// Numbas version: exam_results_page_options
{"percentPass": "40", "timing": {"timeout": {"action": "warn", "message": "<p>il tempo!</p>"}, "allowPause": false, "timedwarning": {"action": "warn", "message": "<p>tra cinque minuti consegna</p>"}}, "showstudentname": true, "name": "test ingresso quarta F", "feedback": {"advicethreshold": 0, "showtotalmark": true, "feedbackmessages": [], "showactualmark": true, "showanswerstate": true, "allowrevealanswer": false, "intro": ""}, "duration": 3600, "question_groups": [{"pickQuestions": 1, "pickingStrategy": "all-shuffled", "name": "Geometria analitica", "questions": [{"name": "riccardo's copy of Julie's copy of Q9 - Coordinate Geometry, Line and Parabola", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "riccardo belle'", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1797/"}], "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "showFeedbackIcon": true, "prompt": "<p>Qual &egrave; il coefficiente angolare di $\\simplify{y={m}x+{yc}}$ ?</p>\n<p>Coefficiente = &nbsp;[[0]]&nbsp;&nbsp;&nbsp; &nbsp; &nbsp;&nbsp; </p>\n<p>What is&nbsp;the $y$-coordinate of the point on the line whose $x$-coordinate is $\\var{xval}$?</p>\n<p>( $\\var{xval}$ , [[2]] )</p>\n<p></p>\n<p></p>", "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "m", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "m", "showCorrectAnswer": false, "allowFractions": true, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "yc", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "yc", "showCorrectAnswer": false, "allowFractions": true, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 1, "answer": "{xval}*{m}+{yc}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": false, "vsetrange": [0, 1], "checkingtype": "absdiff"}], "variableReplacements": []}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "showFeedbackIcon": true, "prompt": "<p>Fill in the table of values for $y=\\simplify[std]{{a}x^2+{c}}$:</p>\n<table id=\"values\">\n<thead>\n<tr><th>$x$</th><th>$-3$</th><th>$-2$</th><th>$-1$</th><th>$0$</th><th>$1$</th><th>$2$</th><th>$3$</th></tr>\n</thead>\n<tbody>\n<tr><th>$y$</th>\n<td>[[0]]</td>\n<td>[[1]]</td>\n<td>[[2]]</td>\n<td>[[3]]</td>\n<td>[[4]]</td>\n<td>[[5]]</td>\n<td>[[6]]</td>\n</tr>\n</tbody>\n</table>\n", "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 0.5, "answer": "{a}*9+{c}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff"}, {"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 0.5, "answer": "{a}*4+{c}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff"}, {"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 0.5, "answer": "{a}*1+{c}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff"}, {"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 0.5, "answer": "{c}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff"}, {"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 0.5, "answer": "{a}*1+{c}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff"}, {"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 0.5, "answer": "{a}*4+{c}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff"}, {"variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "type": "jme", "checkvariablenames": false, "marks": 0.5, "answer": "{a}*9+{c}", "showpreview": true, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "showFeedbackIcon": true, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "vsetrange": [0, 1], "checkingtype": "absdiff"}], "variableReplacements": []}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "showFeedbackIcon": true, "prompt": "<div id=\"dragpoint\"></div>\n<p>Slide the points to the correct $y$ values.</p>\n<p></p>", "type": "gapfill", "marks": 0, "showCorrectAnswer": true, "gaps": [{"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{values[0]}", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "{values[0]}", "showCorrectAnswer": false, "allowFractions": false, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{values[1]}", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "{values[1]}", "showCorrectAnswer": false, "allowFractions": false, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{values[2]}", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "{values[2]}", "showCorrectAnswer": false, "allowFractions": false, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{values[3]}", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "{values[3]}", "showCorrectAnswer": false, "allowFractions": false, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{values[4]}", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "{values[4]}", "showCorrectAnswer": false, "allowFractions": false, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{values[5]}", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "{values[5]}", "showCorrectAnswer": false, "allowFractions": false, "mustBeReduced": false}, {"variableReplacementStrategy": "originalfirst", "type": "numberentry", "marks": "0.5", "correctAnswerStyle": "plain", "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "maxValue": "{values[6]}", "variableReplacements": [], "scripts": {}, "showFeedbackIcon": true, "mustBeReducedPC": 