Weiner, Gord Clement;
@

qu.8.4.question=<p>Solve the absolute value inequality $El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;lt;</mo></mrow></math>$Er. Give your answer using interval notation and enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>if you need it. Enter N if the solution is the null set.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.8.4.maple=grade("$RESPONSE",$ANS);@
qu.8.4.allow2d=0@
qu.8.4.maple_answer=show($ANS);@
qu.8.4.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.8.4.type=maple@
qu.8.4.mode=Maple@
qu.8.4.name=LT Parallel (-infinity,A))@
qu.8.4.comment=<p>Solution:</p>
<p>&nbsp;</p>
<p><strong>Case 1: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo></mrow></math></strong>$condition</p>
<p>&nbsp;</p>
<p>In this case, $El = <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$a2</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$diffdb</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$ans1</p>
<p>&nbsp;</p>
<p>Therefore we need <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo></mrow></math>$condition and <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>$x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$ans1, meaning this case contributes $condition <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$ans1 to the solution.</p>
<p>&nbsp;</p>
<p><strong>Case 2: <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>$a</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>$b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mn>0</mn><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&Rightarrow;</mo><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math></strong>$condition</p>
<p>&nbsp;</p>
<p>In this case&nbsp;$El&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&equals;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></math>.</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$c</mi><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mi mathvariant='normal'>$b</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo><mi mathvariant='normal'>$d</mi></mrow></math></p>
<p>Which is always true, hence this case contributes <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$condition to the solution.</p>
<p>Putting both cases together we get the solution&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&lt;</mo></mrow></math>$ans1 .</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN line is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the absolute value is UNDER the line.</p>
<p>$plot</p>@
qu.8.4.editing=useHTML@
qu.8.4.hint.1=Use&nbsp;cases.@
qu.8.4.solution=@
qu.8.4.algorithm=$a=rint(1,5);
$b=rint(1,5);
$c=-($a);
$d=rint(1,5);
$m = maple("
MathML[ExportPresentation](abs(($a)*x+($b))),MathML[ExportPresentation](($c)*x+($d))
");
$El=switch(0,$m);
$Er=switch(1,$m);

$ANS='"(-infinity,(($d)-($b))/(2*($a)))"';
$plot=plotmaple("plot([abs(($a)*x+($b)),($c)*x+($d)],x=-8..8,y=-8..8,thickness=2),plotdevice='gif', plotoptions='height=250,width=250'");

$a2=2*$a;
$diffdb=$d-$b;
$ans1=mathml("$diffdb/$a2");
$condition=mathml("-$b/$a");@
qu.8.4.uid=117d765e-4ef8-4859-b02c-8b946f88ed97@
qu.8.4.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Linear absolute value inequalities;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
@

qu.9.topic=|ax + b| LT |bx + d|@

qu.9.1.question=<p>Solve the absolute value inequality $El <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></math>$Er. Give your answer using interval notation. Enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union. Enter N if the solution is the null set.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>
<p>&nbsp;</p>@
qu.9.1.maple=grade("$RESPONSE",$ANS);@
qu.9.1.allow2d=0@
qu.9.1.maple_answer=show($ANS);@
qu.9.1.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.9.1.type=maple@
qu.9.1.mode=Maple@
qu.9.1.name=|x + a|  LE  |x + b|@
qu.9.1.comment=<p>Solution:</p>
<p>There are absolute value bars on both sides of the inequalities, the best method is to square both sides.</p>
<p>&nbsp;</p>
<p>$El <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo></mrow></math>$Er</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b</mi></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$a2</mi><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$b2</mi><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bs</mi></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi mathvariant='normal'>$lhs</mi><mi>x</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mi mathvariant='normal'>$rhs</mi></mrow><mrow></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>x</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo>&$symbol;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$sol</mi></mrow></math></p>
<p>Note:</p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the RED absolute value is UNDER or MEETS the GREEN absolute value.</p>
<p>$plot</p>@
qu.9.1.editing=useHTML@
qu.9.1.hint.1=There are absolute value bars on both sides of the equation.@
qu.9.1.hint.2=Square both sides.@
qu.9.1.solution=@
qu.9.1.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition:ne($a,$b);
condition:ne($a, 0);
condition:ne($b,0);
$a2=2*$a;
$b2=2*$b;
$as=$a^2;
$bs=$b^2;
$lhs=2*($b-$a);
$symbol=if(lt($lhs,0),"leq","ge");
$rhs=$as-$bs;
$m=maple("
if (($a) > ($b)) then
MathML[ExportPresentation](abs(x+($a))), MathML[ExportPresentation](abs(x+($b))), 0, convert((-($b)-($a))/2,string)
else
MathML[ExportPresentation](abs(x+($a))), MathML[ExportPresentation](abs(x+($b))), 1, convert((-($b)-($a))/2,string)
end if;
");
$El=switch(0,$m);
$Er=switch(1,$m);
$A=switch(2,$m);
$sol=switch(3,$m);
$ANS=switch($A, '"(-infinity,-(($a)+($b))/2]"', '"[-(($a)+($b))/2,infinity)"');
$plot=plotmaple("plot([abs(x+($a)),abs(x+($b))],x=-10..10,y=-3..15,thickness=2,tickmarks=[2,2]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.1.uid=7b7e13dc-3b8c-4410-b2af-f7c3f315bad9@
qu.9.1.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Solving inequalities with absolute values;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
@

