{"id":18649,"date":"2022-04-22T09:13:56","date_gmt":"2022-04-22T09:13:56","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=18649"},"modified":"2022-04-28T23:18:32","modified_gmt":"2022-04-28T23:18:32","slug":"cr-25-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/cr-25-inequalities\/","title":{"raw":"CR.25: Inequalities","rendered":"CR.25: Inequalities"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Represent inequalities using an inequality symbol and interval notation<\/li>\r\n \t<li>Represent inequalities on a number line<\/li>\r\n \t<li>Solve one-step inequalities<\/li>\r\n \t<li>Solve multi-step inequalities<\/li>\r\n \t<li>Use interval notation to describe intersections and unions<\/li>\r\n \t<li>Use graphs to describe intersections and unions<\/li>\r\n<\/ul>\r\n<\/div>\r\nAn inequality is a mathematical statement that compares two expressions using a phrase such as\u00a0<em>greater than<\/em> or <em>less than<\/em>. Special symbols are used in these statements. In algebra, inequalities are used to describe sets of values, as opposed to single values, of a variable. Sometimes, several numbers will satisfy an inequality, but at other times infinitely many numbers may provide solutions. Rather than try to list a possibly infinitely large set of numbers, mathematicians have developed some efficient ways to describe such large lists.\r\n<h2>Inequality Symbols<\/h2>\r\nOne way to represent such a list of numbers, an inequality, is by using an inequality symbol:\r\n<ul>\r\n \t<li>[latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than\u00a0[latex]9[\/latex]. Since this list is infinite, it would be impossilbe to list all numbers less than [latex]9[\/latex].<\/li>\r\n \t<li>[latex]-5\\le{t}[\/latex] indicates all the numbers that are greater than or equal to [latex]-5[\/latex].<\/li>\r\n<\/ul>\r\nIf you were to read the above statement from left to right, it would translate as <em>[latex]-5[\/latex] is less than or equal to t<\/em>. The direction of the symbol is dependent upon the statement you wish to make. For example, the following statements are equivalent. Both represent the list of all numbers less than 9. Note how the open end of the inequality symbol faces the larger value while the smaller, pointy end points to the smaller of the values:\r\n<ul>\r\n \t<li>[latex]{x}\\lt{9}[\/latex]<\/li>\r\n \t<li>[latex]{9}\\gt{x}[\/latex]<\/li>\r\n<\/ul>\r\nHere's another way of looking at is:\r\n<ul>\r\n \t<li>[latex]x\\lt5[\/latex] means all the real numbers that are less than 5, whereas;<\/li>\r\n \t<li>[latex]5\\lt{x}[\/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\\gt{5}[\/latex]. Note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5 which is easier to interpret than 5 is less than x.<\/li>\r\n<\/ul>\r\nThe box below shows the symbol, meaning, and an example for each inequality sign, as they would be translated reading 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<strong>is<\/strong> <b>not equal<\/b> to 8<em>.<\/em><\/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<strong>is greater than<\/strong>\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], 2<i>\u00a0<\/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], 4<i>\u00a0<\/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\nAnother way to represent an inequality is by graphing it on a number line:\r\n\r\n<img class=\"aligncenter wp-image-3855 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182808\/MITE_Lippman_Arithmetic_pdf__page_356_of_417_-300x58.png\" alt=\"A numberline. It is a long horizontal line with evenly spaced points, the middle of which is zero.\" width=\"300\" height=\"58\" \/>\r\n\r\nBelow are three examples of inequalities and their graphs. \u00a0Graphs are often helpful for visualizing information.\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 [latex]4[\/latex].\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.\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 end point is part of the solution. An open circle is used for <i>greater than<\/i> (&gt;) or <i>less than<\/i> (&lt;). The end point is <i>not <\/i>part of the solution. When the end point is not included in the solution, we often say we have <em>strict inequality<\/em> rather than <em>inequality with equality<\/em>.