$\\simplify{x-{a[0]}<{a[1]}}$
\n$x<$ [[0]]
"}, {"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [], "checkVariableNames": false, "failureRate": 1, "answerSimplification": "all", "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "{a[3]}/{a[2]}"}], "unitTests": [], "type": "gapfill", "prompt": "$\\simplify{{a[2]}x<{a[3]}}$
\n$x<$ [[0]]
"}, {"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [], "checkVariableNames": false, "failureRate": 1, "answerSimplification": "all", "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "({a[5]}+{a[4]})/{a[6]}"}], "unitTests": [], "type": "gapfill", "prompt": "$\\simplify{{a[6]}x-{a[4]}<{a[5]}}$
\n$x<$ [[0]]
"}, {"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [], "checkVariableNames": false, "failureRate": 1, "answerSimplification": "all", "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "({a[5]}+{a[4]})/-{a[6]}"}, {"variableReplacementStrategy": "originalfirst", "distractors": ["", ""], "customName": "", "displayColumns": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "choices": [">
", "<
"], "showCellAnswerState": true, "scripts": {}, "marks": 0, "matrix": ["1", 0], "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "maxMarks": 0, "useCustomName": false, "shuffleChoices": false, "unitTests": [], "type": "1_n_2", "displayType": "dropdownlist", "minMarks": 0}], "unitTests": [], "type": "gapfill", "prompt": "$\\simplify{{-a[6]}x-{a[4]}<{a[5]}}$
\n$x$ [[1]] [[0]]
"}, {"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [], "checkVariableNames": false, "failureRate": 1, "answerSimplification": "all", "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "({b[3]}+{b[1]})/({b[0]}+{b[2]})"}, {"variableReplacementStrategy": "originalfirst", "distractors": ["", ""], "customName": "", "displayColumns": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "choices": [">
", "<
"], "showCellAnswerState": true, "scripts": {}, "marks": 0, "matrix": [0, "1"], "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "maxMarks": 0, "useCustomName": false, "shuffleChoices": false, "unitTests": [], "type": "1_n_2", "displayType": "dropdownlist", "minMarks": 0}], "unitTests": [], "type": "gapfill", "prompt": "$\\simplify{{b[0]}x-{b[1]}<{b[3]}-{b[2]}x}$
\n$x$ [[1]] [[0]]
"}, {"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [{"name": "a", "value": ""}, {"name": "b", "value": ""}], "checkVariableNames": false, "failureRate": 1, "answerSimplification": "all", "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "(-{b[5]}-b+{a[8]}a)/({b[4]}-{a[7]})"}, {"variableReplacementStrategy": "originalfirst", "distractors": ["", ""], "customName": "", "displayColumns": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "choices": [">
", "<
"], "showCellAnswerState": true, "scripts": {}, "marks": 0, "matrix": [0, "1"], "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "maxMarks": 0, "useCustomName": false, "shuffleChoices": false, "unitTests": [], "type": "1_n_2", "displayType": "dropdownlist", "minMarks": 0}], "unitTests": [], "type": "gapfill", "prompt": "$\\simplify{-{b[4]}x+{a[8]}a>{b[5]}+b-{a[7]}x}$
\n$x$ [[1]] [[0]]
"}, {"variableReplacementStrategy": "originalfirst", "customName": "", "showCorrectAnswer": true, "showFeedbackIcon": true, "sortAnswers": false, "variableReplacements": [], "scripts": {}, "marks": 0, "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "useCustomName": false, "gaps": [{"variableReplacementStrategy": "originalfirst", "unitTests": [], "showFeedbackIcon": true, "vsetRangePoints": 5, "variableReplacements": [], "useCustomName": false, "marks": 1, "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "vsetRange": [0, 1], "showCorrectAnswer": true, "type": "jme", "showPreview": true, "valuegenerators": [{"name": "g", "value": ""}, {"name": "h", "value": ""}], "checkVariableNames": false, "failureRate": 1, "answerSimplification": "all", "customName": "", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "answer": "{a[0]}-6h/{c}-g"}, {"variableReplacementStrategy": "originalfirst", "distractors": ["", ""], "customName": "", "displayColumns": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "choices": [">
", "<
"], "showCellAnswerState": true, "scripts": {}, "marks": 0, "matrix": [0, "1"], "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "maxMarks": 0, "useCustomName": false, "shuffleChoices": false, "unitTests": [], "type": "1_n_2", "displayType": "dropdownlist", "minMarks": 0}], "unitTests": [], "type": "gapfill", "prompt": "$\\simplify{-{c}(x+g)>6h-{c}{a[0]}}$
\n$x$ [[1]] [[0]]
"}], "functions": {}, "name": "Solving linear inequalities", "ungrouped_variables": ["a", "b", "c"], "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "In the first three parts, rearrange linear inequalities to make $x$ the subject.
