{"id":532,"date":"2024-04-22T20:45:28","date_gmt":"2024-04-22T20:45:28","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&#038;p=532"},"modified":"2025-07-11T22:05:40","modified_gmt":"2025-07-11T22:05:40","slug":"introduction-to-real-numbers-fresh-take","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/introduction-to-real-numbers-fresh-take\/","title":{"raw":"Introduction to Real Numbers: Fresh Take","rendered":"Introduction to Real Numbers: Fresh Take"},"content":{"raw":"<section class=\"textbox learningGoals\">\r\n<ul>\r\n \t<li>Categorize real numbers as counting numbers, whole numbers, rational numbers, irrational numbers, or integers<\/li>\r\n \t<li>Use the properties of real numbers and the order of operations to accurately calculate and manipulate algebraic expressions.<\/li>\r\n \t<li>Evaluate and simplify algebraic expressions to enhance problem-solving efficiency in mathematics.<\/li>\r\n<\/ul>\r\n<\/section>\r\n<h2>Defining and Identifying Real Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\n<strong>Real numbers<\/strong> are numbers that are either rational or irrational.\r\n\r\nReal numbers are the backbone of our numerical universe, encompassing both rational and irrational numbers. Think of real numbers as a big family gathering where everyone from counting numbers to irrational numbers shows up.\r\n\r\n<\/div>\r\n<section class=\"textbox recall\">\r\n<ul>\r\n \t<li><strong>Natural Numbers or Counting Numbers:<\/strong> Start with [latex]1[\/latex] and continue. [latex] 1,2,3,4,5\u2026[\/latex]<\/li>\r\n \t<li><strong>Whole Numbers:<\/strong> Counting numbers plus zero. [latex] 0,1,2,3,4,5\u2026[\/latex]<\/li>\r\n \t<li><strong>Integers:<\/strong> Whole numbers and their negative counterparts.<\/li>\r\n \t<li><strong>Rational Numbers:<\/strong> Numbers that can be written in the form [latex]{\\Large\\frac{a}{b}}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b\\ne o[\/latex]. In decimal form, the numbers terminate or repeat.<\/li>\r\n \t<li><strong>Irrational Numbers:<\/strong> Numbers that can't be expressed as a simple fraction. In decimal form, the numbers do not repeat or terminate.<\/li>\r\n<\/ul>\r\n<\/section>The following mini-lesson provides more examples of how to classify real numbers.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhchhddd-htP2goe31MM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/htP2goe31MM?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hhchhddd-htP2goe31MM\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843103&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hhchhddd-htP2goe31MM&vembed=0&video_id=htP2goe31MM&video_target=tpm-plugin-hhchhddd-htP2goe31MM'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Identifying+Sets+of+Real+Numbers_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIdentifying Sets of Real Numbers\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Recognizing Properties of Real Numbers<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nReal numbers have a set of rules that they play by, and these rules are called properties. Think of these properties as the \"grammar\" of math, setting the stage for how numbers interact with each other. Whether it's addition, multiplication, or even the use of parentheses, these properties ensure that numbers behave in a predictable way.\r\n<ul>\r\n \t<li><strong>Distributive Property:<\/strong> Imagine you're sharing a pizza equally among friends. If you have [latex]5[\/latex] friends and [latex]3[\/latex] pizzas, each friend gets a share from each pizza. Mathematically, [latex]a\u00d7(b+c)=a\u00d7b+a\u00d7c[\/latex].<\/li>\r\n \t<li><strong>Commutative Properties:<\/strong> In music, a playlist on shuffle plays songs in any order but the music is still enjoyable. Similarly, in math, whether it's addition or multiplication, the order doesn't matter. For addition, [latex]a+b=b+a[\/latex], and for multiplication, [latex]a\u00d7b=b\u00d7a[\/latex].<\/li>\r\n \t<li><strong>Associative Properties:<\/strong> Think of a team huddle in sports. It doesn't matter how players are grouped; the huddle remains the same. In math, whether you're adding or multiplying, the numbers stick together like a team. For addition, [latex]a+(b+c)=(a+b)+c[\/latex], and for multiplication, [latex]a\u00d7(b\u00d7c)=(a\u00d7b)\u00d7c[\/latex].