{"id":1963,"date":"2021-09-16T18:17:00","date_gmt":"2021-09-16T18:17:00","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/?post_type=chapter&#038;p=1963"},"modified":"2022-02-18T05:20:01","modified_gmt":"2022-02-18T05:20:01","slug":"intervals-on-the-real-line","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/introstatscorequisite\/chapter\/intervals-on-the-real-line\/","title":{"raw":"Intervals on the Real Line","rendered":"Intervals on the Real Line"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Represent inequalities using interval notation<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Find the width of a bounded interval<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">Find the midpoint of a bounded interval<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. If there is an infinite collection of solutions to an inequality, we can\u2019t list all of them.\r\n\r\nThere are three ways to write solutions to inequalities:\r\n<ul>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">using inequality notation<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">as a graph<\/li>\r\n \t<li style=\"font-weight: 400;\" aria-level=\"1\">as an interval<\/li>\r\n<\/ul>\r\nWe\u2019ll begin by reviewing inequality symbols and graphing inequalities on the number line, and then describe how to translate to interval notation.\r\n<h3>Inequality Signs<\/h3>\r\nThe box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it is easy to get tangled up in inequalities; just remember to read them from left to right.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Symbol<\/th>\r\n<th>Words<\/th>\r\n<th>Example<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\neq [\/latex]<\/td>\r\n<td>not equal to<\/td>\r\n<td>[latex]{2}\\neq{8}[\/latex], <i>2<\/i>\u00a0<b>is not equal to<\/b> <i>8<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\gt[\/latex]<\/td>\r\n<td>greater than<\/td>\r\n<td>[latex]{5}\\gt{1}[\/latex], <i>5<\/i>\u00a0<b>is greater than<\/b>\u00a0<i>1<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\lt[\/latex]<\/td>\r\n<td>less than<\/td>\r\n<td>[latex]{2}\\lt{11}[\/latex], <i>2<\/i> <b>is less than<\/b>\u00a0<i>11<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex] \\geq [\/latex]<\/td>\r\n<td>greater than or equal to<\/td>\r\n<td>[latex]{4}\\geq{ 4}[\/latex], <i>4<\/i> <b>is greater than or equal to<\/b>\u00a0<i>4<\/i><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]\\leq [\/latex]<\/td>\r\n<td>less than or equal to<\/td>\r\n<td>[latex]{7}\\leq{9}[\/latex], <i>7<\/i>\u00a0<b>is less than or equal to<\/b>\u00a0<i>9<\/i><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe inequality [latex]x&gt;y[\/latex]\u00a0can also be written as [latex]{y}&lt;{x}[\/latex]. The sides of any inequality can be switched, as long as the inequality symbol between them is also reversed.\r\n<h2>Graphing an Inequality<\/h2>\r\nInequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs. Graphs are a very helpful way to visualize information, especially when that information represents an infinite list of numbers!\r\n\r\n[latex]x\\leq -4[\/latex]. This translates to all the real numbers on a number line that are less than or equal to -4.\u00a0This includes -4, so we draw a closed dot at -4 on the number line. This is the endpoint of our solution set. All numbers less than -4 fall to the left of -4 on the number line, so we shade the portion on the number line to the left of -4. The graph of this inequality is shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image034.jpg#fixme#fixme\" alt=\"Number line. Shaded circle on negative 4. Shaded line through all numbers less than negative 4.\" width=\"575\" height=\"31\" \/>\r\n\r\n[latex]{x}\\geq{-3}[\/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3.\u00a0This includes -3 as well as all the numbers greater than -3. So we graph a closed dot at -3 and shade all the numbers to the right of -3.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image035.jpg#fixme#fixme\" alt=\"Number line. Shaded circle on negative 3. Shaded line through all numbers greater than negative 3.\" width=\"575\" height=\"31\" \/>\r\n\r\nEach of these graphs begins with a circle\u2014either an open or closed (shaded) circle. This point is often called the <i>end point<\/i> of the solution. A closed, or shaded, circle is used to represent the inequalities <i>greater than or equal to<\/i>\u00a0[latex] \\displaystyle \\left(\\geq\\right) [\/latex] or <i>less than or equal to<\/i>\u00a0[latex] \\displaystyle \\left(\\leq\\right) [\/latex]. The point is part of the solution. An open circle is used for <i>greater than<\/i> (&gt;) or <i>less than<\/i> (&lt;). The point is <i>not <\/i>part of the solution.