qu.9.2.question=<p>Solve the absolute value inequality $El <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&le;</mo></mrow></math>$Er. Give your answer using interval notation. Enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union. Enter N if the solution is the null set.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.9.2.maple=grade("$RESPONSE",$ANS);@
qu.9.2.allow2d=0@
qu.9.2.maple_answer=show($ANS);@
qu.9.2.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.9.2.type=maple@
qu.9.2.mode=Maple@
qu.9.2.name=|x + a|  LE  |2x + b|@
qu.9.2.comment=<p>Solution: There are absolute value bars on both sides of the inequality, the best method is to square both sides.</p>
<p>$El<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo></mrow></math>$Er</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi></mrow></mfenced><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>4</mn><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$B</mi></mrow></mfenced><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$bs</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$B1</mi></mrow></mfenced><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$B2</mi><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mn>3</mn><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$f1</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$f2</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mn>0</mn><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='auto'/></mrow></math></p>
<p>From here, perform a number line analysis to obtain the answer.</p>
<p>Note:</p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the RED absolute value is UNDER or MEETS the GREEN absolute value.</p>
<p>$plot</p>@
qu.9.2.editing=useHTML@
qu.9.2.hint.1=There are absolute value bars on both sides of the equation.@
qu.9.2.hint.2=Square both sides.@
qu.9.2.solution=@
qu.9.2.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition:ne($b,2*($a));
condition: ne($a,0);
condition: ne($b,0);
$A=2*$a;
$B=4*$b;
$as=$a^2;
$bs=$b^2;
$B1=$B-$A;
$B2=$bs-$as;
$f2=$b-$a;
$f1=$b+$a;
$mpl=maple("
MathML[ExportPresentation](abs(x+($a))),MathML[ExportPresentation](abs(2*x+($b))),convert(min(-(($a)+($b))/3,($a)-($b)),string), convert(max(-(($a)+($b))/3,($a)-($b)), string)
");
$El=switch(0,$mpl);
$Er=switch(1,$mpl);
$m=switch(2,$mpl);
$M=switch(3,$mpl);
$ANS='"(-infinity,$m] U [$M,infinity)"';
$plot=plotmaple("plot([abs(x+($a)),abs(2*x+($b))],x=-10..10,y=-3..12,thickness=2,tickmarks=[2,2]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.2.uid=36295984-6954-4be5-80e3-356775ddd81d@
qu.9.2.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Solving inequalities with absolute value;
  Difficulty=Medium;
  Course=Introduction to Calculus I;
@