<\/li>\r\n \t<li><strong>Identity Properties:<\/strong> Zero and one are like the superheroes of the number world. Zero, when added to any number, doesn't change its identity ([latex]a+0=a[\/latex]). One, when multiplied with any number, keeps it the same ([latex]a\u00d71=a[\/latex]).<\/li>\r\n \t<li><strong>Inverse Properties:<\/strong> These are your \"undo\" buttons in math. For addition, every number has a negative that will bring it back to zero ([latex]a+(\u2212a)=0[\/latex]). For multiplication, every non-zero number has a reciprocal that will bring it back to one ([latex]a\u00d7\\frac{1}{a}=1[\/latex], provided [latex]a \\ne 0[\/latex]).<\/li>\r\n<\/ul>\r\n<\/div>\r\n<section class=\"textbox example\">In each of the following, identify which property of the real numbers is being applied.\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>[latex]5\u00d7(6+19)=5\u00d76+5\u00d719[\/latex]<\/li>\r\n \t<li>[latex]41.7+(\u221241.7)=0[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"214538\"]Show Solution[\/reveal-answer] [hidden-answer a=\"214538\"]\r\n<ol style=\"list-style-type: decimal;\">\r\n \t<li>distributive property<\/li>\r\n \t<li>additive inverse property<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>Watch the following video for more information on the properties of real numbers.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fabhcdcc-8SFm8Os_4C8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/8SFm8Os_4C8?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-fabhcdcc-8SFm8Os_4C8\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843104&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-fabhcdcc-8SFm8Os_4C8&vembed=0&video_id=8SFm8Os_4C8&video_target=tpm-plugin-fabhcdcc-8SFm8Os_4C8'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Properties+of+Real+Numbers_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cProperties of Real Numbers\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h2>Algebraic Expressions<\/h2>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\n<strong>Algebraic expressions<\/strong> are mathematical statements that combine numbers or <strong>constants<\/strong>, <strong>variables <\/strong>(letters that represent unknown numbers), and operations such as addition, subtraction, multiplication, and division. They serve as the foundational language for algebra and allow us to represent and solve complex mathematical problems.\r\n\r\nAn algebraic expression can be as simple as a single variable \"[latex]x[\/latex]\", or as complex as a multi-term expression like \"[latex]3x^2 - 2x + 5[\/latex]\". The parts of the expression separated by addition or subtraction are called terms, each of which can be a combination of numbers (coefficients) and variables raised to a power.\r\n\r\nTo evaluate an algebraic expression, you substitute specific numerical values for the variables and perform the indicated operations. For instance, if we substitute [latex]x=2[\/latex] into the expression \"[latex]3x^2 - 2x + 5[\/latex]\", we would get [latex]3*(2)^2 - 2*2 + 5 = 12 - 4 + 5 = 13[\/latex].\r\n\r\n<\/div>\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdahefcd-OF2GtIinL_s\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/OF2GtIinL_s?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-cdahefcd-OF2GtIinL_s\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=10294974&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-cdahefcd-OF2GtIinL_s&vembed=0&video_id=OF2GtIinL_s&video_target=tpm-plugin-cdahefcd-OF2GtIinL_s'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Algebraic+Expressions+(Basics)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAlgebraic Expressions (Basics)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\">Evaluate the expression [latex]2x - 7[\/latex] for each value for [latex]x[\/latex]<em>.