{"id":16148,"date":"2019-10-01T18:18:41","date_gmt":"2019-10-01T18:18:41","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/wm-developmentalemporium\/chapter\/read-describe-sets-as-intersections-or-unions-2\/"},"modified":"2020-10-22T09:13:25","modified_gmt":"2020-10-22T09:13:25","slug":"read-describe-sets-as-intersections-or-unions-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-rockland-developmentalemporium\/chapter\/read-describe-sets-as-intersections-or-unions-2\/","title":{"raw":"8.2.a - Describing Sets as Intersections or Unions","rendered":"8.2.a &#8211; Describing Sets as Intersections or Unions"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Use interval notation to describe intersections and unions<\/li>\r\n \t<li>Use graphs to describe intersections and\u00a0unions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Use Interval Notation to Describe Sets of Numbers as Intersections and Unions<\/h2>\r\nWhen two inequalities are joined by the word <i>and<\/i>, the solution of the compound inequality occurs when <i>both<\/i> inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word <i>or<\/i>, the solution of the compound inequality occurs when <i>either<\/i> of the inequalities is true. The solution is the combination, or union, of the two individual solutions.\r\n\r\nIn this section, we will learn how to solve compound inequalities that are joined with the words AND and OR. First, it will help to see some examples of inequalities, intervals, and graphs of compound inequalities. This will help you describe the solutions to compound inequalities properly.\r\n\r\nVenn diagrams use the concept of intersections and unions to compare two or more things. For example, this Venn diagram shows the intersection of people who are breaking your heart and those who are shaking your confidence daily. Apparently, Cecilia has both of these qualities; therefore, she is the intersection of the two.\r\n\r\n<img class=\"aligncenter wp-image-3710\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182824\/Screen-Shot-2016-05-06-at-3.25.21-PM-300x234.png\" alt=\"Two circles. One is people who are breaking my heart. The other is people who are shaking my confidence daily. The area where the circles overlap is labeled Cecilia.\" width=\"353\" height=\"275\" \/>\r\n\r\nIn mathematical terms, consider the inequality\u00a0[latex]x\\lt6[\/latex] and\u00a0[latex]x\\gt2[\/latex]. How would we interpret what numbers [latex]x[\/latex] can be, and what would the interval look like?\r\n\r\nIn words, [latex]x[\/latex] must be less than\u00a0[latex]6[\/latex], and at the same time, it must be greater than\u00a0[latex]2[\/latex]. This is much like the Venn diagram above, where Cecilia is at once breaking your heart and shaking your confidence daily. Now look at a graph to see what numbers are possible with these constraints.\r\n\r\n<img class=\"wp-image-3958 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182826\/Screen-Shot-2016-05-10-at-4.43.10-PM-300x46.png\" alt=\"x&gt; 2 and x&lt; 6\" width=\"594\" height=\"91\" \/>\r\n\r\nThe numbers that are shared by both lines on the graph are called the intersection of the two inequalities\u00a0[latex]x\\lt6[\/latex]\u00a0and\u00a0[latex]x\\gt2[\/latex]. This is called a <em>bounded<\/em> inequality, and is written as [latex]2\\lt{x}\\lt6[\/latex]. Think about that one for a minute. [latex]x[\/latex] must be less than\u00a0[latex]6[\/latex] and greater than two\u2014the values for [latex]x[\/latex] will fall <em>between two numbers.<\/em>\u00a0In interval notation, this looks like [latex]\\left(2,6\\right)[\/latex]. The graph would look like this:\r\n\r\n<img class=\"aligncenter wp-image-4014\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182827\/Screen-Shot-2016-05-11-at-4.53.25-PM-300x46.png\" alt=\"Open circle on 2 and open circle on 6 with a line through all numbers between 2 and 6.\" width=\"664\" height=\"102\" \/><span style=\"line-height: 1.5\">On the other hand, if you need to represent two things that\u00a0don't share any common elements or traits, you can use\u00a0a union. The following Venn diagram shows two things that share no similar traits or elements, but are often considered in the same application, such as online shopping or banking.<\/span>\r\n\r\n<img class=\"aligncenter wp-image-3712\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182829\/Screen-Shot-2016-05-06-at-3.26.52-PM-300x150.png\" alt=\"Two circles, one the Internet and the other Privacy.\" width=\"406\" height=\"203\" \/>\r\n\r\nIn mathematical terms, for example, [latex]x&gt;6[\/latex]\u00a0<em>or<\/em>\u00a0[latex]x&lt;2[\/latex] is an inequality joined by the word <em>or<\/em>. Using interval notation, we can describe each of these inequalities separately:\r\n\r\n[latex]x\\gt6[\/latex] is the same as [latex]\\left(6, \\infty\\right)[\/latex] and\u00a0[latex]x&lt;2[\/latex] is the same as\u00a0[latex]\\left(-\\infty, 2\\right)[\/latex]. If we are describing solutions to inequalities, what effect does the\u00a0<em>or\u00a0<\/em>have? \u00a0We are\u00a0saying that solutions are either real numbers less than two\u00a0<em>or<\/em> real numbers greater than\u00a0[latex]6[\/latex]. Can you see why we need to write them as two separate intervals? Let us look at a graph to get a clear picture of what is going on.