\r\n\r\nThe graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex] \\displaystyle x\\geq -3[\/latex] shown above, the end point is [latex]\u22123[\/latex], represented with a closed circle since the inequality is <i>greater than or equal to<\/i> [latex]\u22123[\/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]\u22123[\/latex]. The arrow at the end indicates that the solutions continue infinitely.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGraph the\u00a0inequality [latex]x\\ge 4[\/latex]\r\n[reveal-answer q=\"797241\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797241\"]\r\n\r\nWe can use a number line as shown. Because the values for\u00a0[latex]x[\/latex] include\u00a0[latex]4[\/latex], we place a solid dot on the number line at\u00a0[latex]4[\/latex].\r\n\r\nThen we draw a line that\u00a0begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to\u00a0[latex]4[\/latex].\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182809\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThis video shows an example of how to draw the graph of an inequality.\r\n\r\nhttps:\/\/youtu.be\/-kiAeGbSe5c\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]26182[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Represent Inequalities Using Interval Notation<\/h2>\r\nAnother commonly used, and arguably the most concise, method for describing inequalities and solutions to inequalities is called<strong>\u00a0interval notation.\u00a0<\/strong>With this convention, sets are built\u00a0with parentheses or brackets, each having a distinct meaning. The solutions to [latex]x\\geq 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This method is widely used and will be present in other math courses you may take.\r\n\r\nThe main concept to remember is that parentheses represent solutions greater than or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be \"equaled.\" A few examples of an <strong>interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. The table below\u00a0outlines the possibilities. Remember to read inequalities from left to right, just like text.\r\n\r\nThe table below describes all the possible inequalities that can occur and how to write them using interval notation, where <em>a<\/em> and <em>b<\/em> are real numbers.\r\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th>Inequality<\/th>\r\n<th>Words<\/th>\r\n<th>Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\r\n<td>all real numbers between\u00a0<em>a<\/em> and <em>b<\/em>, not including a and b<\/td>\r\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\gt{a}[\/latex]<\/td>\r\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\r\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\lt{b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\ge{a}[\/latex]<\/td>\r\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\le{b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\r\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers<\/td>\r\n<td>All real numbers<\/td>\r\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDescribe the inequality [latex]x\\ge 4[\/latex] using interval notation\r\n[reveal-answer q=\"817362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"817362\"]\r\n\r\nThe solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex].\r\n\r\nNote the use of a bracket on the left because 4 is included in the solution set.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]72501[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].\r\n[reveal-answer q=\"961990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"961990\"]\r\n\r\nUse a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph.\r\n\r\nhttps:\/\/youtu.be\/X0xrHKgbDT0\r\n<h2>Bounded Intervals: Width and Midpoint<\/h2>\r\nAn interval of the form [latex](a,b), (a,b], [a,b) \\ \\mathrm{or} \\ [a,b],[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are real numbers, is called a\u00a0<strong>bounded<\/strong> interval. Such an interval does not extendd infinitely far in the positive or negative direction. The endpoints, [latex]a[\/latex] and [latex]b[\/latex], bound the interval. Another term for intervals of this form is a finite interval, because its width is finite.