// Numbas version: exam_results_page_options {"name": "Solving linear inequalities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"questions": [{"functions": {}, "variablesTest": {"maxRuns": 100, "condition": ""}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

In parts (a) and (b) rearrange linear inequalities to make $x$ the subject.

\n

In the parts (c) and (d) correctly give the direction of the inequality sign after rearranging an inequality.

"}, "variables": {"c": {"definition": "random(2,3,6)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "c"}, "b": {"definition": "repeat(random(11..20),10)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "b"}, "a": {"definition": "repeat(random(2..10),10)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "a"}}, "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}], "extensions": [], "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.

\n

For example, the following inequality is true:

\n

\$-3 \\lt -2 \$

\n

When we multiply both sides by $-2$, the inequality sign must reverse:

\n

\$6 \\gt 4 \$

\n

a)

\n

To put $x$ on its own, we need to add $\\var{a}$ to both sides of the inequality.

\n

\\begin{align}
\\simplify{x-{a}}&<\\var{a}\\\\[1em]
\\var{x}&<\\simplify[]{{a}+{a}}\\\\[1em]
x&<\\simplify{({a}+{a})}\\text{.}
\\end{align}

\n

b)

\n

In this example we find $x$ by dividing both sides by the coefficient of $x$, $\\var{a}$.

\n

\\begin{align}
\\simplify{{a}}x&<\\var{a}\\\\[1em]
x&<\\simplify{{a}/{a}}\\text{.}
\\end{align}

\n

c)

\n

\\begin{align}
\\simplify{{a}x-{a}}&<\\var{a}\\\\[1em]
\\var{a}x&<\\var{a}+\\var{a} & \\text{Add } 8 \\text{ to get } x \\text{ on its own.}\\\\[1em]
x&<\\simplify[]{({a}+{a})/{a}} & \\text{ Divide by } \\var{a} \\text{.} \\\\[1em]
x&<\\simplify{({a}+{a})/{a}}\\text{.}
\\end{align}

\n

d)

\n

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}x - {a}} &< \\var{a} \\\\[1em]
\\var{-a}x &< \\var{a} + \\var{a} & \\text{Add } \\var{a} \\text{ to both sides.} \\\\[1em]
x &> \\simplify[]{({a}+{a})/-{a}} \\text{ Divide by } \\var{-a} \\text{. The inequality is reversed.} \\\\[1em]
x &> \\simplify{({a}+{a})/-{a}}\\text{.}\\\\
\\end{align}

\n

e)

\n

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}x-{b}}&<\\simplify{{b}-{b}x}\\\\[1em]
\\simplify{({b}+{b})x}&<\\simplify{{b}+{b}}\\\\[1em]
x&<\\simplify{({b}+{b})/({b}+{b})}\\text{.}\\\\[1em]
\\end{align}

\n

f)

\n

In this example, separate the $x$-term from all other terms and remember to reverse the inequality when dividing by $\\simplify{{a}-{b}}$.

\n

\\begin{align}
\\simplify{-{b}x+{a}a}&>\\simplify{{b}+b-{a}x}\\\\[1em]
\\simplify{{a}-{b}}x&>\\simplify{{b}+b-{a}a}\\\\[1em]
x&<\\simplify{(-{b}-b+{a}a)/({b}-{a})}\\text{.}\\\\[1em]
\\end{align}

\n

g)

\n

In 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}}\\\\[1em]
\\simplify{(x+g)}&<\\simplify[]{6h/-{c}+{a}}\\\\[1em]
x&<\\simplify[]{6h/-{c}+{a}-g}\\\\[1em]
x&<\\simplify{{a}-6h/{c}-g}\\text{.}
\\end{align}

", "variable_groups": [], "ungrouped_variables": ["a", "b", "c"], "rulesets": {}, "name": "Solving linear inequalities", "parts": [{"variableReplacements": [], "customMarkingAlgorithm": "", "prompt": "

$\\simplify{{a}x<{a}}$

\n

$x<$ []

", "gaps": [{"variableReplacements": [], "customMarkingAlgorithm": "", "checkingAccuracy": 0.001, "marks": 1, "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showCorrectAnswer": true, "checkVariableNames": false, "showFeedbackIcon": true, "showPreview": true, "unitTests": [], "expectedVariableNames": [], "answerSimplification": "all", "answer": "{a}/{a}", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "scripts": {}, "vsetRange": [0, 1], "type": "jme", "failureRate": 1}], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "sortAnswers": false, "showFeedbackIcon": true, "unitTests": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "gapfill"}, {"variableReplacements": [], "customMarkingAlgorithm": "", "prompt": "

$\\simplify{{b}x-{b}<{b}-{b}x}$

\n

$x$  []  []

", "gaps": [{"variableReplacements": [], "customMarkingAlgorithm": "", "checkingAccuracy": 0.001, "marks": 1, "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "showCorrectAnswer": true, "checkVariableNames": false, "showFeedbackIcon": true, "showPreview": true, "unitTests": [], "expectedVariableNames": [], "answerSimplification": "all", "answer": "({b}+{b})/({b}+{b})", "extendBaseMarkingAlgorithm": true, "checkingType": "absdiff", "scripts": {}, "vsetRange": [0, 1], "type": "jme", "failureRate": 1}, {"variableReplacements": [], "unitTests": [], "marks": 0, "maxMarks": 0, "displayColumns": 0, "showCorrectAnswer": true, "choices": ["

>

", "

<

"], "minMarks": 0, "matrix": [0, "1"], "customMarkingAlgorithm": "", "distractors": ["", ""], "shuffleChoices": false, "variableReplacementStrategy": "originalfirst", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "scripts": {}, "displayType": "dropdownlist", "type": "1_n_2"}], "marks": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "sortAnswers": false, "showFeedbackIcon": true, "unitTests": [], "extendBaseMarkingAlgorithm": true, "scripts": {}, "type": "gapfill"}], "preamble": {"js": "", "css": ""}, "tags": [], "type": "question", "statement": "

Solve the following linear inequalities by finding the set of possible values for $x$. State your answers as fractions where applicable.

"}], "pickingStrategy": "all-ordered"}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}