\r\n\r\n<img class=\"aligncenter wp-image-3960\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182831\/Screen-Shot-2016-05-10-at-4.53.44-PM-300x39.png\" alt=\"Open circle on 2 and line through all numbers less than 2. Open circle on 6 and line through all numbers grater than 6.\" width=\"585\" height=\"76\" \/>\r\n\r\nWhen you place both of these inequalities on a graph, we can see that they share no numbers in common. As mentioned above, this is what we call a union. The interval notation associated with a union is a big U, so instead of writing <em>or<\/em>, we join our intervals with a big U, like this:\r\n<p style=\"text-align: center\">[latex]\\left(-\\infty, 2\\right)\\cup\\left(6, \\infty\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left\">It is common convention to construct intervals starting with the value that is furthest left on the number line as the left value, such as\u00a0[latex]\\left(2,6\\right)[\/latex], where\u00a0[latex]2[\/latex] is less than [latex]6[\/latex]. The number on the right should be greater than the number on the left.<\/p>\r\n\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]2796[\/ohm_question]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDraw the graph of the compound inequality [latex]x\\gt3[\/latex] <em>or<\/em>\u00a0[latex]x\\le4[\/latex] and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n[reveal-answer q=\"641470\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"641470\"]\r\n\r\nThe graph of [latex]x\\gt3[\/latex]\u00a0has an open circle on\u00a0[latex]3[\/latex] and a blue arrow drawn to the right to contain all the numbers greater than\u00a0[latex]3[\/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\/01182832\/image075.jpg\" alt=\"Number line. Open blue circle on 3. Blue highlight on all numbers greater than 3.\" width=\"575\" height=\"53\" \/>\r\n\r\nThe graph of\u00a0[latex]x\\le4[\/latex]\u00a0has a closed circle at 4 and a red arrow to the left to contain all the numbers less than\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\/01182834\/image076.jpg\" alt=\"Number line. Closed red circle on 4. Red highlight on all numbers less than 4.\" width=\"575\" height=\"53\" \/>\r\n\r\nWhat do you notice about the graph that combines these two inequalities?\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182836\/image077.jpg\" alt=\"Number line. Open blue circle on 3 and blue highlight on all numbers greater than 3. Red closed circle on 4 and red highlight through all numbers less than 4. This means that both colored highlights cover the numbers between 3 and 4.\" width=\"575\" height=\"53\" \/>\r\n\r\nSince this compound inequality is an <i>or<\/i> statement, it includes all of the numbers in each of the solutions. In this case, the solution is all the numbers on the number line. (The region of the line greater than\u00a0[latex]3[\/latex] and less than or equal to\u00a0[latex]4[\/latex] is shown in purple because it lies on both of the original graphs.) The solution to the compound inequality [latex]x\\gt3[\/latex] <em>or<\/em> [latex]x\\le4[\/latex] is the set of all real numbers and can be described in interval notation as [latex]\\left(-\\infty, \\infty\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see two examples of how to express inequalities involving <em>or<\/em> graphically and as an interval.\r\n\r\n[embed]https:\/\/youtu.be\/nKarzhZOFIk[\/embed]\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nDraw a graph of the compound inequality:\u00a0[latex]x\\lt5[\/latex]\u00a0<em>and<\/em>\u00a0[latex]x\\ge\u22121[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n[reveal-answer q=\"394627\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"394627\"]\r\n\r\nThe graph of each individual inequality is shown in color.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182838\/image081.jpg\" alt=\"Number line. Open red circle on 5 and red arrow through all numbers less than 5. This red arrow is labeled x is less than 5. Closed blue circle on negative 1 and blue arrow through all numbers greater than negative 1. This blue arrow is labeled x is greater than or equal to negative 1.\" width=\"575\" height=\"53\" \/>\r\n\r\nSince the word <i>and <\/i>joins the two inequalities, the solution is the overlap of the two solutions. This is where both of these statements are true at the same time.\r\n\r\nThe solution to this compound inequality is shown below.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182840\/image082.jpg\" alt=\"Number line. Closed blue circle on negative 1. Open red circle on 5. The numbers between negative 1 and 5 (including negative 1) are colored purple. The purple line is labeled negative 1 is less than or equal to x is less than 5.