qu.9.3.question=<p>Solve the absolute value inequality $El <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow></mrow><mrow><mo lspace='0.0em' rspace='0.0em'>&ge;</mo></mrow></math>$Er. Give your answer using interval notation. Enter infinity for <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>&infin;</mi></mrow></math>and U for union. Enter N if the solution is the null set.</p>
<p>For example, for&nbsp;<math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='(' close=')' separators=','><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mrow><mi>&infin;</mi></mrow><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo lspace='0.2222222em' rspace='0.2222222em'>&minus;</mo><mn>3</mn></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&cup;</mo><mfenced open='[' close=')' separators=','><mrow><mn>4</mn><mo separator='true' lspace='0.0em' rspace='0.3333333em'>&comma;</mo><mo mathvariant='italic' lspace='0.0em' rspace='0.0em'>&infin;</mo></mrow></mfenced></mrow></math> enter (-infinity,-3) U [4,infinity).</p>@
qu.9.3.maple=grade("$RESPONSE",$ANS);@
qu.9.3.allow2d=0@
qu.9.3.maple_answer=show($ANS);@
qu.9.3.libname=__BASE_URI__Library_Intervals/intervalsLib.lib@
qu.9.3.type=maple@
qu.9.3.mode=Maple@
qu.9.3.name=|x + a|  GE  |2x + b|@
qu.9.3.comment=<p>Solution: Since there are absolute value bars on both sides of the inequality we can square both sides of the inequality.</p>
<p>&nbsp;</p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mi>a</mi><mo lspace='0.1111111em' rspace='0.1111111em' stretchy='true'>&verbar;</mo><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mfenced open='|' close='|' separators=','><mrow><mi>b</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em' stretchy='true'>&hArr;</mo><msup><mi>a</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><msup><mi>b</mi><mrow><mn>2</mn></mrow></msup></mrow></math></p>
<p><math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mfenced open='|' close='|' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><mfenced open='|' close='|' separators=','><mrow><mn>2</mn><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfenced><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$a</mi></mrow></mfenced></mrow></mfenced><mrow><mn>2</mn></mrow></msup><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo><msup><mfenced open='(' close=')' separators=','><mrow><mn>2</mn><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$b</mi></mrow></mfenced></mrow></mfenced><mrow><mn>2</mn></mrow></msup></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$A</mi></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mi mathvariant='normal'>$as</mi></mrow><mrow><mo lspace='0.2777778em' rspace='0.2777778em'>&GreaterEqual;</mo></mrow><mrow><mn>4</mn><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$B</mi></mrow></mfenced><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mi mathvariant='normal'>$bs</mi><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mn>3</mn><msup><mi>x</mi><mrow><mn>2</mn></mrow></msup></mrow><mrow><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$B1</mi></mrow></mfenced><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$B2</mi></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>0</mn></mrow><mrow><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mspace height='0.0ex' width='0.0em' depth='0.0ex' linebreak='newline'/><mo lspace='0.0em' rspace='0.0em'>&#32;</mo><mfenced open='(' close=')' separators=','><mrow><mn>3</mn><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$f1</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.0em' rspace='0.0em'>&sdot;</mo><mfenced open='(' close=')' separators=','><mrow><mi>x</mi><mo lspace='0.2222222em' rspace='0.2222222em'>&plus;</mo><mfenced open='(' close=')' separators=','><mrow><mi mathvariant='normal'>$f2</mi></mrow></mfenced></mrow></mfenced><mo lspace='0.2777778em' rspace='0.2777778em'>&leq;</mo><mn>0</mn></mrow><mrow></mrow></math></p>
<p>&nbsp;From there perform a number line analysis to acheive the answer.</p>
<p>&nbsp;</p>
<p>Note:</p>
<p>The RED graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$El and the GREEN graph is <math xmlns='http://www.w3.org/1998/Math/MathML'><mrow><mi>y</mi><mo lspace='0.2777778em' rspace='0.2777778em'>&amp;equals;</mo></mrow></math>$Er. We want to know in this question when the RED absolute value is ABOVE or MEETS the GREEN absolute value.</p>
<p>$plot</p>@
qu.9.3.editing=useHTML@
qu.9.3.hint.1=There are absolute value bars on both sides of the inequality@
qu.9.3.hint.2=Square both sides@
qu.9.3.solution=@
qu.9.3.algorithm=$a=rint(-5,5);
$b=rint(-5,5);
condition: ne($b,2*($a));
condition: ne($a,0);
condition: ne($b,0);
$A=2*$a;
$B=4*$b;
$as=$a^2;
$bs=$b^2;
$B1=$B-$A;
$B2=$bs-$as;
$f2=$b-$a;
$f1=$b+$a;
$mpl=maple(" 
MathML[ExportPresentation](abs(x+($a))),MathML[ExportPresentation](abs(2*x+($b))), convert(min(-(($a)+($b))/3,($a)-($b)),string), convert(max(-(($a)+($b))/3,($a)-($b)),string)
");
$El=switch(0, $mpl);
$Er=switch(1, $mpl);
$m = switch(2, $mpl);
$M= switch(3,$mpl);
$ANS='"[$m,$M]"';
$plot=plotmaple("plot([abs(x+($a)),abs(2*x+($b))],x=-10..10,y=-3..15,thickness=2,tickmarks=[2,2]),plotdevice='gif', plotoptions='height=250,width=250'");@
qu.9.3.uid=77dd6cee-bc5f-43be-8b87-96ef3ece9b71@
qu.9.3.info=  Author=Jack Weiner, Gord Clement;
  Topic=Inequalities and Absolute Value;
  Sub-Topic=Solving absolute value inequalities;
  Course=Introduction to Calculus I;
  Difficulty=Medium;
@

