Match the following inequality symbols with their meanings.
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\nFor example, given $y>2$, we see the large side of the symbol is on the left and the small side is on the right. Therefore, the statement is saying $y$ is greater than 2.
\nWhen there is an extra line on the bottom, e.g. $\\ge$, this gives the look of an 'equals sign' and so $w\\ge 5$ means '$x$ is greater than or equal to 5'. Likewise, $z\\le 0$, means '$z$ is less than or equal to 0'.
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\n\nWhen dealing with negative numbers, it is important to remember that 'less' means further to the left on the number line, and 'greater' means further to the right on the number line. So $-10$ is less than $-5$, since $-10$ is five more places to the left than $-5$ on the number line.
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"g", "ans2", "h", "j", "k", "ans3", "l", "m", "n", "ans4", "p", "q", "r", "ans5", "s", "t", "u", "ans6"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Given $\\var{a}x+\\var{b}<\\var{c}$, solving for $x$ gives $x<$ [[0]].
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Solve like an equation except when dividing or multiplying by a negative we must swap the direction of the inequality.
\n| $\\var{a}x+\\var{b}$ | \n$<$ | \n$\\var{c}$ | \n
| \n | \n | \n |
| $\\var{a}x+\\var{b}-\\var{b}$ | \n$<$ | \n$\\var{c}-\\var{b}$ | \n
| \n | \n | \n |
| $\\var{a}x$ | \n$<$ | \n$\\var{c-b}$ | \n
| \n | \n | \n |
| $\\displaystyle{\\frac{\\var{a}x}{\\var{a}}}$ | \n$<$ | \n$\\displaystyle{\\frac{\\var{c-b}}{\\var{a}}}$ | \n
| \n | \n | \n |
| $x$ | \n$<$ | \n$\\displaystyle{\\simplify{{c-b}/{a}}}$ | \n
To satisfy $\\var{d}-\\var{f}y\\le\\var{g}$, we require $y\\ge$ [[0]].
", "stepsPenalty": "1", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Solve like an equation except when dividing or multiplying by a negative we must swap the direction of the inequality.
\n\n| $\\var{d}-\\var{f}y$ | \n$\\le$ | \n$\\var{g}$ | \n
| \n | \n | \n |
| $\\var{d}-\\var{f}y-\\var{d}$ | \n$\\le$ | \n$\\var{g}-\\var{d}$ | \n
| \n | \n | \n |
| $-\\var{f}y$ | \n$\\le$ | \n$\\var{g-d}$ | \n
| \n | \n | \n |
| $\\displaystyle{\\frac{\\var{-f}y}{\\var{-f}}}$ | \n$\\ge$ | \n$\\displaystyle{\\frac{\\var{g-d}}{\\var{-f}}}$ | \n
| \n | \n | \n |
| $y$ | \n$\\ge$ | \n$\\displaystyle{\\simplify{{g-d}/{-f}}}$ | \n
Rearrange $\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}<\\var{k}$ to determine the value of $z$.
\n$z>$ [[0]]
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\n| $\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}$ | \n$<$ | \n$\\var{k}$ | \n
| \n | \n | \n |
| $\\displaystyle{-\\frac{z}{\\var{h}}}-\\var{j}+\\var{j}$ | \n$<$ | \n$\\var{k}+\\var{j}$ | \n
| \n | \n | \n |
| $\\displaystyle{-\\frac{z}{\\var{h}}}$ | \n$<$ | \n$\\var{k+j}$ | \n
| \n | \n | \n |
| $\\displaystyle{-\\frac{z}{\\var{h}}\\times\\var{h}}$ | \n$<$ | \n$\\var{k+j}\\times \\var{h}$ | \n
| \n | \n | \n |
| $-z$ | \n$<$ | \n$\\var{-ans3}$ | \n
| \n | \n | \n |
| $z$ | \n$>$ | \n$\\var{ans3}$ | \n
Solve $\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}\\le\\var{n}$ for $a$.
\n$a\\le$ [[0]]
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\n| $\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}$ | \n$\\le$ | \n$\\var{n}$ | \n
| \n | \n | \n |
| $\\displaystyle{\\frac{a-\\var{l}}{\\var{m}}}\\times \\var{m}$ | \n$\\le$ | \n$\\var{n}\\times\\var{m}$ | \n
| \n | \n | \n |
| $a-\\var{l}$ | \n$\\le$ | \n$\\var{n*m}$ | \n
| \n | \n | \n |
| $a-\\var{l}+\\var{l}$ | \n$\\le$ | \n$\\var{n*m}+\\var{l}$ | \n
| \n | \n | \n |
| $a$ | \n$\\le$ | \n$\\var{ans4}$ | \n
Solve $\\var{p}>\\var{q}(\\var{r}+b)$.