0, "minValue": "{values[6]}", "showCorrectAnswer": false, "allowFractions": false, "mustBeReduced": false}], "variableReplacements": []}], "preamble": {"css": "table#values th {\n  background: none;\n  text-align: center;\n}", "js": "\nfunction dragpoint_board() {\n\n  var scope = question.scope; \n  var a = scope.variables.a.value;\n\n  var c = scope.variables.c.value;\n  var maxy = Math.max(Math.abs(a*9+c),Math.abs(c));\n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('250px','400px',{boundingBox:[-5,maxy+3,5,-maxy-3],grid:true});\n  $(question.display.html).find('#dragpoint').append(div);\n  \n  var board = div.board;\n  \n  //shorthand to evaluate a mathematical expression to a number\n  function evaluate(expression) {\n    try {\n      var val = Numbas.jme.evaluate(expression,question.scope);\n      return Numbas.jme.unwrapValue(val);\n    }\n    catch(e) {\n      // if there's an error, return no number\n      return NaN;\n    }\n  }\n  \n  // set up points array\n  var num_points = 7;\n  var points = [];\n  \n  \n  // this function sets up the i^th point\n  function make_point(i) {\n    \n    // calculate initial coordinates\n    var x = i-(num_points-1)/2;\n    \n    // create an invisible vertical line for the point to slide along\n    var line = board.create('line',[[x,0],[x,1]],{visible: false});\n    \n    // create the point\n    var point = points[i] = board.create(\n      'glider',\n      [i-(num_points-1)/2,0,line],\n      {\n        name:'',\n        size:2,\n        snapSizeY: 0.1,  // the point will snap to y-coordinates which are multiples of 0.1\n        snapToGrid: true\n      }\n    );\n    \n    // the contents of the input box for this point\n    var studentAnswer = question.parts[2].gaps[i].display.studentAnswer;\n    \n    //Here I have commented out the functions which connect the student input to the graph and the filling in of the answer fields\n    //when the student drags the points on the graph.\n    \n    // watch the student's input and reposition the point when it changes.    \n   // ko.computed(function() {\n     // y = evaluate(studentAnswer());\n      //if(!(isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n      //  point.moveTo([x,y],100);\n     // }\n   // });\n    \n    // when the student drags the point, update the gapfill input\n   point.on('drag',function(){\n   var y = Numbas.math.niceNumber(point.Y());\n    studentAnswer(y);\n    });\n    \n  }\n  \n  // create each point\n  for(var i=0;i<num_points;i++) {\n    make_point(i);\n  }\n  \n  // create the parabola\n  var parabool = board.create('spline',points,{strokeWidth:3,strokeColor:'red'});\n}\nfunction showgraph(){\n  var scope = question.scope; \n  var a = scope.variables.a.value;\n  var v1=[];\n \n  var c = scope.variables.c.value;\n  var maxy = Math.max(Math.abs(a*9+c),Math.abs(c));\n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('250px','400px',{boundingBox:[-5,maxy+3,5,-maxy-3],grid:true});\n  $(question.display.html).find('#showgraph').append(div);\n  \n  var board = div.board;\n   for(var i=-3;i<4;i++){\n    v1[i+3]=board.create('point', [i,a*Math.pow(i,2)+c], {size: 4, name: ''});\n   }\n  var parab = board.create('spline',v1,{strokeWidth:3,strokeColor:'red'});\n}\n\nquestion.signals.on('HTMLAttached',function() {\n  dragpoint_board();\n  showgraph();\n}); "}, "rulesets": {"std": ["all", "fractionNumbers"]}, "statement": "<p>Data la seguente funzione</p>\n<p>$y=\\simplify[std]{{a}x^2+{c}}$</p>", "advice": "<p>This is the graph you should have obtained.</p>\n<div id=\"showgraph\"></div>", "functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "ungrouped_variables": ["a", "c", "values", "v1", "m", "yc", "xval"], "tags": [], "variables": {"c": {"definition": "random(-4..4 except 0)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "c"}, "xval": {"definition": "random(-7..7 except [1,0,-1,m])", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "xval"}, "yc": {"definition": "random(-5..5 except [0,m])", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "yc"}, "values": {"definition": "map({a*x^2+c},x,-3..3)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "values"}, "m": {"definition": "random(-2,-1.5,-1,-0.5,0.5,1,1.5,2)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "m"}, "v1": {"definition": "map([x-3,values[x]],x,0..6)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "v1"}, "a": {"definition": "random(-2,-1,-0.5,0.5,1,2)", "group": "Ungrouped variables", "description": "", "templateType": "anything", "name": "a"}}, "metadata": {"description": "<p>Compute a table of values for a quadratic function. The student input is now disconnected from the graph so that they slide the points on the graph after they input the values and the answer fields are not updated. Now includes a graph in advice.