{"id":4673,"date":"2023-06-19T17:10:26","date_gmt":"2023-06-19T17:10:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&#038;p=4673"},"modified":"2024-10-18T20:50:58","modified_gmt":"2024-10-18T20:50:58","slug":"numbers-and-their-applications-background-youll-need-1","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/numbers-and-their-applications-background-youll-need-1\/","title":{"raw":"Numbers and Their Applications: Background You\u2019ll Need 1","rendered":"Numbers and Their Applications: Background You\u2019ll Need 1"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n\t<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Combine like terms&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12929,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Combine like terms<\/span><\/li>\r\n<\/ul>\r\n<\/section>\r\n<section class=\"textbox recall\">Some important terminology before we begin:\r\n\r\n\r\n<ul>\r\n\t<li><strong>operations\/operators:<\/strong>\u00a0In mathematics we call things like multiplication, division, addition, and subtraction operations. \u00a0They are the verbs of the math world, doing work on numbers and variables. The symbols used to denote operations are called operators, such as [latex]+{, }-{, }\\times{, }\\div[\/latex]. As you learn more math, you will learn more operators.<\/li>\r\n\t<li><strong>term:\u00a0<\/strong>Examples of terms would be [latex]2x[\/latex] and [latex]-\\Large\\frac{3}{2}[\/latex] or [latex]a^3[\/latex]. Even lone integers can be a term, like [latex]0[\/latex].<\/li>\r\n\t<li><strong>expression:\u00a0<\/strong>A\u00a0mathematical expression is one that connects terms with mathematical operators.\u00a0For example \u00a0[latex]\\Large\\frac{1}{2}\\normalsize +\\left(2^2\\right)- 9\\div\\Large\\frac{6}{7}[\/latex] is an expression.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Combining Like Terms<\/h2>\r\n<p>One way we can simplify expressions is to combine like terms. <strong>Like terms<\/strong> are terms where the variables match exactly (exponents included). <br \/>\r\n<br \/>\r\nExamples of like terms would be [latex]5xy[\/latex] and [latex]-3xy[\/latex], or [latex]8a^2b[\/latex] and [latex]a^2b[\/latex], or [latex]-3[\/latex] and [latex]8[\/latex]. \u00a0<br \/>\r\n<br \/>\r\nIf we have like terms we are\u00a0allowed to add (or subtract) the\u00a0numbers in front of the variables, then keep the variables the same. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.<\/p>\r\n<section class=\"textbox example\">Combine like terms: <center>\u00a0[latex]5x-2y-8x+7y[\/latex]<\/center>[reveal-answer q=\"647512\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"647512\"]The like terms in this expression are:\r\n\r\n\r\n<p style=\"text-align: center;\">[latex]5x[\/latex] and [latex]-8x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-2y[\/latex] and [latex]7y[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Note how we kept the sign in front of each term.<\/p>\r\n<p style=\"text-align: left;\">Combine like terms:<\/p>\r\n<p style=\"text-align: center;\">[latex]5x-8x = -3x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-2y+7y = 5y[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Note how signs become operations when you combine like terms.<\/p>\r\n<p style=\"text-align: left;\">Simplified Expression:<\/p>\r\n<p style=\"text-align: center;\">[latex]5x-2y-8x+7y=-3x+5y[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8354[\/ohm2_question]<\/section>\r\n<section class=\"textbox example\">Combine like terms: \u00a0<center>[latex]x^2-3x+9-5x^2+3x-1[\/latex]<\/center>[reveal-answer q=\"533174\"]Show Answer[\/reveal-answer]<br \/>\r\n[hidden-answer a=\"533174\"]The like terms in this expression are:\r\n\r\n\r\n<p style=\"text-align: center;\">[latex]x^2[\/latex] and [latex]-5x^2[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]-3x[\/latex] and [latex]3x[\/latex]<\/p>\r\n<p style=\"text-align: center;\">[latex]9[\/latex] and [latex]-1[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Combine like terms:<\/p>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x^2-5x^2 = -4x^2\\\\-3x+3x=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\9-1=8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n<p>Simplified Expression:<\/p>\r\n<p style=\"text-align: center;\">[latex]-4x^2+8[\/latex]<\/p>\r\n<p>[\/hidden-answer]<\/p>\r\n<\/section>\r\n<section class=\"textbox tryIt\">[ohm2_question hide_question_numbers=1]8363[\/ohm2_question]<\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li><span data-sheets-value=\"{&quot;1&quot;:2,&quot;2&quot;:&quot;Combine