\r\n\r\nThe\u00a0<strong>width<\/strong> of an interval of the form [latex](a,b), (a,b], [a,b) \\ \\mathrm{or} \\ [a,b],[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are real numbers, is the distance between its endpoints,\r\n<p style=\"text-align: center;\">[latex]\\mathrm{width} \\ =b-a[\/latex].<\/p>\r\nThe\u00a0<strong>midpoint<\/strong> of an interval of the form [latex](a,b), (a,b], [a,b) \\ \\mathrm{or} \\ [a,b],[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are real numbers, is the point which is the same distance from each of its endpoints,\r\n<p style=\"text-align: center;\">[latex]\\mathrm{midpoint} \\ = \\frac{a+b}{2}[\/latex].<\/p>\r\n\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nFind the width and the midpoint of the interval [latex](3,5)[\/latex].\r\n\r\n[reveal-answer q=\"247552\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"247552\"]\r\n\r\nThe endpoints of the interval are [latex]a=3[\/latex] and [latex]b=5[\/latex].\r\n\r\nThe width of the interval is [latex]b-a=5-3=2[\/latex].\r\n\r\nThe midpoint of the interval is [latex]\\frac{a+b}{2} = \\frac{3+5}{2} = \\frac{8}{2}=4[\/latex].\r\n\r\nNotice that the distance of the midpoint from each endpoint is the same, in the case 3 is 1 unit to the left of 4 and 5 is 1 unit to the right of 4.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Represent inequalities using interval notation<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Find the width of a bounded interval<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">Find the midpoint of a bounded interval<\/li>\n<\/ul>\n<\/div>\n<p>An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. If there is an infinite collection of solutions to an inequality, we can\u2019t list all of them.<\/p>\n<p>There are three ways to write solutions to inequalities:<\/p>\n<ul>\n<li style=\"font-weight: 400;\" aria-level=\"1\">using inequality notation<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">as a graph<\/li>\n<li style=\"font-weight: 400;\" aria-level=\"1\">as an interval<\/li>\n<\/ul>\n<p>We\u2019ll begin by reviewing inequality symbols and graphing inequalities on the number line, and then describe how to translate to interval notation.<\/p>\n<h3>Inequality Signs<\/h3>\n<p>The box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it is easy to get tangled up in inequalities; just remember to read them from left to right.<\/p>\n<table>\n<thead>\n<tr>\n<th>Symbol<\/th>\n<th>Words<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\neq[\/latex]<\/td>\n<td>not equal to<\/td>\n<td>[latex]{2}\\neq{8}[\/latex], <i>2<\/i>\u00a0<b>is not equal to<\/b> <i>8<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\gt[\/latex]<\/td>\n<td>greater than<\/td>\n<td>[latex]{5}\\gt{1}[\/latex], <i>5<\/i>\u00a0<b>is greater than<\/b>\u00a0<i>1<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\lt[\/latex]<\/td>\n<td>less than<\/td>\n<td>[latex]{2}\\lt{11}[\/latex], <i>2<\/i> <b>is less than<\/b>\u00a0<i>11<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\geq[\/latex]<\/td>\n<td>greater than or equal to<\/td>\n<td>[latex]{4}\\geq{ 4}[\/latex], <i>4<\/i> <b>is greater than or equal to<\/b>\u00a0<i>4<\/i><\/td>\n<\/tr>\n<tr>\n<td>[latex]\\leq[\/latex]<\/td>\n<td>less than or equal to<\/td>\n<td>[latex]{7}\\leq{9}[\/latex], <i>7<\/i>\u00a0<b>is less than or equal to<\/b>\u00a0<i>9<\/i><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The inequality [latex]x>y[\/latex]\u00a0can also be written as [latex]{y}<{x}[\/latex]. The sides of any inequality can be switched, as long as the inequality symbol between them is also reversed.\n\n\n<h2>Graphing an Inequality<\/h2>\n<p>Inequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs. Graphs are a very helpful way to visualize information, especially when that information represents an infinite list of numbers!<\/p>\n<p>[latex]x\\leq -4[\/latex]. This translates to all the real numbers on a number line that are less than or equal to -4.\u00a0This includes -4, so we draw a closed dot at -4 on the number line. This is the endpoint of our solution set. All numbers less than -4 fall to the left of -4 on the number line, so we shade the portion on the number line to the left of -4. The graph of this inequality is shown below.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image034.jpg#fixme#fixme\" alt=\"Number line. Shaded circle on negative 4. Shaded line through all numbers less than negative 4.\" width=\"575\" height=\"31\" \/><\/p>\n<p>[latex]{x}\\geq{-3}[\/latex]. This translates to all the real numbers on the number line that are greater than or equal to -3.