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question"}, {"name": "Unit 4 - Which quadratic function is shown by the parabola?", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Richard Miles", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/882/"}], "rulesets": {}, "parts": [{"showFeedbackIcon": true, "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "type": "1_n_2", "matrix": "[2,0,0,0,0]", "steps": [{"variableReplacements": [], "showFeedbackIcon": true, "scripts": {}, "variableReplacementStrategy": "originalfirst", "marks": 0, "prompt": "<p>You can narrow down the choices by checking points where the curve intersects the axes, in particular, the $y$-axis intercept. In addition, the $x$-coordinate of the vertex of the parabola is $x=\\frac{-b}{2a}$, when the curve is written in the standard form $y=ax^2+bx+c$. The line $x=\\frac{-b}{2a}$ is also the vertical line of symmetry for the curve.</p>", "showCorrectAnswer": true, "type": "information"}], "displayType": "radiogroup", "stepsPenalty": "1", "variableReplacements": [], "choices": "[function1,function2,function3,function4,function5]", "scripts": {}, "prompt": "<p>Decide which of the following expressions for $f(x)$ corresponds to the parabola shown.</p>", "marks": 0, "minMarks": 0, "shuffleChoices": true, "showCorrectAnswer": true}], "functions": {"drawgraph": {"parameters": [], "language": "javascript", "type": "html", "definition": "      var miny = Numbas.jme.unwrapValue(scope.variables.miny);\n      var maxy = Numbas.jme.unwrapValue(scope.variables.maxy);\n      var minx = Numbas.jme.unwrapValue(scope.variables.minx);\n      var maxx = Numbas.jme.unwrapValue(scope.variables.maxx);\n      var a_inverse = Numbas.jme.unwrapValue(scope.variables.a_inverse);\n      var b = Numbas.jme.unwrapValue(scope.variables.b);\n      var c = Numbas.jme.unwrapValue(scope.variables.c);\n      \n      var div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n      {boundingBox:[minx,maxy,maxx,miny],\n       axis:false,\n       showNavigation:false,\n       grid:true});\n        JXG.Options.text.useMathJax = true;\n        var brd = div.board;          \n        var xaxis=brd.createElement('axis', [[minx,0],[maxx,0]]);\n        var yaxis=brd.createElement('axis', [[0,miny],[0,maxy]]);\n        var xlabel = brd.create('text',[maxx-1,0,function(){return '\\\\[x\\\\]';}]);\n        var ylabel = brd.create('text',[-2,maxy,function(){return '\\\\[y\\\\]';}]);\n        var graph = brd.create('functiongraph',[function(x){ return (1/a_inverse)*x*x+b*x+c;},minx,maxx]);\n\nreturn div;\n      "}}, "tags": [], "preamble": {"css": "", "js": ""}, "ungrouped_variables": [], "variables": {"c": {"definition": "random(-7..7 except [b,0,-1,1])", "name": "c", "templateType": "anything", "description": "", "group": "Graph parameters"}, "function4": {"definition": "\"<p>$\\\\var{a_inverse}\\\\simplify{x^2+{b}}\\\\simplify{x+{c}}$</p>\"", "name": "function4", "templateType": "long string", "description": "", "group": "Graph parameters"}, "function1": {"definition": "\"<p>$\\\\frac{1}{\\\\var{a_inverse}}\\\\simplify{x^2+{b}}\\\\simplify{x+{c}}$</p>\"", "name": "function1", "templateType": "long string", "description": "", "group": "Graph parameters"}, "function3": {"definition": "\"<p>$\\\\var{a_inverse}\\\\simplify{x^2+{b}}\\\\simplify{x-{c}}$</p>\"", "name": "function3", "templateType": "long string", "description": "", "group": "Graph parameters"}, "b": {"definition": "random([-4,-3,-2,2,3,4] except a_inverse)", "name": "b", "templateType": "anything", "description": "", "group": "Graph parameters"}, "maxx": {"definition": "20", "name": "maxx", "templateType": "anything", "description": "", "group": "Graph parameters"}, "maxy": {"definition": "15", "name": "maxy", "templateType": "anything", "description": "", "group": "Graph parameters"}, "minx": {"definition": "-20", "name": "minx", "templateType": "anything", "description": "", "group": "Graph parameters"}, "miny": {"definition": "-15", "name": "miny", "templateType": "anything", "description": "", "group": "Graph parameters"}, "function2": {"definition": "\"<p>$\\\\frac{1}{\\\\var{a_inverse}}\\\\simplify{x^2+{b}*3}\\\\simplify{x+{c}}$</p>\"", "name": "function2", "templateType": "long string", "description": "", "group": "Graph parameters"}, "function5": {"definition": "\"<p>$(\\\\simplify{x+{c}})^2+\\\\simplify{x+{b}}$</p>\"", "name": "function5", "templateType": "long string", "description": "", "group": "Graph parameters"}, "a_inverse": {"definition": "random(2..5)", "name": "a_inverse", "templateType": "anything", "description": "", "group": "Graph parameters"}}, "variable_groups": [{"variables": ["minx", "maxx", "miny", "maxy", "a_inverse", "b", "c", "function1", "function2", "function3", "function4", "function5"], "name": "Graph parameters"}], "advice": "", "statement": "<p><span style=\"font-weight: 400;\">The curve $y=f(x)$ is shown below.