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.
", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["a", "b", "c", "d", "g", "f"], "type": "question", "extensions": ["geogebra"], "advice": "This example 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 $f(x)>0$.\\[\\\\[0.1em]\\]
\n{geogebra_applet('Hk5dTptY',[[\"a\",a],[\"b\",b]])}
\n\\[
\\begin{align}
\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}&>0\\\\
\\simplify{(x+{a})(x-{b})}&=0\\text{.}
\\end{align}
\\]
Therefore
\n\\[ \\simplify{x<{-a}}\\text{ or }\\simplify{x>{b}}\\text{.}\\]
\nWe use the same method in this example but this time we use our graph to visualise where $g(x)<0$.\\[\\\\[0.1em]\\]
\n{geogebra_applet('PUFStTNa',[[\"c\",c],[\"d\",d]])}
\n\\[
\\begin{align}
\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}&<0\\\\
\\simplify{(x+{c})(x-{d})}&=0\\text{.}
\\end{align}
\\]
Therefore
\n\\[ \\simplify{x>{-c}}\\text{ and }\\simplify{x<{d}}\\text{.}\\]
\nThe notable difference with solving this equation is the requirement to rearrange the inequality before factorisation.\\[\\\\[0.1em]\\]
\n{geogebra_applet('CZsSCdH6',[[\"e\",g],[\"f\",f]])}
\n\\[
\\begin{align}
\\simplify{({g}+{f})x-{g}{f}}&>x^2\\\\
\\simplify{x^2-({g}+{f})x+{g}{f}}&<0\\\\
\\simplify{(x-{g})(x-{f})}&=0\\text{.}
\\end{align}
\\]
Therefore
\n\\[ \\simplify{x>{g}}\\text{ and }\\simplify{x<{f}}\\text{.}\\]
\nAlternatively, we can plot the graph of $x^2$ against $\\simplify{({g}+{f})x-{g}{f}}$ to visualise the same result.
\n{geogebra_applet('Ez697SNE',[[\"g\",g],[\"f\",f]])}
\nFrom this graph we can see that the values of $x$ where $\\simplify{({g}+{f})x-{g}{f}}>x^2$ are the same as the values of $x$ where $\\simplify{x^2-({g}+{f})x+{g}{f}}<0$.
", "variable_groups": [], "rulesets": {}, "statement": "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 quadratic.
", "name": "Solve quadratic inequalities", "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": ["AND
", "OR
"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "Find the range of values for $x$ such that $\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}>0$.
\nFactorise $f(x)$:
\n$\\simplify{f(x)=x^2+({a}-{b})x-{a}{b}}=$ [[0]] $=0$.
\nHence,
\n$x>$ [[1]]
\n[[3]]
\n$x<$ [[2]]
"}, {"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": ["AND
", "OR
"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "Find the range of values for $x$ such that $\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}<0$.
\nFactorise $f(x)$:
\n$\\simplify{f(x)=x^2+({c}-{d})x-{c}{d}}=$ [[0]] $=0$.
\nHence,
\n$x<$ [[1]]
\n[[3]]
\n$x>$ [[2]]
"}, {"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": ["AND
", "OR
"], "scripts": {}, "distractors": ["", ""], "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist"}], "showFeedbackIcon": true, "prompt": "Find the range of vaues such that: $\\simplify{({g}+{f})x-{g}{f}>x^2}$
\nRearrange then factorise the inequality:
\n[[0]] $<0$.
\nUse the above result to find the range of values for $x$ such that $\\simplify{({g}+{f})x-{g}{f}>x^2}$.
\n$x>$ [[1]]
\n[[3]]
\n$x<$ [[2]]
"}], "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": ""}, "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/"}]}]}], "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/"}]}