\u00a0This includes -3 as well as all the numbers greater than -3. So we graph a closed dot at -3 and shade all the numbers to the right of -3.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image035.jpg#fixme#fixme\" alt=\"Number line. Shaded circle on negative 3. Shaded line through all numbers greater than negative 3.\" width=\"575\" height=\"31\" \/><\/p>\n<p>Each of these graphs begins with a circle\u2014either an open or closed (shaded) circle. This point is often called the <i>end point<\/i> of the solution. A closed, or shaded, circle is used to represent the inequalities <i>greater than or equal to<\/i>\u00a0[latex]\\displaystyle \\left(\\geq\\right)[\/latex] or <i>less than or equal to<\/i>\u00a0[latex]\\displaystyle \\left(\\leq\\right)[\/latex]. The point is part of the solution. An open circle is used for <i>greater than<\/i> (&gt;) or <i>less than<\/i> (&lt;). The point is <i>not <\/i>part of the solution.<\/p>\n<p>The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex]\\displaystyle x\\geq -3[\/latex] shown above, the end point is [latex]\u22123[\/latex], represented with a closed circle since the inequality is <i>greater than or equal to<\/i> [latex]\u22123[\/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]\u22123[\/latex]. The arrow at the end indicates that the solutions continue infinitely.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Graph the\u00a0inequality [latex]x\\ge 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797241\">Show Solution<\/span><\/p>\n<div id=\"q797241\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can use a number line as shown. Because the values for\u00a0[latex]x[\/latex] include\u00a0[latex]4[\/latex], we place a solid dot on the number line at\u00a0[latex]4[\/latex].<\/p>\n<p>Then we draw a line that\u00a0begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to positive infinity, which illustrates that the solution set includes all real numbers greater than or equal to\u00a0[latex]4[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182809\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>This video shows an example of how to draw the graph of an inequality.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Graph Linear Inequalities in One Variable (Basic)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/-kiAeGbSe5c?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm26182\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=26182&theme=oea&iframe_resize_id=ohm26182&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Represent Inequalities Using Interval Notation<\/h2>\n<p>Another commonly used, and arguably the most concise, method for describing inequalities and solutions to inequalities is called<strong>\u00a0interval notation.\u00a0<\/strong>With this convention, sets are built\u00a0with parentheses or brackets, each having a distinct meaning. The solutions to [latex]x\\geq 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This method is widely used and will be present in other math courses you may take.<\/p>\n<p>The main concept to remember is that parentheses represent solutions greater than or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be &#8220;equaled.&#8221; A few examples of an <strong>interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. The table below\u00a0outlines the possibilities. Remember to read inequalities from left to right, just like text.<\/p>\n<p>The table below describes all the possible inequalities that can occur and how to write them using interval notation, where <em>a<\/em> and <em>b<\/em> are real numbers.<\/p>\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\n<thead>\n<tr>\n<th>Inequality<\/th>\n<th>Words<\/th>\n<th>Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\n<td>all real numbers between\u00a0<em>a<\/em> and <em>b<\/em>, not including a and b<\/td>\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\gt{a}[\/latex]<\/td>\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\lt{b}[\/latex]<\/td>\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\ge{a}[\/latex]<\/td>\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\le{b}[\/latex]<\/td>\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers<\/td>\n<td>All real numbers<\/td>\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Describe the inequality [latex]x\\ge 4[\/latex] using interval notation<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q817362\">Show Solution<\/span><\/p>\n<div id=\"q817362\" class=\"hidden-answer\" style=\"display: none\">\n<p>The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex].<\/p>\n<p>Note the use of a bracket on the left because 4 is included in the solution set.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm72501\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=72501&theme=oea&iframe_resize_id=ohm72501&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q961990\">Show Solution<\/span><\/p>\n<div id=\"q961990\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Ex: Graph Basic Inequalities and Express Using Interval Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/X0xrHKgbDT0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Bounded Intervals: Width and Midpoint<\/h2>\n<p>An interval of the form [latex](a,b), (a,b], [a,b) \\ \\mathrm{or} \\ [a,b],[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are real numbers, is called a\u00a0<strong>bounded<\/strong> interval. Such an interval does not extendd infinitely far in the positive or negative direction. The endpoints, [latex]a[\/latex] and [latex]b[\/latex], bound the interval. Another term for intervals of this form is a finite interval, because its width is finite.<\/p>\n<p>The\u00a0<strong>width<\/strong> of an interval of the form [latex](a,b), (a,b], [a,b) \\ \\mathrm{or} \\ [a,b],[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are real numbers, is the distance between its endpoints,<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{width} \\ =b-a[\/latex].<\/p>\n<p>The\u00a0<strong>midpoint<\/strong> of an interval of the form [latex](a,b), (a,b], [a,b) \\ \\mathrm{or} \\ [a,b],[\/latex] where [latex]a[\/latex] and [latex]b[\/latex] are real numbers, is the point which is the same distance from each of its endpoints,<\/p>\n<p style=\"text-align: center;\">[latex]\\mathrm{midpoint} \\ = \\frac{a+b}{2}[\/latex].<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Find the width and the midpoint of the interval [latex](3,5)[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q247552\">Show Answer<\/span><\/p>\n<div id=\"q247552\" class=\"hidden-answer\" style=\"display: none\">\n<p>The endpoints of the interval are [latex]a=3[\/latex] and [latex]b=5[\/latex].<\/p>\n<p>The width of the interval is [latex]b-a=5-3=2[\/latex].<\/p>\n<p>The midpoint of the interval is [latex]\\frac{a+b}{2} = \\frac{3+5}{2} = \\frac{8}{2}=4[\/latex].<\/p>\n<p>Notice that the distance of the midpoint from each endpoint is the same, in the case 3 is 1 unit to the left of 4 and 5 is 1 unit to the right of 4.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1963\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Graph Linear Inequalities in One Variable (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-kiAeGbSe5c\">https:\/\/youtu.be\/-kiAeGbSe5c<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>QID 26182, 72501. <strong>Authored by<\/strong>: Jones,k; Lumen Learning. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Ex: Graph Basic Inequalities and Express Using Interval Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/X0xrHKgbDT0\">https:\/\/youtu.be\/X0xrHKgbDT0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Jay Abramson, et al. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-describe-solutions-to-inequalities-2\/\">https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-describe-solutions-to-inequalities-2\/<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program. <strong>Provided by<\/strong>: Monterey Institute of Technology and Education. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/nrocnetwork.org\/dm-opentext\">http:\/\/nrocnetwork.org\/dm-opentext<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":169134,"menu_order":3,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc\",\"description\":\"Ex: Graph Basic Inequalities and Express Using Interval Notation\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/X0xrHKgbDT0\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Jay Abramson, et al\",\"organization\":\"Lumen Learning\",\"url\":\"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-describe-solutions-to-inequalities-2\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\"http:\/\/nrocnetwork.org\/dm-opentext\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Graph Linear Inequalities in One Variable (Basic)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen Learning\",\"organization\":\"\",\"url\":\"https:\/\/youtu.be\/-kiAeGbSe5c\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"QID 26182, 72501\",\"author\":\"Jones,k; 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