<br/></span></p>\n<table>\n<tbody>\n<tr>\n<td>{drawgraph()}</td>\n</tr>\n</tbody>\n</table>", "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"description": "<p>Tests the ability to match a quadratic function to a given parabola. </p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question"}]}, {"pickQuestions": 1, "pickingStrategy": "all-shuffled", "name": "Disequazioni", "questions": [{"name": "Arithmetic operations: Inequalities", "extensions": [], "custom_part_types": [], "resources": [["question-resources/drawingresize_grbP9s8.svg", "/srv/numbas/media/question-resources/drawingresize_grbP9s8.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}], "type": "question", "statement": "<p>Choose the correct symbols to describe the relations between each of these pairs of numbers.</p>", "variablesTest": {"condition": "", "maxRuns": "100"}, "variables": {"neg": {"description": "<p>Random negative integers.</p>", "name": "neg", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(-300..-1),8)"}, "c": {"description": "", "name": "c", "group": "Ungrouped variables", "templateType": "anything", "definition": "-b"}, "random": {"description": "", "name": "random", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(0..0.01#0.001),3)"}, "d": {"description": "", "name": "d", "group": "Ungrouped variables", "templateType": "anything", "definition": "-a"}, "pos": {"description": "<p>Random positive integers.</p>", "name": "pos", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(1..300),5)"}, "a": {"description": "", "name": "a", "group": "Ungrouped variables", "templateType": "anything", "definition": "neg[7] + random2"}, "b": {"description": "", "name": "b", "group": "Ungrouped variables", "templateType": "anything", "definition": "neg[7] + 0.9 + random[2]"}, "dec": {"description": "<p>Random decimals.</p>", "name": "dec", "group": "Ungrouped variables", "templateType": "anything", "definition": "repeat(random(0..50 #0.01 except 0..50), 7)"}, "random2": {"description": "", "name": "random2", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(0.5..0.8#0.001)"}}, "functions": {}, "tags": ["inequality", "taxonomy"], "variable_groups": [], "parts": [{"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "choices": ["<p>&nbsp;&gt;</p>", "<p>&nbsp;&lt;</p>", "<p>&nbsp;=</p>"], "showFeedbackIcon": true, "shuffleChoices": true, "displayColumns": 0, "variableReplacements": [], "marks": 0, "matrix": ["1", 0, 0], "showCorrectAnswer": true, "maxMarks": 0, "type": "1_n_2"}], "showFeedbackIcon": true, "prompt": "<p><em>&nbsp;</em></p>\n<p style=\"text-align: left; padding-left: 120px;\">$\\var{dec[6] + 0.001 +&nbsp;random[0]}$ &nbsp;[[0]]&nbsp; $\\var{dec[6] - random[1]}$</p>\n<p style=\"text-align: left;\"></p>", "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "choices": ["<p>&nbsp;&lt;</p>", "<p>&nbsp;&gt;</p>", "<p>&nbsp;=</p>"], "showFeedbackIcon": true, "shuffleChoices": true, "displayColumns": 0, "variableReplacements": [], "marks": 0, "matrix": ["1", 0, 0], "showCorrectAnswer": true, "maxMarks": 0, "type": "1_n_2"}], "showFeedbackIcon": true, "prompt": "<p><em>&nbsp;</em></p>\n<p style=\"text-align: left; padding-left: 90px;\">$\\var{neg[7] + random2}$&nbsp; [[0]]&nbsp; $\\var{neg[7]&nbsp;+&nbsp;0.9 + random[2]}$</p>", "marks": 0}, {"scripts": {}, "variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"scripts": {}, "minMarks": 0, "distractors": ["", "", ""], "variableReplacementStrategy": "originalfirst", "displayType": "dropdownlist", "choices": ["<p>&nbsp;=</p>", "<p>&nbsp;&gt;</p>", "<p>&nbsp;&lt;</p>"], "showFeedbackIcon": true, "shuffleChoices": true, "displayColumns": 0, "variableReplacements": [], "marks": 0, "matrix": [0, "1", 0], "showCorrectAnswer": true, "maxMarks": 0, "type": "1_n_2"}], "showFeedbackIcon": true, "prompt": "<p><em>&nbsp;</em></p>\n<p style=\"text-align: left; padding-left: 60px;\">$(\\var{neg[3]}) \\times (\\var{neg[2]})$&nbsp; [[0]]&nbsp;&nbsp; $\\var{-neg[3]*neg[2]}$</p>", "marks": 0}], "ungrouped_variables": ["dec", "neg", "pos", "random2", "random", "a", "b", "c", "d"], "rulesets": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Complete the inequality relationships by selecting the correct symbol from a drop down box</p>"}, "preamble": {"css": "", "js": ""}, "advice": "<p>\\[\\begin{align} \\text{Symbol }&amp;\\lt&nbsp; \\text{ denotes \"less than\".} \\\\&nbsp;\\text{Symbol }&amp;\\gt \\text{denotes \"greater than\".} \\end{align}\\]</p>\n<h4>a)</h4>\n<p>$\\var{dec[6] + 0.001 +&nbsp;random[0]}$ is greater than&nbsp;$\\var{dec[6] - random[1]}$ so</p>\n<p>\\[\\var{dec[6] + 0.001 +&nbsp;random[0]} \\gt \\var{dec[6] - random[1]} \\text{.