\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\nhttps:\/\/youtu.be\/-kiAeGbSe5c\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nWrite an inequality describing all the real numbers on the number line that are strictly less than\u00a0[latex]2[\/latex]. Then draw the corresponding graph.\r\n[reveal-answer q=\"867890\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"867890\"]\r\n\r\nWe need to start from the left and work right, so we start from negative infinity and end at [latex]2[\/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]2[\/latex].\r\n\r\nInequality: [latex]x\\lt2[\/latex]\r\n\r\nTo draw the graph, place an open dot on the number line first, and then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image033.jpg#fixme#fixme\" alt=\"Number line. Unshaded circle around 2 and shaded line through all numbers less than 2.\" width=\"575\" height=\"31\" \/>\r\n\r\n[\/hidden-answer]\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> interval notation. <\/strong>With this convention, sets are built with 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 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 \u201cequaled.\u201d 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 outlines 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 <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<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q817362\">Show Solution<\/span>\r\n<div id=\"q817362\" class=\"hidden-answer\" style=\"display: none;\">\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<\/div>\r\n<\/div>\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<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q961990\">Show Solution<\/span>\r\n<div id=\"q961990\" class=\"hidden-answer\" style=\"display: none;\">\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<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nIn the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and asked to write the inequality and draw the graph.\r\n\r\nGiven [latex]\\left(-\\infty,10\\right)[\/latex], write the associated inequality and draw the graph.\r\n\r\nIn the box below, write down whether you think it will be easier to draw the graph first or write the inequality first.\r\n\r\n&nbsp;\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q15120\">Show Solution<\/span>\r\n<div id=\"q15120\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nWe will draw the graph first.\r\n\r\nThe interval reads \u201call real numbers less than 10,\u201d so we will start by placing an open dot on 10 and drawing a line to the left with an arrow indicating the solution continues to negative infinity.\r\n<div class=\"wp-nocaption aligncenter\"><img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182811\/4.png\" alt=\"An open circle on 10 and a line going from 10 to all numbers below 10.\" width=\"339\" height=\"95\" \/><\/div>\r\nTo write the inequality, we will use &lt; since the parentheses indicate that 10 is not included. [latex]x&lt;10[\/latex]\r\n\r\n<\/div>\r\n<\/div>\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\n<iframe src=\"https:\/\/www.youtube.com\/embed\/X0xrHKgbDT0?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\n<!-- .entry-content -->\r\n\r\n<!-- #post-## -->\r\n<h2 style=\"text-align: left;\">Multiplication and Division Properties of Inequality<\/h2>\r\nSolving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer, you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similar to but not exactly as we treat equations. We can use the <strong>addition property<\/strong> and the <strong>multiplication property<\/strong> to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.