// 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, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "preamble": {"js": "", "css": ""}, "advice": "<p><strong></strong>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.</p>\n<p>For example, the following inequality is true:</p>\n<p>\\[ -3 \\lt -2 \\]</p>\n<p>When we multiply both sides by $-2$, the inequality sign must reverse:</p>\n<p>\\[ 6 \\gt 4 \\]</p>\n<h4>a)</h4>\n<p>To put $x$ on its own, we need to add $\\var{a[0]}$ to&nbsp;both sides of the inequality.</p>\n<p>\\begin{align}<br/>\\simplify{x-{a[0]}}&amp;&lt;\\var{a[1]}\\\\[1em]<br/>\\var{x}&amp;&lt;\\simplify[]{{a[1]}+{a[0]}}\\\\[1em]<br/>x&amp;&lt;\\simplify{({a[1]}+{a[0]})}\\text{.}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>In this example we find&nbsp;$x$ by dividing both sides by the coefficient of $x$, $\\var{a[2]}$.</p>\n<p>\\begin{align}<br/>\\simplify{{a[2]}}x&amp;&lt;\\var{a[3]}\\\\[1em]<br/>x&amp;&lt;\\simplify{{a[3]}/{a[2]}}\\text{.}<br/>\\end{align}</p>\n<h4>c)</h4>\n<p>\\begin{align}<br/>\\simplify{{a[6]}x-{a[4]}}&amp;&lt;\\var{a[5]}\\\\[1em]<br/>\\var{a[6]}x&amp;&lt;\\var{a[5]}+\\var{a[4]} &amp; \\text{Add } 8 \\text{ to get } x \\text{ on its own.}\\\\[1em]<br/>x&amp;&lt;\\simplify[]{({a[5]}+{a[4]})/{a[6]}} &amp; \\text{ Divide by } \\var{a[6]} \\text{.} \\\\[1em]<br/>x&amp;&lt;\\simplify{({a[5]}+{a[4]})/{a[6]}}\\text{.}<br/>\\end{align}</p>\n<h4>d)</h4>\n<p>In this example, take the constants&nbsp;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.</p>\n<p>\\begin{align}<br/>\\simplify{{-a[6]}x - {a[4]}} &amp;&lt; \\var{a[5]} \\\\[1em]<br/>\\var{-a[6]}x &amp;&lt; \\var{a[5]} + \\var{a[4]} &amp; \\text{Add } \\var{a[4]} \\text{ to both sides.} \\\\[1em]<br/>x &amp;&gt; \\simplify[]{({a[5]}+{a[4]})/-{a[6]}} \\text{ Divide by } \\var{-a[6]} \\text{. The inequality is reversed.} \\\\[1em]<br/>x &amp;&gt; \\simplify{({a[5]}+{a[4]})/-{a[6]}}\\text{.}\\\\<br/>\\end{align}</p>\n<h4>e)</h4>\n<p>In this example, separate the constants&nbsp;and the $x$-term, then divide by the $x$-coefficient to find an inequality for $x$.</p>\n<p>\\begin{align}<br/>\\simplify{{b[0]}x-{b[1]}}&amp;&lt;\\simplify{{b[3]}-{b[2]}x}\\\\[1em]<br/>\\simplify{({b[0]}+{b[2]})x}&amp;&lt;\\simplify{{b[3]}+{b[1]}}\\\\[1em]<br/>x&amp;&lt;\\simplify{({b[3]}+{b[1]})/({b[0]}+{b[2]})}\\text{.}\\\\[1em]<br/>\\end{align}</p>\n<h4>f)</h4>\n<p>In this example, separate the $x$-term from all other terms and remember to reverse the inequality when dividing by $\\simplify{{a[7]}-{b[4]}}$.</p>\n<p>\\begin{align}<br/>\\simplify{-{b[4]}x+{a[8]}a}&amp;&gt;\\simplify{{b[5]}+b-{a[7]}x}\\\\[1em]<br/>\\simplify{{a[7]}-{b[4]}}x&amp;&gt;\\simplify{{b[5]}+b-{a[8]}a}\\\\[1em]<br/>x&amp;&lt;\\simplify{(-{b[5]}-b+{a[8]}a)/({b[4]}-{a[7]})}\\text{.}\\\\[1em]<br/>\\end{align}</p>\n<p><strong>g)</strong></p>\n<p>In this example, a simple way to solve for $x$ is&nbsp;to divide by $-\\var{c}$ before rearranging the rest of the equation by subtracting $g$ from both sides.</p>\n<p>\\begin{align}<br/>\\simplify{-{c}(x+g)}&amp;&gt;\\simplify{6h-{c}{a[0]}}\\\\[1em]<br/>\\simplify{(x+g)}&amp;&lt;\\simplify[]{6h/-{c}+{a[0]}}\\\\[1em]<br/>x&amp;&lt;\\simplify[]{6h/-{c}+{a[0]}-g}\\\\[1em]<br/>x&amp;&lt;\\simplify{{a[0]}-6h/{c}-g}\\text{.}<br/>\\end{align}</p>", "variables": {"a": {"templateType": "anything", "name": "a", "description": "", "definition": "repeat(random(2..10),10)", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "name": "c", 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correctly give the direction of the inequality sign after rearranging an inequality.</p>"}, "statement": "<p>Solve the following linear inequalities by finding the set of possible values for $x$. State your answers as fractions where applicable.</p>", "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/"}]}