\nIn the last four parts, correctly give the direction of the inequality sign after rearranging an inequality.
"}, "extensions": [], "statement": "Solve the following linear inequalities by finding the set of possible values for $x$. State your answers as fractions where applicable.
", "advice": "As with regular linear equations, we aim to isolate the variable by subtracting any constants when dividing by the $x$ coefficient. The only major difference is that when we divide or multiply by a negative number, the inequality sign is reversed.
\nFor example, the following inequality is true:
\n\\[ -3 \\lt -2 \\]
\nWhen we multiply both sides by $-2$, the inequality sign must reverse:
\n\\[ 6 \\gt 4 \\]
\nTo put $x$ on its own, we need to add $\\var{a[0]}$ to both sides of the inequality.
\n\\begin{align}
\\simplify{x-{a[0]}}&<\\var{a[1]}\\\\[1em]
\\var{x}&<\\simplify[]{{a[1]}+{a[0]}}\\\\[1em]
x&<\\simplify{({a[1]}+{a[0]})}\\text{.}
\\end{align}
In this example we find $x$ by dividing both sides by the coefficient of $x$, $\\var{a[2]}$.
\n\\begin{align}
\\simplify{{a[2]}}x&<\\var{a[3]}\\\\[1em]
x&<\\simplify{{a[3]}/{a[2]}}\\text{.}
\\end{align}
\\begin{align}
\\simplify{{a[6]}x-{a[4]}}&<\\var{a[5]}\\\\[1em]
\\var{a[6]}x&<\\var{a[5]}+\\var{a[4]} & \\text{Add } 8 \\text{ to get } x \\text{ on its own.}\\\\[1em]
x&<\\simplify[]{({a[5]}+{a[4]})/{a[6]}} & \\text{ Divide by } \\var{a[6]} \\text{.} \\\\[1em]
x&<\\simplify{({a[5]}+{a[4]})/{a[6]}}\\text{.}
\\end{align}
In this example, take the constants to one side, and keep the $x$ term on the other. Divide through by the negative $x$-coefficient to find an inequality for $x$. Notice that where you divide (or multiply) an equality by a negative value, the inequality sign is reversed.
\n\\begin{align}
\\simplify{{-a[6]}x - {a[4]}} &< \\var{a[5]} \\\\[1em]
\\var{-a[6]}x &< \\var{a[5]} + \\var{a[4]} & \\text{Add } \\var{a[4]} \\text{ to both sides.} \\\\[1em]
x &> \\simplify[]{({a[5]}+{a[4]})/-{a[6]}} \\text{ Divide by } \\var{-a[6]} \\text{. The inequality is reversed.} \\\\[1em]
x &> \\simplify{({a[5]}+{a[4]})/-{a[6]}}\\text{.}\\\\
\\end{align}
In this example, separate the constants and the $x$-term, then divide by the $x$-coefficient to find an inequality for $x$.
\n\\begin{align}
\\simplify{{b[0]}x-{b[1]}}&<\\simplify{{b[3]}-{b[2]}x}\\\\[1em]
\\simplify{({b[0]}+{b[2]})x}&<\\simplify{{b[3]}+{b[1]}}\\\\[1em]
x&<\\simplify{({b[3]}+{b[1]})/({b[0]}+{b[2]})}\\text{.}\\\\[1em]
\\end{align}
In this example, separate the $x$-term from all other terms and remember to reverse the inequality when dividing by $\\simplify{{a[7]}-{b[4]}}$.
\n\\begin{align}
\\simplify{-{b[4]}x+{a[8]}a}&>\\simplify{{b[5]}+b-{a[7]}x}\\\\[1em]
\\simplify{{a[7]}-{b[4]}}x&>\\simplify{{b[5]}+b-{a[8]}a}\\\\[1em]
x&<\\simplify{(-{b[5]}-b+{a[8]}a)/({b[4]}-{a[7]})}\\text{.}\\\\[1em]
\\end{align}
g)
\nIn this example, a simple way to solve for $x$ is to divide by $-\\var{c}$ before rearranging the rest of the equation by subtracting $g$ from both sides.
\n\\begin{align}
\\simplify{-{c}(x+g)}&>\\simplify{6h-{c}{a[0]}}\\\\[1em]
\\simplify{(x+g)}&<\\simplify[]{6h/-{c}+{a[0]}}\\\\[1em]
x&<\\simplify[]{6h/-{c}+{a[0]}-g}\\\\[1em]
x&<\\simplify{{a[0]}-6h/{c}-g}\\text{.}
\\end{align}