<\/em>\r\n<ol>\r\n \t<li>[latex]x=0[\/latex]<\/li>\r\n \t<li>[latex]x=1[\/latex]<\/li>\r\n \t<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\r\n \t<li>[latex]x=-4[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"421675\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"421675\"]\r\n<ol>\r\n \t<li>Substitute 0 for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(0\\right)-7 \\\\ &amp; =0-7 \\\\ &amp; =-7\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute 1 for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(1\\right)-7 \\\\ &amp; =2-7 \\\\ &amp; =-5\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]\\dfrac{1}{2}[\/latex] for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(\\frac{1}{2}\\right)-7 \\\\ &amp; =1-7 \\\\ &amp; =-6\\end{align}[\/latex]<\/div><\/li>\r\n \t<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\r\n<div>[latex]\\begin{align}2x-7 &amp; = 2\\left(-4\\right)-7 \\\\ &amp; =-8-7 \\\\ &amp; =-15\\end{align}[\/latex]<\/div><\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/section>In the following video we present more examples of how to evaluate an expression for a given value.\r\n\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hbfcaahe-MkRdwV4n91g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MkRdwV4n91g?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-hbfcaahe-MkRdwV4n91g\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=12843105&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-hbfcaahe-MkRdwV4n91g&vembed=0&video_id=MkRdwV4n91g&video_target=tpm-plugin-hbfcaahe-MkRdwV4n91g'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Evaluate+Various+Algebraic+Expressions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluate Various Algebraic Expressions\u201d here (opens in new window).<\/a>\r\n\r\n<\/section>\r\n<h3>Simplify Algebraic Expressions<\/h3>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\u00a0<\/strong>\r\n\r\nSimplifying algebraic expressions is the process of making an expression as concise as possible without changing its value. It involves several steps, often relying on mathematical properties and operations to combine like terms and eliminate unnecessary parts.\r\n\r\nOne fundamental process is the combination of like terms, which are terms in the expression that have the same variables and exponents. For example, in the expression [latex]3x + 2y - 5x + y[\/latex], you can combine the [latex]3x[\/latex] and [latex]-5x[\/latex] to get [latex]-2x[\/latex], and the [latex]2y[\/latex] and [latex]y[\/latex] to get [latex]3y[\/latex]. Thus, the expression simplifies to [latex]-2x + 3y[\/latex].\r\n\r\nThe distributive property is also frequently used in simplification. It allows us to remove parentheses in expressions like [latex]3(x + 2)[\/latex], which becomes [latex]3x + 6[\/latex] after distribution.\r\n\r\n<\/div>\r\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api \"><\/script>\r\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-abffdafa-uqKY7dK_DFQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uqKY7dK_DFQ?enablejsapi=1 \" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\r\n\r\n<div id=\"3p-plugin-target-abffdafa-uqKY7dK_DFQ\" class=\"p3sdk-target\"><\/div>\r\n<p class=\"cc-media-iframe-container\"><script type='text\/javascript' src='\/\/plugin.3playmedia.com\/ajax.js?cc=1&cc_minimizable=1&cc_minimize_on_load=0&cc_multi_text_track=0&cc_overlay=1&cc_searchable=0&embed=ajax&mf=10294975&p3sdk_version=1.11.7&p=20361&player_type=youtube&plugin_skin=dark&target=3p-plugin-target-abffdafa-uqKY7dK_DFQ&vembed=0&video_id=uqKY7dK_DFQ&video_target=tpm-plugin-abffdafa-uqKY7dK_DFQ'><\/script><\/p>\r\nYou can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Algebraic+Expressions+(Advanced)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAlgebraic Expressions (Advanced)\u201d here (opens in new window).<\/a>\r\n\r\n<\/section><section class=\"textbox example\">A rectangle with length [latex]L[\/latex] and width [latex]W[\/latex] has a perimeter [latex]P[\/latex] given by [latex]P=L+W+L+W[\/latex]. Simplify this expression.