// Numbas version: exam_results_page_options {"name": "Ida's copy of Solving linear inequalities", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "preamble": {"js": "", "css": ""}, "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 $\\var{a[0]}$ 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}

\n

#### b)

\n

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}

\n

#### c)

\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 } 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}

\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[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}

\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[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}

\n

#### f)

\n

In this example, separate the $x$-term from all other terms and remember to reverse the inequality when dividing by $\\simplify{{a[7]}-{b[4]}}$.

\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}

\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[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}

", "variables": {"a": {"templateType": "anything", "name": "a", "description": "", "definition": "repeat(random(2..10),10)", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "name": "c", "description": "", "definition": "random(2,3,6)", "group": "Ungrouped variables"}, "b": {"templateType": "anything", "name": "b", "description": "", "definition": "repeat(random(11..20),10)", "group": "Ungrouped variables"}}, "variable_groups": [], "name": "Ida's copy of Solving linear inequalities", "tags": ["inequalities", "linear inequalities", "solving linear inequalities", "taxonomy"], "rulesets": {}, "parts": [{"type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrange": [0, 1], "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "checkingtype": "absdiff", "vsetrangepoints": 5, "answer": "({a[1]}+{a[0]})", "type": "jme", "variableReplacements": [], "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showpreview": true}], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "

$\\simplify{x-{a[0]}<{a[1]}}$

\n

$x<$ [[0]]

"}, {"type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrange": [0, 1], "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "checkingtype": "absdiff", "answersimplification": "all", "answer": "{a[3]}/{a[2]}", "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showpreview": true}], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "

$\\simplify{{a[2]}x<{a[3]}}$

\n

$x<$ [[0]]

"}, {"type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrange": [0, 1], "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "checkingtype": "absdiff", "answersimplification": "all", "answer": "({a[5]}+{a[4]})/{a[6]}", "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showpreview": true}], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "

$\\simplify{{a[6]}x-{a[4]}<{a[5]}}$

\n

$x<$ [[0]]

"}, {"type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrange": [0, 1], "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "checkingtype": "absdiff", "answersimplification": "all", "answer": "({a[5]}+{a[4]})/-{a[6]}", "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showpreview": true}, {"displayColumns": 0, "shuffleChoices": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "variableReplacements": [], "matrix": ["1", 0], "displayType": "dropdownlist", "showFeedbackIcon": true, "choices": ["

>

", "

<

"], "maxMarks": 0, "marks": 0, "distractors": ["", ""]}], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "

$\\simplify{{-a[6]}x-{a[4]}<{a[5]}}$

\n

$x$ [[1]]  [[0]]

"}, {"type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrange": [0, 1], "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "checkingtype": "absdiff", "answersimplification": "all", "answer": "({b[3]}+{b[1]})/({b[0]}+{b[2]})", "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showpreview": true}, {"displayColumns": 0, "shuffleChoices": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist", "showFeedbackIcon": true, "choices": ["

>

", "

<

"], "maxMarks": 0, "marks": 0, "distractors": ["", ""]}], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "

$\\simplify{{b[0]}x-{b[1]}<{b[3]}-{b[2]}x}$

\n

$x$  [[1]]  [[0]]

"}, {"type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrange": [0, 1], "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "checkingtype": "absdiff", "answersimplification": "all", "answer": "(-{b[5]}-b+{a[8]}a)/({b[4]}-{a[7]})", "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showpreview": true}, {"displayColumns": 0, "shuffleChoices": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist", "showFeedbackIcon": true, "choices": ["

>

", "

<

"], "maxMarks": 0, "marks": 0, "distractors": ["", ""]}], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "

$\\simplify{-{b[4]}x+{a[8]}a>{b[5]}+b-{a[7]}x}$

\n

$x$ [[1]] [[0]]

"}, {"type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"vsetrange": [0, 1], "expectedvariablenames": [], "checkvariablenames": false, "checkingaccuracy": 0.001, "showCorrectAnswer": true, "scripts": {}, "checkingtype": "absdiff", "answersimplification": "all", "answer": "{a[0]}-6h/{c}-g", "type": "jme", "variableReplacements": [], "vsetrangepoints": 5, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "marks": 1, "showpreview": true}, {"displayColumns": 0, "shuffleChoices": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "variableReplacements": [], "matrix": [0, "1"], "displayType": "dropdownlist", "showFeedbackIcon": true, "choices": ["

>

", "

<

"], "maxMarks": 0, "marks": 0, "distractors": ["", ""]}], "showFeedbackIcon": true, "showCorrectAnswer": true, "scripts": {}, "marks": 0, "prompt": "

$\\simplify{-{c}(x+g)>6h-{c}{a[0]}}$

\n

$x$ [[1]]  [[0]]

"}], "ungrouped_variables": ["a", "b", "c"], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

In the first three parts, rearrange linear inequalities to make $x$ the subject.

\n

In the last four parts, correctly give the direction of the inequality sign after rearranging an inequality.

"}, "statement": "

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

", "functions": {}, "type": "question", "contributors": [{"name": "Ida Landg\u00e4rds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2336/"}]}]}], "contributors": [{"name": "Ida Landg\u00e4rds", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2336/"}]}