}&nbsp;\\]</p>\n<h4>b)</h4>\n<p>When both of the numbers that you are comparing are negative, it may be tempting to ignore the negative signs and make an incorrect assumption. For example, when we have -5 and -4 we might ignore the signs and assume -5 is larger than -4 since +5 is larger than +4. This is however wrong, -5&nbsp;&lt; -4.</p>\n<p>To understand this a bit better, look at the following number line:</p>\n<p style=\"text-align: center;\"><object width=\"967\" height=\"967\" data=\"resources/question-resources/drawingresize_grbP9s8.svg\" type=\"image/svg+xml\"></object></p>\n<p><strong></strong>Following the number line from left to right, we can see that $\\var{neg[7] + random2}$ is less than $\\var{neg[7]&nbsp;+&nbsp;0.9 + random[2]}$, so</p>\n<p>\\[\\var{neg[7] + random2} \\lt \\var{neg[7]&nbsp;+&nbsp;0.9 + random[2]} \\text{.}\\]</p>\n<p></p>\n<h4>c)</h4>\n<p>Multiplying two negative numbers results in a positive number. Therefore we can see without performing any calculation that $(\\var{neg[3]}) \\times (\\var{neg[2]}) \\gt&nbsp;\\var{-neg[3]*neg[2]}$ as positive numbers are always larger than negative numbers.</p>\n<p>\\[(\\var{neg[3]}&nbsp;\\times \\var{neg[2]}) \\gt&nbsp;\\var{-neg[3]*neg[2]}\\]</p>\n<p>\\[\\var{neg[3] * neg[2]} \\gt&nbsp;\\var{-neg[3]*neg[2]}\\]</p>"}, {"name": "Solve quadratic inequalities", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}], "metadata": {"description": "<p>This question takes the student through variety of examples of quadratic inequalities by asking them for the range(s) for which $x$ meets the inequality.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "b", "c", "d", "g", "f"], "type": "question", "advice": "<h3>a)</h3>\n<p>This example&nbsp;is best illustrated by the graph of $\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}$ below. By finding the roots of the equation we can find the $x$ coordinates where the line crosses the $x$ axis and then we can use a sketch or visualise the graph to work out the set of values for $x$ where $f(x)&gt;0$.\\[\\\\[0.1em]\\]</p>\n<p style=\"text-align: center;\">{geogebra_applet('Hk5dTptY',[[\"a\",a],[\"b\",b]])}</p>\n<p>\\[ <br/>\\begin{align}<br/>\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}&amp;&gt;0\\\\<br/>\\simplify{(x+{a})(x-{b})}&amp;=0\\text{.}<br/>\\end{align}<br/>\\]</p>\n<p>Therefore</p>\n<p>\\[ \\simplify{x&lt;{-a}}\\text{ or&nbsp;}\\simplify{x&gt;{b}}\\text{.}\\]</p>\n<h4>b)</h4>\n<p>We use the same method in this example but this time we use&nbsp;our graph to visualise where $g(x)&lt;0$.\\[\\\\[0.1em]\\]</p>\n<p style=\"text-align: center;\">{geogebra_applet('PUFStTNa',[[\"c\",c],[\"d\",d]])}</p>\n<p>\\[ <br/>\\begin{align}<br/>\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}&amp;&lt;0\\\\<br/>\\simplify{(x+{c})(x-{d})}&amp;=0\\text{.}<br/>\\end{align}<br/>\\]</p>\n<p>Therefore</p>\n<p>\\[ \\simplify{x&gt;{-c}}\\text{ and }\\simplify{x&lt;{d}}\\text{.}\\]</p>\n<h4>c)</h4>\n<p style=\"text-align: left;\">The notable difference with solving this equation is the requirement to rearrange the inequality&nbsp;before factorisation.\\[\\\\[0.1em]\\]</p>\n<p style=\"text-align: center;\">{geogebra_applet('CZsSCdH6',[[\"e\",g],[\"f\",f]])}</p>\n<p>\\[ <br/>\\begin{align}<br/>\\simplify{({g}+{f})x-{g}{f}}&amp;&gt;x^2\\\\<br/>\\simplify{x^2-({g}+{f})x+{g}{f}}&amp;&lt;0\\\\<br/>\\simplify{(x-{g})(x-{f})}&amp;=0\\text{.}<br/>\\end{align}<br/>\\]</p>\n<p>Therefore</p>\n<p>\\[ \\simplify{x&gt;{g}}\\text{ and }\\simplify{x&lt;{f}}\\text{.}\\]</p>\n<p>Alternatively, we can plot the graph of $x^2$ against $\\simplify{({g}+{f})x-{g}{f}}$ to visualise the same result.</p>\n<p style=\"text-align: center;\">{geogebra_applet('Ez697SNE',[[\"g\",g],[\"f\",f]])}</p>\n<p style=\"text-align: left;\">From this graph we can see that the values of $x$ where&nbsp;$\\simplify{({g}+{f})x-{g}{f}}&gt;x^2$ are the same as the values of $x$ where $\\simplify{x^2-({g}+{f})x+{g}{f}}&lt;0$.</p>", "variable_groups": [], "rulesets": {}, "statement": "<p>Solve the following quadratic inequalities by firstly factorising $f(x)$ and then solving for $x$ when $f(x)=0$. It may be helpful to sketch each&nbsp;quadratic.</p>", "parts": [{"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "(x-{b})(x+{a})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{b}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "-{a}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["<p>AND</p>", "<p>OR</p>"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "<p>Find the range&nbsp;of values for $x$ such that&nbsp;$\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}&gt;0$.</p>\n<p>Factorise $f(x)$:</p>\n<p>$\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}=$&nbsp;[[0]] $=0$.