\" width=\"575\" height=\"53\" \/>\r\n\r\nNotice that this is a bounded inequality. You can rewrite [latex]x\\ge\u22121\\,\\text{and }x\\le5[\/latex] as [latex]\u22121\\le x\\le 5[\/latex]\u00a0since the solution is between [latex]\u22121[\/latex] and\u00a0[latex]5[\/latex], including [latex]\u22121[\/latex]. You read [latex]\u22121\\le x\\lt{5}[\/latex]\u00a0as \u201c<i>x<\/i> is greater than or equal to [latex]\u22121[\/latex]\u00a0<i>and<\/i> less than\u00a0[latex]5[\/latex].\u201d You can rewrite an <i>and<\/i> statement this way only if the answer is <i>between<\/i> two numbers. The set of solutions to this inequality can be written in interval notation like this: [latex]\\left[{-1},{5}\\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Examples<\/h3>\r\nDraw the graph of the compound inequality [latex]x\\lt{-3}[\/latex] <em>and<\/em> [latex]x\\gt{3}[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n[reveal-answer q=\"870500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"870500\"]\r\n\r\nFirst, draw a graph. We are looking for values for <em>x<\/em> that will satisfy <strong>both\u00a0<\/strong>inequalities since they are joined with the word <em>and<\/em>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182842\/image091.jpg\" alt=\"Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/>\r\n\r\nIn this case, there are no shared <em>x<\/em>-values, and therefore, there is no intersection for these two inequalities. We can write \"no solution,\" or DNE.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video presents two examples of how to draw inequalities involving <em>and<\/em>,\u00a0as well as write the corresponding intervals.\r\n\r\nhttps:\/\/youtu.be\/LP3fsZNjJkc\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]197132[\/ohm_question]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Use interval notation to describe intersections and unions<\/li>\n<li>Use graphs to describe intersections and\u00a0unions<\/li>\n<\/ul>\n<\/div>\n<h2>Use Interval Notation to Describe Sets of Numbers as Intersections and Unions<\/h2>\n<p>When two inequalities are joined by the word <i>and<\/i>, the solution of the compound inequality occurs when <i>both<\/i> inequalities are true at the same time. It is the overlap, or intersection, of the solutions for each inequality. When the two inequalities are joined by the word <i>or<\/i>, the solution of the compound inequality occurs when <i>either<\/i> of the inequalities is true. The solution is the combination, or union, of the two individual solutions.<\/p>\n<p>In this section, we will learn how to solve compound inequalities that are joined with the words AND and OR. First, it will help to see some examples of inequalities, intervals, and graphs of compound inequalities. This will help you describe the solutions to compound inequalities properly.<\/p>\n<p>Venn diagrams use the concept of intersections and unions to compare two or more things. For example, this Venn diagram shows the intersection of people who are breaking your heart and those who are shaking your confidence daily. Apparently, Cecilia has both of these qualities; therefore, she is the intersection of the two.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3710\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182824\/Screen-Shot-2016-05-06-at-3.25.21-PM-300x234.png\" alt=\"Two circles. One is people who are breaking my heart. The other is people who are shaking my confidence daily. The area where the circles overlap is labeled Cecilia.\" width=\"353\" height=\"275\" \/><\/p>\n<p>In mathematical terms, consider the inequality\u00a0[latex]x\\lt6[\/latex] and\u00a0[latex]x\\gt2[\/latex]. How would we interpret what numbers [latex]x[\/latex] can be, and what would the interval look like?<\/p>\n<p>In words, [latex]x[\/latex] must be less than\u00a0[latex]6[\/latex], and at the same time, it must be greater than\u00a0[latex]2[\/latex]. This is much like the Venn diagram above, where Cecilia is at once breaking your heart and shaking your confidence daily. Now look at a graph to see what numbers are possible with these constraints.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3958 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182826\/Screen-Shot-2016-05-10-at-4.43.10-PM-300x46.png\" alt=\"x&gt; 2 and x&lt; 6\" width=\"594\" height=\"91\" \/><\/p>\n<p>The numbers that are shared by both lines on the graph are called the intersection of the two inequalities\u00a0[latex]x\\lt6[\/latex]\u00a0and\u00a0[latex]x\\gt2[\/latex]. This is called a <em>bounded<\/em> inequality, and is written as [latex]2\\lt{x}\\lt6[\/latex]. Think about that one for a minute. [latex]x[\/latex] must be less than\u00a0[latex]6[\/latex] and greater than two\u2014the values for [latex]x[\/latex] will fall <em>between two numbers.<\/em>\u00a0In interval notation, this looks like [latex]\\left(2,6\\right)[\/latex]. The graph would look like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-4014\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182827\/Screen-Shot-2016-05-11-at-4.53.25-PM-300x46.png\" alt=\"Open circle on 2 and open circle on 6 with a line through all numbers between 2 and 6.