// Numbas version: exam_results_page_options {"name": "Ida Friestad's copy of Solving linear inequalities/inequations: two step", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["a", "b", "c", "ans1", "d", "f", "g", "ans2", "h", "j", "k", "ans3", "l", "m", "n", "ans4", "p", "q", "r", "ans5", "s", "t", "u", "ans6"], "name": "Ida Friestad's copy of Solving linear inequalities/inequations: two step", "tags": ["algebra", "balancing equations", "inequalities", "inequality", "inequation", "inequations", "Linear equations", "linear equations", "rearranging equations", "solving equations", "Solving equations", "two step equations"], "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "

Given $\\var{a}x+\\var{b}<\\var{c}$, solving for $x$ gives $x<$ [[0]].

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Solve like an equation except when dividing or multiplying by a negative we must swap the direction of the inequality.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{a}x+\\var{b}$$<$$\\var{c}$ 
 
$\\var{a}x+\\var{b}-\\var{b}$$<$$\\var{c}-\\var{b}$
 
$\\var{a}x$$<$$\\var{c-b}$
 
$\\displaystyle{\\frac{\\var{a}x}{\\var{a}}}$ $<$$\\displaystyle{\\frac{\\var{c-b}}{\\var{a}}}$
 
$x$$<$$\\displaystyle{\\simplify{{c-b}/{a}}}$
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Given $\\var{d}-\\var{f}y\\le\\var{g}$,  $y\\ge$ [[0]].

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Solve like an equation except when dividing or multiplying by a negative we must swap the direction of the inequality.

\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{d}-\\var{f}y$$\\le$$\\var{g}$ 
 
$\\var{d}-\\var{f}y-\\var{d}$$\\le$$\\var{g}-\\var{d}$
 
$-\\var{f}y$$\\le$$\\var{g-d}$
 
$\\displaystyle{\\frac{\\var{-f}y}{\\var{-f}}}$ $\\ge$$\\displaystyle{\\frac{\\var{g-d}}{\\var{-f}}}$
 
$y$$\\ge$$\\displaystyle{\\simplify{{g-d}/{-f}}}$
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Rearrange $\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}<\\var{k}$ to determine the value of $z$.

\n

$z>$ [[0]]

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Solve like an equation except when dividing or multiplying by a negative we must swap the direction of the inequality.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}$$<$$\\var{k}$ 
 
$\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}+\\var{j}$$<$$\\var{k}+\\var{j}$
 
$\\displaystyle{-\\frac{z}{\\var{h}}}$$<$$\\var{k+j}$
 
$\\displaystyle{-\\frac{z}{\\var{h}}\\times\\var{h}}$ $<$$\\var{k+j}\\times \\var{h}$
 
$-z$$<$$\\var{-ans3}$
 
$z$$>$$\\var{ans3}$
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Solve $\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}\\le\\var{n}$ for $a$.

\n

$a\\le$ [[0]]

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Solve like an equation except when dividing or multiplying by a negative we must swap the direction of the inequality.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}$$\\le$$\\var{n}$ 
 
$\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}\\times \\var{m}$$\\le$$\\var{n}\\times\\var{m}$
 
$a-\\var{l}$$\\le$$\\var{n*m}$
 
$a-\\var{l}+\\var{l}$ $\\le$$\\var{n*m}+\\var{l}$
 
$a$$\\le$$\\var{ans4}$
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Solve $\\var{p}>\\var{q}(\\var{r}+b)$.

\n

$b>$ [[0]]

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Solve like an equation except when dividing or multiplying by a negative we must swap the direction of the inequality.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\var{p}$$>$$\\var{q}(\\var{r}+b)$ 
 
$\\displaystyle{\\frac{\\var{p}}{\\var{q}}}$$<$$\\displaystyle{\\frac{\\var{q}(\\var{r}+b)}{\\var{q}}}$
 
$\\displaystyle{\\simplify{{p}/{q}}}$$<$$\\var{r}+b$
 
$\\displaystyle{\\simplify{{p}/{q}}}-\\var{r}$ $<$$\\var{r}+b-\\var{r}$
 
$\\displaystyle{\\simplify{{p-r*q}/{q}}}$$<$$b$
 
$b$$>$$\\displaystyle{\\simplify{{p-r*q}/{q}}}$
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Solve $\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}\\ge\\var{u}$.

\n

$w\\le$ [[0]]

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Solve like an equation except when dividing or multiplying by a negative we must swap the direction of the inequality.

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
$\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}$$\\ge$$\\var{u}$ 
 
$\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}\\times\\var{t}$$\\ge$$\\var{u}\\times\\var{t}$
 
$\\var{s}w$$\\ge$$\\var{u*t}$
 
$\\displaystyle{\\frac{\\var{s}w}{\\var{s}}}$ $\\le$$\\displaystyle{\\frac{\\var{u*t}}{\\var{s}}}$
 
$w$$\\le$$\\displaystyle{\\simplify{{u*t}/{s}}}$
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