</p>\n<p>Hence,</p>\n<p>$x&gt;$&nbsp;[[1]]&nbsp;</p>\n<p>[[3]]</p>\n<p>$x&lt;$&nbsp;[[2]]</p>"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "(x-{d})(x+{c})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{d}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "-{c}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["<p>AND</p>", "<p>OR</p>"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "<p>Find the range&nbsp;of values for $x$ such that&nbsp;$\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}&lt;0$.</p>\n<p>Factorise $f(x)$:</p>\n<p>$\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}=$ [[0]]&nbsp;$=0$.</p>\n<p>Hence,</p>\n<p>$x&lt;$&nbsp;[[1]]</p>\n<p>[[3]]&nbsp;</p>\n<p>$x&gt;$&nbsp;[[2]]</p>"}, {"scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "marks": 0, "gaps": [{"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "(x-{g})(x-{f})", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{g}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showCorrectAnswer": true, "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{f}", "showpreview": true, "variableReplacementStrategy": "originalfirst", "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}, {"shuffleChoices": false, "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["<p>AND</p>", "<p>OR</p>"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "<p>Find the range of vaues such that:&nbsp;$\\simplify{({g}+{f})x-{g}{f}&gt;x^2}$ &nbsp;&nbsp;</p>\n<p>Rearrange&nbsp;then factorise&nbsp;the inequality:</p>\n<p>[[0]]&nbsp;$&lt;0$.</p>\n<p>Use the above result&nbsp;to find the range&nbsp;of values for $x$ such that $\\simplify{({g}+{f})x-{g}{f}&gt;x^2}$.</p>\n<p>$x&gt;$&nbsp;[[1]]&nbsp;</p>\n<p>[[3]]&nbsp;&nbsp;</p>\n<p>$x&lt;$&nbsp;[[2]]</p>"}], "tags": ["factorise a quadratic equation", "factorising", "inequalities", "quadratic inequalities", "range of values for x", "taxonomy"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"a": {"description": "", "group": "Ungrouped variables", "definition": "random(1..5)", "name": "a", "templateType": "anything"}, "f": {"description": "", "group": "Ungrouped variables", "definition": "random(5..9)", "name": "f", "templateType": "anything"}, "b": {"description": "", "group": "Ungrouped variables", "definition": "random(1..5)", "name": "b", "templateType": "anything"}, "c": {"description": "", "group": "Ungrouped variables", "definition": "random(3..6)", "name": "c", "templateType": "anything"}, "g": {"description": "", "group": "Ungrouped variables", "definition": "random(1..4)", "name": "g", "templateType": "anything"}, "d": {"description": "", "group": "Ungrouped variables", "definition": "random(2..4)", "name": "d", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}}, {"name": "Inequalities that involve a single absolute value", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "ungrouped_variables": ["choice", "sym", "a", "c", "d", "b", "leadCoeff", "prod", "one", "rootList", "r1", "r2", "strict", "backChoice", "backsym", "extremeY", "backprod", "dr", "hole", "nonhole"], "metadata": {"description": "<p>Inequality involving a single absolute value, question solution uses the piecewise nature of the absolute value function.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>You are given the inequality \\[|\\simplify{{a}x+{b}}|&nbsp;\\var{latex(sym)} \\var{c}.\\]</p>\n<p></p>", "preamble": {"js": "", "css": ""}, "parts": [{"scripts": {"constructor": {"order": "after", "script": "// override the default 'submit' method for the part\n// we only want to submit gaps 1,2,3,4 if the answer to gap 0 is \"one\" and we only want to submit gaps 5,6,7,8 if the answer to gap 0 is \"two\"\nthis.submit = Numbas.util.extend(function() {\n  this.gaps[0].submit();\n  if(this.gaps[0].one) {\n    this.gaps[1].submit();\n    this.gaps[2].submit();\n    this.gaps[3].submit();\n    this.gaps[4].submit();\n    this.gaps[5].answered = true;\n    this.gaps[6].answered = true;\n    this.gaps[7].answered = true;\n    this.gaps[8].answered = true;\n  } else {\n    this.gaps[1].answered = true;\n    this.gaps[2].answered = true;\n    this.gaps[3].answered = true;\n    this.gaps[4].answered = true;\n    this.gaps[5].submit();\n    this.gaps[6].submit();\n    this.gaps[7].submit();\n    this.gaps[8].submit();  \n  }\n},Part.prototype.submit);\n"}, "mark": {"order": "instead", "script": "var gaps = this.gaps;\n// if the student said \"one\" to gap 0, then mark gaps 0 to 4\nif(gaps[0].one) {\n  this.setCredit((gaps[0].credit*gaps[0].marks+gaps[1].credit*gaps[1].marks+gaps[2].credit*gaps[2].marks+gaps[3].credit*gaps[3].marks+gaps[4].credit*gaps[4].marks)/(gaps[0].marks+gaps[1].marks+gaps[2].marks+gaps[3].marks+gaps[4].marks));\n// otherwise, work out the credit based on the gaps 0, 5 to 8\n} else {\n  this.setCredit((gaps[0].credit*gaps[0].marks+gaps[5].credit*gaps[5].marks+gaps[6].credit*gaps[6].marks+gaps[7].credit*gaps[7].marks+gaps[8].credit*gaps[8].marks)/(gaps[0].marks+gaps[5].marks+gaps[6].marks+gaps[7].marks+gaps[8].marks));\n}\n\n"}}, "variableReplacementStrategy": "originalfirst", "gaps": [{"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["<p>one interval.