\n$b>$ [[0]]
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\n| $\\var{p}$ | \n$>$ | \n$\\var{q}(\\var{r}+b)$ | \n
| \n | \n | \n |
| $\\displaystyle{\\frac{\\var{p}}{\\var{q}}}$ | \n$<$ | \n$\\displaystyle{\\frac{\\var{q}(\\var{r}+b)}{\\var{q}}}$ | \n
| \n | \n | \n |
| $\\displaystyle{\\simplify{{p}/{q}}}$ | \n$<$ | \n$\\var{r}+b$ | \n
| \n | \n | \n |
| $\\displaystyle{\\simplify{{p}/{q}}}-\\var{r}$ | \n$<$ | \n$\\var{r}+b-\\var{r}$ | \n
| \n | \n | \n |
| $\\displaystyle{\\simplify{{p-r*q}/{q}}}$ | \n$<$ | \n$b$ | \n
| \n | \n | \n |
| $b$ | \n$>$ | \n$\\displaystyle{\\simplify{{p-r*q}/{q}}}$ | \n
Solve $\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}\\ge\\var{u}$.
\n$w\\le$ [[0]]
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\n| $\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}$ | \n$\\ge$ | \n$\\var{u}$ | \n
| \n | \n | \n |
| $\\displaystyle{\\frac{\\var{s}w}{\\var{t}}}\\times\\var{t}$ | \n$\\ge$ | \n$\\var{u}\\times\\var{t}$ | \n
| \n | \n | \n |
| $\\var{s}w$ | \n$\\ge$ | \n$\\var{u*t}$ | \n
| \n | \n | \n |
| $\\displaystyle{\\frac{\\var{s}w}{\\var{s}}}$ | \n$\\le$ | \n$\\displaystyle{\\frac{\\var{u*t}}{\\var{s}}}$ | \n
| \n | \n | \n |
| $w$ | \n$\\le$ | \n$\\displaystyle{\\simplify{{u*t}/{s}}}$ | \n
Given that $x>\\var{left_ba}$ and $x<\\var{right_ba}$, we can write [[0]] $<x<$ [[1]].
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", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "1", "prompt": "Given $\\var{left_bb}<\\frac{x}{\\var{c}}<\\var{right_bb}$, solving for $x$ gives [[0]]$<x<$ [[1]].
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "{bleft}", "minValue": "{bleft}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": false, "variableReplacements": [], "maxValue": "{bright}", "minValue": "{bright}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "Solving inequalities is similar to solving equations, ensure you do the same thing to all sides. Note that the operations we do are to get $x$ by itself.
\n| $\\var{left_bb}$ | \n$<$ | \n$\\displaystyle \\frac{x}{\\var{c}}$ | \n$<$ | \n$\\var{right_bb}$ | \n
| \n | \n | \n | \n | \n |
| $\\var{left_bb}\\times \\var{c}$ | \n$<$ | \n$\\displaystyle \\frac{x}{\\var{c}}\\times \\var{c}$ | \n$<$ | \n$\\var{right_bb}\\times \\var{c}$ | \n
| \n | \n | \n | \n | \n |
| $\\var{bleft}$ | \n$<$ | \n$x$ | \n$<$ | \n$\\var{bright}$ | \n
Given $\\var{left_bc}<\\frac{\\simplify{{a}x+{b}}}{\\var{c}}<\\var{right_bc}$, solving for $x$ gives [[0]]$<x<$ [[1]].
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "{left_bc*c-b}/{a}", "minValue": "{left_bc*c-b}/{a}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"allowFractions": true, "variableReplacements": [], "maxValue": "{right_bc*c-b}/{a}", "minValue": "{right_bc*c-b}/{a}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "Solving inequalities is similar to solving equations, ensure you do the same thing to all sides. Note that the operations we do are to get $x$ by itself.
\n| $\\var{left_bc}$ | \n$<$ | \n$\\displaystyle \\frac{\\simplify{{a}x+{b}}}{\\var{c}}$ | \n$<$ | \n$\\var{right_bc}$ | \n
| \n | \n | \n | \n | \n |
| $\\var{left_bc}\\times \\var{c}$ | \n$<$ | \n$\\displaystyle \\frac{\\simplify{{a}x+{b}}}{\\var{c}}\\times \\var{c}$ | \n$<$ | \n$\\var{right_bc}\\times \\var{c}$ | \n
| \n | \n | \n | \n | \n |
| $\\var{left_bc*c}$ | \n$<$ | \n$\\simplify{{a}x+{b}}$ | \n$<$ | \n$\\var{right_bc*c}$ | \n
| \n | \n | \n | \n | \n |
| $\\simplify[basic]{{left_bc*c}-{b}}$ | \n$<$ | \n$\\simplify[basic]{{a}x+{b}-{b}}$ | \n$<$ | \n$\\simplify[basic]{{right_bc*c}-{b}}$ | \n
| \n | \n | \n | \n | \n |
| $\\var{left_bc*c-b}$ | \n$<$ | \n$\\var{a}x$ | \n$<$ | \n$\\var{right_bc*c-b}$ | \n
| \n | \n | \n | \n | \n |
| $\\displaystyle\\frac{\\var{left_bc*c-b}}{\\var{a}}$ | \n$<$ | \n$\\displaystyle\\frac{\\var{a}x}{\\var{a}}$ | \n$<$ | \n$\\displaystyle\\frac{\\var{right_bc*c-b}}{\\var{a}}$ | \n
| \n | \n | \n | \n | \n |
| $\\displaystyle\\simplify{{left_bc*c-b}/{a}}$ | \n$<$ | \n$x$ | \n$<$ | \n$\\displaystyle\\simplify{{right_bc*c-b}/{a}}$ | \n