\" width=\"664\" height=\"102\" \/><span style=\"line-height: 1.5\">On the other hand, if you need to represent two things that\u00a0don&#8217;t share any common elements or traits, you can use\u00a0a union. The following Venn diagram shows two things that share no similar traits or elements, but are often considered in the same application, such as online shopping or banking.<\/span><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3712\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182829\/Screen-Shot-2016-05-06-at-3.26.52-PM-300x150.png\" alt=\"Two circles, one the Internet and the other Privacy.\" width=\"406\" height=\"203\" \/><\/p>\n<p>In mathematical terms, for example, [latex]x>6[\/latex]\u00a0<em>or<\/em>\u00a0[latex]x<2[\/latex] is an inequality joined by the word <em>or<\/em>. Using interval notation, we can describe each of these inequalities separately:<\/p>\n<p>[latex]x\\gt6[\/latex] is the same as [latex]\\left(6, \\infty\\right)[\/latex] and\u00a0[latex]x<2[\/latex] is the same as\u00a0[latex]\\left(-\\infty, 2\\right)[\/latex]. If we are describing solutions to inequalities, what effect does the\u00a0<em>or\u00a0<\/em>have? \u00a0We are\u00a0saying that solutions are either real numbers less than two\u00a0<em>or<\/em> real numbers greater than\u00a0[latex]6[\/latex]. Can you see why we need to write them as two separate intervals? Let us look at a graph to get a clear picture of what is going on.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3960\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182831\/Screen-Shot-2016-05-10-at-4.53.44-PM-300x39.png\" alt=\"Open circle on 2 and line through all numbers less than 2. Open circle on 6 and line through all numbers grater than 6.\" width=\"585\" height=\"76\" \/><\/p>\n<p>When you place both of these inequalities on a graph, we can see that they share no numbers in common. As mentioned above, this is what we call a union. The interval notation associated with a union is a big U, so instead of writing <em>or<\/em>, we join our intervals with a big U, like this:<\/p>\n<p style=\"text-align: center\">[latex]\\left(-\\infty, 2\\right)\\cup\\left(6, \\infty\\right)[\/latex]<\/p>\n<p style=\"text-align: left\">It is common convention to construct intervals starting with the value that is furthest left on the number line as the left value, such as\u00a0[latex]\\left(2,6\\right)[\/latex], where\u00a0[latex]2[\/latex] is less than [latex]6[\/latex]. The number on the right should be greater than the number on the left.<\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm2796\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=2796&theme=oea&iframe_resize_id=ohm2796&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Draw the graph of the compound inequality [latex]x\\gt3[\/latex] <em>or<\/em>\u00a0[latex]x\\le4[\/latex] and describe the set of <em>x<\/em>-values that will satisfy it with an interval.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q641470\">Show Solution<\/span><\/p>\n<div id=\"q641470\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph of [latex]x\\gt3[\/latex]\u00a0has an open circle on\u00a0[latex]3[\/latex] and a blue arrow drawn to the right to contain all the numbers greater than\u00a0[latex]3[\/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\/01182832\/image075.jpg\" alt=\"Number line. Open blue circle on 3. Blue highlight on all numbers greater than 3.\" width=\"575\" height=\"53\" \/><\/p>\n<p>The graph of\u00a0[latex]x\\le4[\/latex]\u00a0has a closed circle at 4 and a red arrow to the left to contain all the numbers less than\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\/01182834\/image076.jpg\" alt=\"Number line. Closed red circle on 4. Red highlight on all numbers less than 4.\" width=\"575\" height=\"53\" \/><\/p>\n<p>What do you notice about the graph that combines these two inequalities?<\/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\/01182836\/image077.jpg\" alt=\"Number line. Open blue circle on 3 and blue highlight on all numbers greater than 3. Red closed circle on 4 and red highlight through all numbers less than 4. This means that both colored highlights cover the numbers between 3 and 4.\" width=\"575\" height=\"53\" \/><\/p>\n<p>Since this compound inequality is an <i>or<\/i> statement, it includes all of the numbers in each of the solutions. In this case, the solution is all the numbers on the number line. (The region of the line greater than\u00a0[latex]3[\/latex] and less than or equal to\u00a0[latex]4[\/latex] is shown in purple because it lies on both of the original graphs.) The solution to the compound inequality [latex]x\\gt3[\/latex] <em>or<\/em> [latex]x\\le4[\/latex] is the set of all real numbers and can be described in interval notation as [latex]\\left(-\\infty, \\infty\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see two examples of how to express inequalities involving <em>or<\/em> graphically and as an interval.