// Numbas version: exam_results_page_options
{"name": "1.3.2. Removing Brackets from a(b+c) and a(b-c)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "1.3.2. Removing Brackets from a(b+c) and a(b-c)", "tags": [], "metadata": {"description": "<p>Remove the brackets from algebraic expressions. Part of HELM Book 1.3</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<h3>Removing Brackets from expressions $a(b+c)$ and $a(b-c)$</h3>\n<p>Removing brackets means <strong>multiplying out</strong>. For example</p>\n<p style=\"text-align: center;\">\\[5(2+4) = 5\\times 2+5\\times 4 = 10+20 = 30.\\]</p>\n<p>In this simple example we could alternatively get the same result as follows:</p>\n<p style=\"text-align: center;\">\\[5(2 + 4) = 5 \\times 6 = 30.\\]</p>\n<p>That is:</p>\n<p style=\"text-align: center;\">\\[5(2 + 4) = 5 \\times 2 + 5 \\times 4\\]</p>\n<p>In an expression such as $5(x+y)$ it is intended that the $5$ multiplies both $x$ and $y$ to produce $5x+5y$.</p>\n<p>Thus the expressions $5(x + y)$ and $5x + 5y$ are equivalent. In general we have the following rules known as <strong>distributive laws</strong>:</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h4 style=\"text-align: center;\">Key Point 14</h4>\n<p style=\"text-align: center;\">\\(a(b+c)=ab+ac\\)</p>\n<p style=\"text-align: center;\">\\(a(b-c)=ab-ac\\)</p>\n<p>Note that when the brackets are removed both terms in the brackets are multiplied by $a$.</p>\n</div>\n<p></p>\n<p>As we have noted above, if you insert numbers instead of letters into these expressions you will see that both left and right hand sides are equivalent. For example</p>\n<p style=\"text-align: center;\">$4(3 + 5)$ has the same value as $4(3) + 4(5)$; that is $32$</p>\n<p style=\"text-align: left;\">and</p>\n<p style=\"text-align: center;\">$7(8 - 3)$ has the same value as $7(8) - 7(3)$; that is $35$</p>\n<p><strong>Example 31</strong></p>\n<p style=\"padding-left: 40px;\">Remove the brackets from (a) $9(2+y),\\quad$ (b) $9(2y)$</p>\n<p><strong>Solution</strong><strong></strong></p>\n<p style=\"padding-left: 40px;\">(a) In the expression $9(2 + y)$ the $9$ must multiply both terms in the brackets:</p>\n<p style=\"padding-left: 40px;\">\\[ \\begin{align*} 9(2+y)&amp;=9(2)+9(y) \\\\ &amp;=18+9y \\end{align*} \\]</p>\n<p style=\"padding-left: 40px;\">(b) Recall that $9(2y)$ means $9 \\times (2 \\times y)$ and that when multiplying numbers together the presence of brackets is irrelevant. Thus $9(2y) = 9 \\times 2 \\times y = 18y$</p>\n<p>The crucial distinction between the role of the factor $9$ in the two expressions $9(2 + y)$ and $9(2y)$ in Example 31 should be noted.</p>\n<p><strong>Example 32</strong></p>\n<p style=\"padding-left: 40px;\">Remove the brackets from $9(x+2y)$.</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">In the expression $9(x+2y)$ the $9$ must multiply both the $x$ and the $2y$ in the brackets. 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"suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><strong>Example 33</strong></p>\n<p style=\"text-align: left; padding-left: 40px;\">Remove the brackets from $-3(5x-z)$.</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">The number $-3$ must multiply both the $5x$ and the $z$.</p>\n<p style=\"padding-left: 40px;\">\\[ \\begin{align*} -3(-5x-z) &amp;= (-3)(5x) - (-3)(z)\\\\ &amp;= -15x+3z \\end{align*} \\]</p>"}, {"type": "jme", "useCustomName": true, "customName": "Task 2", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Remove the brackets from $\\var{q2expr}$.</p>\n<p>Remember to type * when two variables are multipled together.</p>", "answer": "{q2ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`! m_anywhere(?*(? + ?`+))", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}, {"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": "<p><strong>Example 34</strong></p>\n<p style=\"padding-left: 40px;\">Remove the brackets from $-(3x+1)$.</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">Although the $1$ is unwritten, the minus sign outside the brackets stands for $-1$. We must therefore consider the expression $-(3x+1)$.</p>\n<p style=\"padding-left: 40px;\">\\[ \\begin{align*} -1(3x-1) &amp;= (-1)(3x) + (-1)(1)\\\\ &amp;= -3x + (-1) \\\\ &amp;= -3x -1 \\end{align*} \\]</p>"}, {"type": "jme", "useCustomName": true, "customName": "Task 3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Remove the brackets from $\\var{q3expr}$.</p>", "answer": "{q3ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`! m_anywhere(?*(? + ?`+))", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "Task 4", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Remove the brackets from $\\var{q4expr}$.</p>", "answer": "{q4ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "`! m_anywhere(?*(? + ?`+))", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}, {"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": "<p><strong>Example 35</strong></p>\n<p style=\"padding-left: 40px;\">Remove the brackets from the expression $5x-(3x+1)$ and simplify the result by collecting like terms.</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">The brackets in $-(3x+1)$ were removed in Example 34.</p>\n<p style=\"padding-left: 40px;\">\\[ \\begin{align*} 5x-(3x+1) &amp;= 5x -1(3x+1)\\\\ &amp;= 5x-3x-1\\\\&amp;= 2x-1 \\end{align*} \\]</p>\n<p><strong>Example 36</strong></p>\n<p style=\"padding-left: 40px;\">Show that $\\displaystyle{\\frac{-x-1}{4}}$, $\\displaystyle{\\frac{-(x+1)}{4}}$ and $\\displaystyle {-\\frac{x+1}{4}}$ are all equivalent expressions.</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">Consider $-(x+1)$. Removing the brackets we obtain $-x-1$ and so \\[\\frac{-x-1}{4} = \\frac{-(x+1)}{4}\\]</p>\n<p style=\"padding-left: 40px;\">A negative quantity divided by a positive quantity will be negative. Hence \\[ \\frac{-(x+1)}{4} = -\\frac{x+1}{4} =\\]</p>\n<p style=\"padding-left: 40px;\">You should study all three expressions carefully to recognise the variety of equivalent ways in which we can write an algebraic expression.</p>\n<p>Sometimes the bracketed expression can appear on the left, as in (a + b)c. To remove the brackets here we use the following rules:</p>\n<div style=\"border-style: solid; border-color: black; border-radius: 25px; padding-left: 10px;\">\n<h4 style=\"text-align: center;\">Key Point 15</h4>\n<p>\\[(a+b)c = ac + bc\\]</p>\n<p>\\[(a-b)c = ac-bc\\]</p>\n</div>\n<p></p>\n<p>Note that when the brackets are removed both the terms in the brackets multiply $c$.</p>\n<p><strong>Example 37</strong></p>\n<p style=\"padding-left: 40px;\">Remove the brackets from $(2x+3y)x$.</p>\n<p><strong>Solution</strong></p>\n<p style=\"padding-left: 40px;\">Both terms in the brackets multiply the $x$ outside. 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