\r\n\r\nThe following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Start With<\/strong><\/td>\r\n<td><strong>Multiply By<\/strong><\/td>\r\n<td><strong>Final Inequality<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]ac&gt;bc[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5&gt;3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]15&gt;9[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]-c[\/latex]<\/td>\r\n<td>[latex]-ac&lt;-bc[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5&gt;3[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]-15&lt;-9[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe following table illustrates how the division property is applied to inequalities, and how dividing by a negative reverses the inequality:\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Start With<\/strong><\/td>\r\n<td><strong>Divide By<\/strong><\/td>\r\n<td><strong>Final Inequality<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{a}{c}&gt;\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4&gt;2[\/latex]<\/td>\r\n<td>[latex]2[\/latex]<\/td>\r\n<td>[latex] \\displaystyle \\frac{4}{2}&gt;\\frac{2}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]-c[\/latex]<\/td>\r\n<td>[latex] \\displaystyle -\\frac{a}{c}&lt;-\\frac{b}{c}[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]4&gt;2[\/latex]<\/td>\r\n<td>[latex]-2[\/latex]<\/td>\r\n<td>[latex] \\displaystyle -\\frac{4}{2}&lt;-\\frac{2}{2}[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIn the first example, we will show how to apply the multiplication and division properties of equality to solve some inequalities.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nIllustrate the multiplication property for inequalities by solving each of the following:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]3x&lt;6[\/latex]<\/li>\r\n \t<li>[latex]-2x - 1\\ge 5[\/latex]<\/li>\r\n \t<li>[latex]5-x&gt;10[\/latex]<\/li>\r\n<\/ol>\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q432848\">Show Solution<\/span>\r\n<div id=\"q432848\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\na.\r\n[latex]\\begin{array}{cc}\\hfill3x&lt;6 \\hfill\\\\\\dfrac{1}{3}\\normalsize\\left(3x\\right)&lt;\\left(6\\right)\\dfrac{1}{3} \\\\ \\hfill{x}&lt;2 \\hfill\\end{array}[\/latex]\r\n\r\n&nbsp;\r\n\r\nb.\r\n[latex]\\begin{array}{rr}-2x - 1\\ge 5\\\\ \\hfill\\hfill-2x\\ge 6\\end{array}[\/latex]\r\n\r\nMultiply both sides by [latex]-\\dfrac{1}{2}[\/latex].\r\n\r\n[latex]\\begin{array}{ll}\\hfill\\hfill\\left(-\\dfrac{1}{2}\\normalsize\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\dfrac{1}{2}\\normalsize\\right)\\end{array}[\/latex]\r\n\r\nReverse the inequality.\r\n\r\n[latex]\\begin{array}{l}\\hfill&amp;\\hfill&amp;\\hfill&amp;\\hfill&amp;\\hfill x\\le -3\\end{array}[\/latex]\r\n\r\nc.\r\n[latex]\\begin{array}{ll}5-x&gt;10\\\\ -x&gt;5\\hfill &amp;\\hfill\\end{array}[\/latex]\r\n\r\nMultiply both sides by [latex] -1[\/latex].\r\n\r\n[latex]\\left(-1\\right)\\left(-x\\right)&gt;\\left(5\\right)\\left(-1\\right)[\/latex]\r\n\r\nReverse the inequality\r\n\r\n[latex]x&lt;-5[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<h2 style=\"text-align: left;\">Solve Inequalities Using the Addition Property<\/h2>\r\nWhen we solve equations we may need to add or subtract in order to isolate the variable, the same is true for inequalities. There are no special behaviors to watch out for when using the addition property to solve inequalities.\r\n\r\nThe following table illustrates how the addition property applies to inequalities.\r\n<table style=\"width: 20%;\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Start With<\/strong><\/td>\r\n<td><strong>Add<\/strong><\/td>\r\n<td><strong>Final Inequality<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]c[\/latex]<\/td>\r\n<td>[latex]a+c&gt;b+c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5&gt;3[\/latex]<\/td>\r\n<td>[latex]3[\/latex]<\/td>\r\n<td>[latex]8&gt;6[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]a&gt;b[\/latex]<\/td>\r\n<td>[latex]-c[\/latex]<\/td>\r\n<td>[latex]a-c&gt;b-c[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]5&gt;3[\/latex]<\/td>\r\n<td>[latex]-3[\/latex]<\/td>\r\n<td>[latex]2&gt;0[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThese properties also apply to [latex]a\\le b[\/latex], [latex]a&gt;b[\/latex], and [latex]a\\ge b[\/latex].\r\n\r\nIn our next example we will use the addition property to solve inequalities.