[reveal-answer q=\"921194\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"921194\"]\r\n<div style=\"text-align: center;\">[latex]\\begin{align}&amp;P=L+W+L+W \\\\ &amp;P=L+L+W+W &amp;&amp; \\text{Commutative property of addition} \\\\ &amp;P=2L+2W &amp;&amp; \\text{Simplify} \\\\ &amp;P=2\\left(L+W\\right) &amp;&amp; \\text{Distributive property}\\end{align}[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/section>\r\n<h3>Evaluating Formulas<\/h3>\r\n<div class=\"textbox shaded\">\r\n\r\n<strong>The Main Idea\r\n<\/strong>\r\nA <strong>formula<\/strong> is an equation expressing a relationship between constant and variable quantities.\r\n\r\n<\/div>\r\n<section class=\"textbox example\">\r\n\r\n[caption id=\"\" align=\"alignnone\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223819\/CNX_CAT_Figure_01_01_005.jpg\" alt=\"\/ An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.\" width=\"487\" height=\"407\" \/> Framed Artwork with Dimensions[\/caption]\r\n\r\nA photograph with length <em>L<\/em> and width <em>W<\/em> is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm<sup>2<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a mat for a photograph with length [latex]32[\/latex] cm and width [latex]24[\/latex] cm.<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[reveal-answer q=\"846181\"]Show Solution[\/reveal-answer]\r\n<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[hidden-answer a=\"846181\"]<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]1,152[\/latex] cm<\/span><sup style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">2<\/sup><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[\/hidden-answer]<\/span><\/section><section><iframe src=\"https:\/\/lumenlearning.h5p.com\/content\/1290624917813565478\/embed\" width=\"1088\" height=\"637\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>","rendered":"<section class=\"textbox learningGoals\">\n<ul>\n<li>Categorize real numbers as counting numbers, whole numbers, rational numbers, irrational numbers, or integers<\/li>\n<li>Use the properties of real numbers and the order of operations to accurately calculate and manipulate algebraic expressions.<\/li>\n<li>Evaluate and simplify algebraic expressions to enhance problem-solving efficiency in mathematics.<\/li>\n<\/ul>\n<\/section>\n<h2>Defining and Identifying Real Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Real numbers<\/strong> are numbers that are either rational or irrational.<\/p>\n<p>Real numbers are the backbone of our numerical universe, encompassing both rational and irrational numbers. Think of real numbers as a big family gathering where everyone from counting numbers to irrational numbers shows up.<\/p>\n<\/div>\n<section class=\"textbox recall\">\n<ul>\n<li><strong>Natural Numbers or Counting Numbers:<\/strong> Start with [latex]1[\/latex] and continue. [latex]1,2,3,4,5\u2026[\/latex]<\/li>\n<li><strong>Whole Numbers:<\/strong> Counting numbers plus zero. [latex]0,1,2,3,4,5\u2026[\/latex]<\/li>\n<li><strong>Integers:<\/strong> Whole numbers and their negative counterparts.<\/li>\n<li><strong>Rational Numbers:<\/strong> Numbers that can be written in the form [latex]{\\Large\\frac{a}{b}}[\/latex], where [latex]a[\/latex] and [latex]b[\/latex] are integers and [latex]b\\ne o[\/latex]. In decimal form, the numbers terminate or repeat.<\/li>\n<li><strong>Irrational Numbers:<\/strong> Numbers that can&#8217;t be expressed as a simple fraction. In decimal form, the numbers do not repeat or terminate.<\/li>\n<\/ul>\n<\/section>\n<p>The following mini-lesson provides more examples of how to classify real numbers.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hhchhddd-htP2goe31MM\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/htP2goe31MM?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hhchhddd-htP2goe31MM\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843103&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hhchhddd-htP2goe31MM&#38;vembed=0&#38;video_id=htP2goe31MM&#38;video_target=tpm-plugin-hhchhddd-htP2goe31MM\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Identifying+Sets+of+Real+Numbers_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cIdentifying Sets of Real Numbers\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Recognizing Properties of Real Numbers<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Real numbers have a set of rules that they play by, and these rules are called properties. Think of these properties as the &#8220;grammar&#8221; of math, setting the stage for how numbers interact with each other. Whether it&#8217;s addition, multiplication, or even the use of parentheses, these properties ensure that numbers behave in a predictable way.