</p>", "<p>two intervals.</p>"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {"mark": {"order": "instead", "script": "// apply the normal marking algorithm for this part\nthis.__proto__.mark.apply(this);\n// store whether the student said \"one interval\" in an attribute that's easier to access\nthis.one = this.ticks[0][0];"}}, "displayType": "dropdownlist", "matrix": ["if(one=true,1,0)", "if(one=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "r1", "variableReplacements": [], "marks": 1, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "maxValue": "r1"}, {"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["<p>&lt;</p>", "<p>&le;</p>"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "displayType": "dropdownlist", "matrix": ["if(one=true and strict=true,1,0)", "if(one=true and strict=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["<p>&lt;</p>", "<p>&le;</p>"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "displayType": "dropdownlist", "matrix": ["if(one=true and strict=true,1,0)", "if(one=true and strict=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "r2", "variableReplacements": [], "marks": 1, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "maxValue": "r2"}, {"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["<p>&lt;</p>", "<p>&le;</p>"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "displayType": "dropdownlist", "matrix": ["if(one=false and strict=true,1,0)", "if(one=false and strict=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "r1", "variableReplacements": [], "marks": 1, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "maxValue": "r1"}, {"minMarks": 0, "variableReplacementStrategy": "originalfirst", "choices": ["<p>&gt;</p>", "<p>&ge;</p>"], "showFeedbackIcon": true, "distractors": ["", ""], "maxMarks": 0, "variableReplacements": [], "shuffleChoices": false, "marks": 0, "showCorrectAnswer": true, "scripts": {}, "displayType": "dropdownlist", "matrix": ["if(one=false and strict=true,1,0)", "if(one=false and strict=false,1,0)"], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "mustBeReduced": false, "showFeedbackIcon": true, "minValue": "r2", "variableReplacements": [], "marks": 1, "notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "type": "numberentry", "maxValue": "r2"}], "prompt": "<p>The solution to the above inequality</p>\n<p>is made up of &nbsp;[[0]]</p>\n<div data-bind=\"fadeVisible: part.gaps[0].display.studentAnswer()=='0'\">\n<p>Given by the condition&nbsp;[[1]] [[2]] $x$&nbsp;[[3]]&nbsp;[[4]].</p>\n</div>\n<div data-bind=\"fadeVisible: part.gaps[0].display.studentAnswer()=='1'\">\n<p>Given by the condition,&nbsp;$x$&nbsp;[[5]]&nbsp;[[6]] or $x$&nbsp;[[7]]&nbsp;[[8]].</p>\n</div>", "showFeedbackIcon": true, "variableReplacements": [], "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "rulesets": {}, "advice": "<h3><span data-jme-visible=\"choice=3 or choice=4\"><strong>Short-cut Method</strong></span></h3>\n<p></p>\n<p><span data-jme-visible=\"choice=3 or choice=4\">For questions of this type (where the absolute value is 'less than' or 'less than and equal to' something positive) we can set out our working in a shorter way:</span></p>\n<p><span data-jme-visible=\"a&gt;0 and (choice=3 or choice=4)\">\\begin{alignat}{2} &amp;&amp;|\\simplify{{a}x+{b}}| &amp;\\var{latex(sym)} \\var{c}\\\\ \\var{-c} &amp;\\var{latex(sym)}&amp;\\simplify{{a}x+{b}}&amp;\\var{latex(sym)}\\var{c}\\\\ \\var{-c-b} &amp;\\var{latex(sym)} &amp;\\simplify{{a}x}\\quad&amp;\\var{latex(sym)}\\var{c-b}\\\\ \\var[fractionNumbers]{r1}&amp;\\var{latex(sym)}&amp;x\\quad&amp;\\var{latex(sym)}\\var[fractionNumbers]{r2}.\\end{alignat}</span></p>\n<p><span data-jme-visible=\"a=0 and (choice=3 or choice=4)\">\\begin{alignat}{2} &amp;&amp;|\\simplify{{a}x+{b}}| &amp;\\var{latex(sym)} \\var{c}\\\\ \\var{-c} &amp;\\var{latex(sym)}&amp;\\simplify{{a}x+{b}}&amp;\\var{latex(sym)}\\var{c}\\\\ \\var{-c-b} &amp;\\var{latex(sym)} &amp;\\simplify{{a}x}\\quad&amp;\\var{latex(sym)}\\var{c-b}.\\end{alignat}</span></p>\n<p><span data-jme-visible=\"a&lt;0 and (choice=3 or choice=4)\">\\begin{alignat}{2} &amp;&amp;|\\simplify{{a}x+{b}}| &amp;\\var{latex(sym)} \\var{c}\\\\ \\var{-c} &amp;\\var{latex(sym)}&amp;\\simplify{{a}x+{b}}&amp;\\var{latex(sym)}\\var{c}\\\\ \\var{-c-b} &amp;\\var{latex(sym)} &amp;\\simplify{{a}x}\\quad&amp;\\var{latex(sym)}\\var{c-b}\\\\ \\var[fractionNumbers]{r2}&amp;\\var{latex(backsym)}&amp;x\\quad&amp;\\var{latex(backsym)}\\var[fractionNumbers]{r1}\\\\\\var[fractionNumbers]{r1}&amp;\\var{latex(sym)}&amp;x\\quad&amp;\\var{latex(sym)}\\var[fractionNumbers]{r2}.