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-1\" title=\"Solutions to Basic OR Compound Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/nKarzhZOFIk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Draw a graph of the compound inequality:\u00a0[latex]x\\lt5[\/latex]\u00a0<em>and<\/em>\u00a0[latex]x\\ge\u22121[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q394627\">Show Solution<\/span><\/p>\n<div id=\"q394627\" class=\"hidden-answer\" style=\"display: none\">\n<p>The graph of each individual inequality is shown in color.<\/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\/01182838\/image081.jpg\" alt=\"Number line. Open red circle on 5 and red arrow through all numbers less than 5. This red arrow is labeled x is less than 5. Closed blue circle on negative 1 and blue arrow through all numbers greater than negative 1. This blue arrow is labeled x is greater than or equal to negative 1.\" width=\"575\" height=\"53\" \/><\/p>\n<p>Since the word <i>and <\/i>joins the two inequalities, the solution is the overlap of the two solutions. This is where both of these statements are true at the same time.<\/p>\n<p>The solution to this compound inequality is shown below.<\/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\/01182840\/image082.jpg\" alt=\"Number line. Closed blue circle on negative 1. Open red circle on 5. The numbers between negative 1 and 5 (including negative 1) are colored purple. The purple line is labeled negative 1 is less than or equal to x is less than 5.\" width=\"575\" height=\"53\" \/><\/p>\n<p>Notice that this is a bounded inequality. You can rewrite [latex]x\\ge\u22121\\,\\text{and }x\\le5[\/latex] as [latex]\u22121\\le x\\le 5[\/latex]\u00a0since the solution is between [latex]\u22121[\/latex] and\u00a0[latex]5[\/latex], including [latex]\u22121[\/latex]. You read [latex]\u22121\\le x\\lt{5}[\/latex]\u00a0as \u201c<i>x<\/i> is greater than or equal to [latex]\u22121[\/latex]\u00a0<i>and<\/i> less than\u00a0[latex]5[\/latex].\u201d You can rewrite an <i>and<\/i> statement this way only if the answer is <i>between<\/i> two numbers. The set of solutions to this inequality can be written in interval notation like this: [latex]\\left[{-1},{5}\\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Examples<\/h3>\n<p>Draw the graph of the compound inequality [latex]x\\lt{-3}[\/latex] <em>and<\/em> [latex]x\\gt{3}[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q870500\">Show Solution<\/span><\/p>\n<div id=\"q870500\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, draw a graph. We are looking for values for <em>x<\/em> that will satisfy <strong>both\u00a0<\/strong>inequalities since they are joined with the word <em>and<\/em>.<\/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\/01182842\/image091.jpg\" alt=\"Number line. Open red circle on negative 3 and red arrow through all numbers less than negative 3. Red arrow represents x is less than negative 3. Open blue circle on 3 and blue arrow through all numbers greater than 3. Blue arrow represents x is greater than 3.\" width=\"575\" height=\"53\" \/><\/p>\n<p>In this case, there are no shared <em>x<\/em>-values, and therefore, there is no intersection for these two inequalities. We can write &#8220;no solution,&#8221; or DNE.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video presents two examples of how to draw inequalities involving <em>and<\/em>,\u00a0as well as write the corresponding intervals.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-2\" title=\"Solutions to Basic AND Compound Inequalities\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/LP3fsZNjJkc?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=\"ohm197132\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=197132&theme=oea&iframe_resize_id=ohm197132&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\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-16148\">\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>Cecilia Venn Diagram Image. <strong>Authored 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>Internet Privacy Venn Diagram. <strong>Authored 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>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>Solutions to Basic OR Compound Inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nKarzhZOFIk\">https:\/\/youtu.be\/nKarzhZOFIk<\/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>Solutions to Basic AND Compound Inequalities Mathispower4u . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LP3fsZNjJkc\">https:\/\/youtu.be\/LP3fsZNjJkc<\/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 class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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.candelalearning.com\/collegealgebra1xmaster\/\">https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/<\/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 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