<\/p>\n<ul>\n<li><strong>Distributive Property:<\/strong> Imagine you&#8217;re sharing a pizza equally among friends. If you have [latex]5[\/latex] friends and [latex]3[\/latex] pizzas, each friend gets a share from each pizza. Mathematically, [latex]a\u00d7(b+c)=a\u00d7b+a\u00d7c[\/latex].<\/li>\n<li><strong>Commutative Properties:<\/strong> In music, a playlist on shuffle plays songs in any order but the music is still enjoyable. Similarly, in math, whether it&#8217;s addition or multiplication, the order doesn&#8217;t matter. For addition, [latex]a+b=b+a[\/latex], and for multiplication, [latex]a\u00d7b=b\u00d7a[\/latex].<\/li>\n<li><strong>Associative Properties:<\/strong> Think of a team huddle in sports. It doesn&#8217;t matter how players are grouped; the huddle remains the same. In math, whether you&#8217;re adding or multiplying, the numbers stick together like a team. For addition, [latex]a+(b+c)=(a+b)+c[\/latex], and for multiplication, [latex]a\u00d7(b\u00d7c)=(a\u00d7b)\u00d7c[\/latex].<\/li>\n<li><strong>Identity Properties:<\/strong> Zero and one are like the superheroes of the number world. Zero, when added to any number, doesn&#8217;t change its identity ([latex]a+0=a[\/latex]). One, when multiplied with any number, keeps it the same ([latex]a\u00d71=a[\/latex]).<\/li>\n<li><strong>Inverse Properties:<\/strong> These are your &#8220;undo&#8221; buttons in math. For addition, every number has a negative that will bring it back to zero ([latex]a+(\u2212a)=0[\/latex]). For multiplication, every non-zero number has a reciprocal that will bring it back to one ([latex]a\u00d7\\frac{1}{a}=1[\/latex], provided [latex]a \\ne 0[\/latex]).<\/li>\n<\/ul>\n<\/div>\n<section class=\"textbox example\">In each of the following, identify which property of the real numbers is being applied.<\/p>\n<ol style=\"list-style-type: decimal;\">\n<li>[latex]5\u00d7(6+19)=5\u00d76+5\u00d719[\/latex]<\/li>\n<li>[latex]41.7+(\u221241.7)=0[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q214538\">Show Solution<\/button> <\/p>\n<div id=\"q214538\" class=\"hidden-answer\" style=\"display: none\">\n<ol style=\"list-style-type: decimal;\">\n<li>distributive property<\/li>\n<li>additive inverse property<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>Watch the following video for more information on the properties of real numbers.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-fabhcdcc-8SFm8Os_4C8\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/8SFm8Os_4C8?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-fabhcdcc-8SFm8Os_4C8\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843104&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-fabhcdcc-8SFm8Os_4C8&#38;vembed=0&#38;video_id=8SFm8Os_4C8&#38;video_target=tpm-plugin-fabhcdcc-8SFm8Os_4C8\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Properties+of+Real+Numbers_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cProperties of Real Numbers\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h2>Algebraic Expressions<\/h2>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p><strong>Algebraic expressions<\/strong> are mathematical statements that combine numbers or <strong>constants<\/strong>, <strong>variables <\/strong>(letters that represent unknown numbers), and operations such as addition, subtraction, multiplication, and division. They serve as the foundational language for algebra and allow us to represent and solve complex mathematical problems.<\/p>\n<p>An algebraic expression can be as simple as a single variable &#8220;[latex]x[\/latex]&#8220;, or as complex as a multi-term expression like &#8220;[latex]3x^2 - 2x + 5[\/latex]&#8220;. The parts of the expression separated by addition or subtraction are called terms, each of which can be a combination of numbers (coefficients) and variables raised to a power.