\\end{alignat}</span></p>\n<p></p>\n<h3><strong>Cases Method</strong></h3>\n<p>&nbsp;</p>\n<p>Recall that the absolute value is defined as the piecewise function&nbsp;\\[|x|=\\begin{cases}x, &amp;\\text{ for } x\\ge 0\\\\ -x, &amp;\\text{ for } x&lt;0. \\end{cases}\\]</p>\n<p>This means that $|\\simplify{{a}x+{b}}|$ is actually $\\simplify{{a}x+{b}}$&nbsp;when $\\simplify{{a}x+{b}}\\ge 0$ but it is $-(\\simplify{{a}x+{b}})$ when $\\simplify{{a}x+{b}}&lt;0$. We have two cases:</p>\n<p></p>\n<p><strong>Case 1:&nbsp;$\\simplify{{a}x+{b}}\\ge 0$</strong></p>\n<p>Rearranging&nbsp;$\\simplify{{a}x+{b}}\\ge 0$ for $x$ gives <span data-jme-visible=\"a&gt;0\">$x \\ge \\simplify[fractionNumbers]{{-b/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x \\le \\simplify[fractionNumbers]{{-b/a}}$</span> and so case 1 is only&nbsp;relevant for <span data-jme-visible=\"a&gt;0\">$x \\ge \\simplify[fractionNumbers]{{-b/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x \\le \\simplify[fractionNumbers]{{-b/a}}$</span>.</p>\n<p>In case 1, our inequality is simply&nbsp;$\\simplify{{a}x+{b}} \\var{latex(sym)} \\var{c}$, and so rearranging it for $x$ gives <span data-jme-visible=\"a&gt;0\">$x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,!unitDenominator]{{(c-b)/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x \\var{latex(backsym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$</span>.&nbsp;</p>\n<p><strong></strong>So in conclusion for case 1, we require that <span data-jme-visible=\"a&gt;0\">$x \\ge \\simplify[fractionNumbers]{{-b/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x \\le \\simplify[fractionNumbers]{{-b/a}}$</span>&nbsp;&nbsp;<strong> and&nbsp;</strong> <span data-jme-visible=\"a&gt;0\">$x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x \\var{latex(backsym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$</span> which is the same as <span data-jme-visible=\"prod=3 or prod=4\">$\\simplify[fractionNumbers]{-{b/a}}\\le x\\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$</span> <span data-jme-visible=\"prod=-3 or prod=-4\">$\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}\\var{latex(sym)} x\\le \\simplify[fractionNumbers]{{-b/a}}$</span> <span data-jme-visible=\"prod=1 or prod=2\">$x\\var{latex(sym)}\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$</span> <span data-jme-visible=\"prod=-1 or prod=-2\">$x\\var{latex(backsym)}\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(c-b)/a}}$</span>.</p>\n<p></p>\n<p><strong>Case 2:&nbsp;$\\simplify{{a}x+{b}}&lt; 0$</strong></p>\n<p>Rearranging&nbsp;$\\simplify{{a}x+{b}}&lt; 0$ for $x$ gives <span data-jme-visible=\"a&gt;0\">$x &lt;&nbsp;\\simplify[fractionNumbers]{{-b/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x &gt;&nbsp;\\simplify[fractionNumbers]{{-b/a}}$</span> and so case 2&nbsp;is only&nbsp;relevant for <span data-jme-visible=\"a&gt;0\">$x &lt;&nbsp;\\simplify[fractionNumbers]{{-b/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x &gt;&nbsp;\\simplify[fractionNumbers]{{-b/a}}$</span>.</p>\n<p>In case 2, our inequality is actually $-(\\simplify{{a}x+{b}}) \\var{latex(sym)} \\var{c}$, and so rearranging it for $x$ gives <span data-jme-visible=\"a&gt;0\">$x \\var{latex(backsym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$</span>.&nbsp;</p>\n<p><strong></strong>So in conclusion for case 2, we require that <span data-jme-visible=\"a&gt;0\">$x &lt;&nbsp;\\simplify[fractionNumbers]{-{b/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x &gt;&nbsp;\\simplify[fractionNumbers]{-{b/a}}$</span><strong>&nbsp; and&nbsp;</strong> <span data-jme-visible=\"a&gt;0\">$x \\var{latex(backsym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$</span> <span data-jme-visible=\"a&lt;0\">$x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$</span> which is the same as&nbsp;<span data-jme-visible=\"prod=3 or prod=4\">$\\simplify[fractionNumbers]{{-(b+c)/a}}\\var{latex(sym)} x&lt; \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{-b}/{a}}$</span> <span data-jme-visible=\"prod=-3 or prod=-4\">$\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{-b/a}}&lt; x \\var{latex(sym)} \\simplify[fractionNumbers,simplifyFractions,unitDenominator]{-{(b+c)/a}}$</span> <span data-jme-visible=\"prod=1 or prod=2\">$x\\var{latex(backsym)}\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$</span> <span data-jme-visible=\"prod=-1 or prod=-2\">$x\\var{latex(sym)}\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{{(-c-b)/a}}$</span>.</p>\n<p></p>\n<p>Therefore the solution to our original inequality $|\\simplify{{a}x+{b}}|&nbsp;\\var{latex(sym)} \\var{c}$ is&nbsp;</p>\n<p><span data-jme-visible=\"one=true and strict=true\">\\[\\var[fractionNumbers]{r1}&lt;x&lt;\\var[fractionNumbers]{r2}.\\]</span></p>\n<p><span data-jme-visible=\"one=true and strict=false\">\\[\\var[fractionNumbers]{r1}\\le 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