like terms&quot;}\" data-sheets-userformat=\"{&quot;2&quot;:12929,&quot;3&quot;:{&quot;1&quot;:0},&quot;10&quot;:0,&quot;12&quot;:0,&quot;15&quot;:&quot;Calibri&quot;,&quot;16&quot;:10}\">Combine like terms<\/span><\/li>\n<\/ul>\n<\/section>\n<section class=\"textbox recall\">Some important terminology before we begin:<\/p>\n<ul>\n<li><strong>operations\/operators:<\/strong>\u00a0In mathematics we call things like multiplication, division, addition, and subtraction operations. \u00a0They are the verbs of the math world, doing work on numbers and variables. The symbols used to denote operations are called operators, such as [latex]+{, }-{, }\\times{, }\\div[\/latex]. As you learn more math, you will learn more operators.<\/li>\n<li><strong>term:\u00a0<\/strong>Examples of terms would be [latex]2x[\/latex] and [latex]-\\Large\\frac{3}{2}[\/latex] or [latex]a^3[\/latex]. Even lone integers can be a term, like [latex]0[\/latex].<\/li>\n<li><strong>expression:\u00a0<\/strong>A\u00a0mathematical expression is one that connects terms with mathematical operators.\u00a0For example \u00a0[latex]\\Large\\frac{1}{2}\\normalsize +\\left(2^2\\right)- 9\\div\\Large\\frac{6}{7}[\/latex] is an expression.<\/li>\n<\/ul>\n<\/section>\n<h2>Combining Like Terms<\/h2>\n<p>One way we can simplify expressions is to combine like terms. <strong>Like terms<\/strong> are terms where the variables match exactly (exponents included). <\/p>\n<p>Examples of like terms would be [latex]5xy[\/latex] and [latex]-3xy[\/latex], or [latex]8a^2b[\/latex] and [latex]a^2b[\/latex], or [latex]-3[\/latex] and [latex]8[\/latex]. \u00a0<\/p>\n<p>If we have like terms we are\u00a0allowed to add (or subtract) the\u00a0numbers in front of the variables, then keep the variables the same. As we combine like terms we need to interpret subtraction signs as part of the following term. This means if we see a subtraction sign, we treat the following term like a negative term. The sign always stays with the term.<\/p>\n<section class=\"textbox example\">Combine like terms: <\/p>\n<div style=\"text-align: center;\">\u00a0[latex]5x-2y-8x+7y[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q647512\">Show Answer<\/button><\/p>\n<div id=\"q647512\" class=\"hidden-answer\" style=\"display: none\">The like terms in this expression are:<\/p>\n<p style=\"text-align: center;\">[latex]5x[\/latex] and [latex]-8x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-2y[\/latex] and [latex]7y[\/latex]<\/p>\n<p style=\"text-align: left;\">Note how we kept the sign in front of each term.<\/p>\n<p style=\"text-align: left;\">Combine like terms:<\/p>\n<p style=\"text-align: center;\">[latex]5x-8x = -3x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-2y+7y = 5y[\/latex]<\/p>\n<p style=\"text-align: left;\">Note how signs become operations when you combine like terms.<\/p>\n<p style=\"text-align: left;\">Simplified Expression:<\/p>\n<p style=\"text-align: center;\">[latex]5x-2y-8x+7y=-3x+5y[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8354\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8354&theme=lumen&iframe_resize_id=ohm8354&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n<section class=\"textbox example\">Combine like terms: \u00a0<\/p>\n<div style=\"text-align: center;\">[latex]x^2-3x+9-5x^2+3x-1[\/latex]<\/div>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q533174\">Show Answer<\/button><\/p>\n<div id=\"q533174\" class=\"hidden-answer\" style=\"display: none\">The like terms in this expression are:<\/p>\n<p style=\"text-align: center;\">[latex]x^2[\/latex] and [latex]-5x^2[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]-3x[\/latex] and [latex]3x[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]9[\/latex] and [latex]-1[\/latex]<\/p>\n<p style=\"text-align: left;\">Combine like terms:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x^2-5x^2 = -4x^2\\\\-3x+3x=0\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\9-1=8\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Simplified Expression:<\/p>\n<p style=\"text-align: center;\">[latex]-4x^2+8[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section class=\"textbox tryIt\"><iframe loading=\"lazy\" id=\"ohm8363\" class=\"resizable\" src=\"https:\/\/ohm.one.lumenlearning.com\/multiembedq.php?id=8363&theme=lumen&iframe_resize_id=ohm8363&source=tnh\" width=\"100%\" height=\"150\"><\/iframe><\/section>\n","protected":false},"author":16,"menu_order":2,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen 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