<\/p>\n<p>To evaluate an algebraic expression, you substitute specific numerical values for the variables and perform the indicated operations. For instance, if we substitute [latex]x=2[\/latex] into the expression &#8220;[latex]3x^2 - 2x + 5[\/latex]&#8220;, we would get [latex]3*(2)^2 - 2*2 + 5 = 12 - 4 + 5 = 13[\/latex].<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-cdahefcd-OF2GtIinL_s\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/OF2GtIinL_s?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-cdahefcd-OF2GtIinL_s\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=10294974&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-cdahefcd-OF2GtIinL_s&#38;vembed=0&#38;video_id=OF2GtIinL_s&#38;video_target=tpm-plugin-cdahefcd-OF2GtIinL_s\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Algebraic+Expressions+(Basics)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAlgebraic Expressions (Basics)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">Evaluate the expression [latex]2x - 7[\/latex] for each value for [latex]x[\/latex]<em>.<\/em><\/p>\n<ol>\n<li>[latex]x=0[\/latex]<\/li>\n<li>[latex]x=1[\/latex]<\/li>\n<li>[latex]x=\\dfrac{1}{2}[\/latex]<\/li>\n<li>[latex]x=-4[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q421675\">Show Solution<\/button><\/p>\n<div id=\"q421675\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>Substitute 0 for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(0\\right)-7 \\\\ & =0-7 \\\\ & =-7\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute 1 for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(1\\right)-7 \\\\ & =2-7 \\\\ & =-5\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]\\dfrac{1}{2}[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(\\frac{1}{2}\\right)-7 \\\\ & =1-7 \\\\ & =-6\\end{align}[\/latex]<\/div>\n<\/li>\n<li>Substitute [latex]-4[\/latex] for [latex]x[\/latex].\n<div>[latex]\\begin{align}2x-7 & = 2\\left(-4\\right)-7 \\\\ & =-8-7 \\\\ & =-15\\end{align}[\/latex]<\/div>\n<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<p>In the following video we present more examples of how to evaluate an expression for a given value.<\/p>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-hbfcaahe-MkRdwV4n91g\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/MkRdwV4n91g?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-hbfcaahe-MkRdwV4n91g\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=12843105&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-hbfcaahe-MkRdwV4n91g&#38;vembed=0&#38;video_id=MkRdwV4n91g&#38;video_target=tpm-plugin-hbfcaahe-MkRdwV4n91g\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Evaluate+Various+Algebraic+Expressions_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cEvaluate Various Algebraic Expressions\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<h3>Simplify Algebraic Expressions<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea\u00a0<\/strong><\/p>\n<p>Simplifying algebraic expressions is the process of making an expression as concise as possible without changing its value. It involves several steps, often relying on mathematical properties and operations to combine like terms and eliminate unnecessary parts.<\/p>\n<p>One fundamental process is the combination of like terms, which are terms in the expression that have the same variables and exponents. For example, in the expression [latex]3x + 2y - 5x + y[\/latex], you can combine the [latex]3x[\/latex] and [latex]-5x[\/latex] to get [latex]-2x[\/latex], and the [latex]2y[\/latex] and [latex]y[\/latex] to get [latex]3y[\/latex]. Thus, the expression simplifies to [latex]-2x + 3y[\/latex].<\/p>\n<p>The distributive property is also frequently used in simplification. It allows us to remove parentheses in expressions like [latex]3(x + 2)[\/latex], which becomes [latex]3x + 6[\/latex] after distribution.<\/p>\n<\/div>\n<section class=\"textbox watchIt\"><script type=\"text\/javascript\" src=\"https:\/\/www.youtube.com\/iframe_api\"><\/script><\/p>\n<p class=\"cc-media-iframe-container\"><iframe id=\"tpm-plugin-abffdafa-uqKY7dK_DFQ\" class=\"cc-media-iframe\" src=\"https:\/\/www.youtube.com\/embed\/uqKY7dK_DFQ?enablejsapi=1\" frameborder=\"0\" data-mce-fragment=\"1\"><\/iframe><\/p>\n<div id=\"3p-plugin-target-abffdafa-uqKY7dK_DFQ\" class=\"p3sdk-target\"><\/div>\n<p class=\"cc-media-iframe-container\"><script type=\"text\/javascript\" src=\"\/\/plugin.3playmedia.com\/ajax.js?cc=1&#38;cc_minimizable=1&#38;cc_minimize_on_load=0&#38;cc_multi_text_track=0&#38;cc_overlay=1&#38;cc_searchable=0&#38;embed=ajax&#38;mf=10294975&#38;p3sdk_version=1.11.7&#38;p=20361&#38;player_type=youtube&#38;plugin_skin=dark&#38;target=3p-plugin-target-abffdafa-uqKY7dK_DFQ&#38;vembed=0&#38;video_id=uqKY7dK_DFQ&#38;video_target=tpm-plugin-abffdafa-uqKY7dK_DFQ\"><\/script><\/p>\n<p>You can view the\u00a0<a href=\"https:\/\/course-building.s3.us-west-2.amazonaws.com\/College+Algebra+Corequisite\/Transcripts\/Algebraic+Expressions+(Advanced)_transcript.txt\" target=\"_blank\" rel=\"noopener\">transcript for \u201cAlgebraic Expressions (Advanced)\u201d here (opens in new window).<\/a><\/p>\n<\/section>\n<section class=\"textbox example\">A rectangle with length [latex]L[\/latex] and width [latex]W[\/latex] has a perimeter [latex]P[\/latex] given by [latex]P=L+W+L+W[\/latex]. Simplify this expression.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q921194\">Show Solution<\/button><\/p>\n<div id=\"q921194\" class=\"hidden-answer\" style=\"display: none\">\n<div style=\"text-align: center;\">[latex]\\begin{align}&P=L+W+L+W \\\\ &P=L+L+W+W && \\text{Commutative property of addition} \\\\ &P=2L+2W && \\text{Simplify} \\\\ &P=2\\left(L+W\\right) && \\text{Distributive property}\\end{align}[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/section>\n<h3>Evaluating Formulas<\/h3>\n<div class=\"textbox shaded\">\n<p><strong>The Main Idea<br \/>\n<\/strong><br \/>\nA <strong>formula<\/strong> is an equation expressing a relationship between constant and variable quantities.<\/p>\n<\/div>\n<section class=\"textbox example\">\n<figure style=\"width: 487px\" class=\"wp-caption alignnone\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/21223819\/CNX_CAT_Figure_01_01_005.jpg\" alt=\"\/ An art frame with a piece of artwork in the center. The frame has a width of 8 centimeters. The artwork itself has a length of 32 centimeters and a width of 24 centimeters.\" width=\"487\" height=\"407\" \/><figcaption class=\"wp-caption-text\">Framed Artwork with Dimensions<\/figcaption><\/figure>\n<p>A photograph with length <em>L<\/em> and width <em>W<\/em> is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm<sup>2<\/sup>) is found to be [latex]A=\\left(L+16\\right)\\left(W+16\\right)-L\\cdot W[\/latex]. Find the area of a mat for a photograph with length [latex]32[\/latex] cm and width [latex]24[\/latex] cm.<span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><button class=\"show-answer show-answer-button collapsed\" data-target=\"q846181\">Show Solution<\/button><br \/>\n<\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><\/p>\n<div id=\"q846181\" class=\"hidden-answer\" style=\"display: none\"><\/span><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">[latex]1,152[\/latex] cm<\/span><sup style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\">2<\/sup><span style=\"font-family: 'Public Sans', -apple-system, BlinkMacSystemFont, 'Segoe UI', Roboto, Oxygen-Sans, Ubuntu, Cantarell, 'Helvetica Neue', sans-serif;\"><\/div>\n<\/div>\n<p><\/span><\/section>\n<section><iframe loading=\"lazy\" src=\"https:\/\/lumenlearning.h5p.com\/content\/1290624917813565478\/embed\" width=\"1088\" height=\"637\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/section>\n","protected":false},"author":12,"menu_order":12,"template":"","meta":{"_candela_citation":"[{\"type\":\"copyrighted_video\",\"description\":\"Identifying Sets of Real Numbers \",\"author\":\"\",\"organization\":\"Mathispower4u\",\"url\":\"https:\/\/www.youtube.com\/watch?v=htP2goe31MM\",\"project\":\"\",\"license\":\"arr\",\"license_terms\":\"Standard YouTube License\"},{\"type\":\"copyrighted_video\",\"description\":\"Properties of Real Numbers 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