\r\n<div class=\"textbox exercises\" style=\"text-align: left;\">\r\n<h3>Example<\/h3>\r\nIllustrate the addition property for inequalities by solving each of the following:\r\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]x - 15&lt;4[\/latex]<\/li>\r\n \t<li>[latex]6\\ge x - 1[\/latex]<\/li>\r\n \t<li>[latex]x+7&gt;9[\/latex]<\/li>\r\n<\/ol>\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q399605\">Show Solution<\/span>\r\n<div id=\"q399605\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nThe addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.\r\na.\r\n[latex]\\begin{array}{rr}\\hfill x - 15&lt;4\\hfill\\hfill \\\\ \\hfill x - 15+15&lt;4+15\\hfill&amp; \\text{Add 15 to both sides.}\\hfill\\\\\\hfill\\quad x&lt;19 \\hfill\\end{array}[\/latex]\r\n\r\nb.\r\n[latex]\\begin{array}{rr}\\hfill 6\u2265 x - 1\\hfill\\hfill \\\\\\hfill 6+1\\ge x - 1+1\\hfill &amp; \\text{Add 1 to both sides}.\\hfill \\\\\\quad\\quad 7\u2265 x\\hfill \\end{array}[\/latex]\r\n\r\nc.\r\n[latex]\\begin{array}{rr}\\hfill x+7&gt;9\\hfill\\hfill\\\\\\hfill x+7 - 7&gt;9 - 7\\hfill &amp; \\text{Subtract 7 from both sides}.\\hfill\\quad \\\\\\hfill x&gt;2\\hfill \\end{array}[\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe following video shows examples of solving single-step inequalities using the multiplication and addition properties.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/1Z22Xh66VFM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n\r\nThe following video shows examples of solving inequalities with the variable on the right side.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/RBonYKvTCLU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<h2 style=\"text-align: left;\">Solve Multi-Step Inequalities<\/h2>\r\nAs the previous examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations. To isolate the variable and solve, we combine like terms and perform operations with the multiplication and addition properties.\r\n<div class=\"textbox exercises\" style=\"text-align: left;\">\r\n<h3>Example<\/h3>\r\nSolve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q532189\">Show Solution<\/span>\r\n<div id=\"q532189\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nSolving this inequality is similar to solving an equation up until the last step.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{rr}13 - 7x\\ge 10x - 4\\hfill &amp; \\\\ 13 - 17x\\ge -4\\hfill &amp; \\text{Move variable terms to one side of the inequality}.\\hfill&amp;\\quad \\\\-17x\\ge -17\\hfill&amp;\\text{Isolate the variable term}.\\hfill&amp;\\quad \\\\x\\le 1\\hfill &amp; \\text{Dividing both sides by }-17\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is given by the interval [latex]\\left(-\\infty ,1\\right][\/latex], or all real numbers less than and including 1.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\" style=\"text-align: left;\">\r\n<h3>Example<\/h3>\r\nSolve the inequality. Write the inequality in interval notation. [latex]8(x+6)-9x\\lt 6(2x+1)-7x[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q532190\">Show Solution<\/span>\r\n<div id=\"q532190\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nSolving this inequality is similar to solving an equation up until the last step.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{rr}8(x+6)-9x\\lt 6(2x+1)-7x\\hfill &amp; \\\\ 8x+48-9x\\lt 12x+6-7x\\hfill &amp; \\text{Distribute}.\\hfill&amp;\\quad \\\\ 48-x\\lt 5x+6\\hfill &amp; \\text{Add like terms}.\\hfill&amp;\\quad \\\\ 48 - 6x\\lt 6\\hfill &amp; \\text{Move variable terms to one side of the inequality}.\\hfill&amp;\\quad \\\\-6x\\lt -42\\hfill&amp;\\text{Isolate the variable term}.\\hfill&amp;\\quad \\\\x\\gt 7\\hfill &amp; \\text{Dividing both sides by }-6\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is given by the interval [latex]\\left[7,\\infty\\right)[\/latex], or all real numbers greater than 7.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nIn the next example we solve an inequality that contains fractions, not how we need to reverse the inequality sign at the end because we multiply by a negative.\r\n<div class=\"textbox exercises\" style=\"text-align: left;\">\r\n<h3>Example<\/h3>\r\nSolve the following inequality and write the answer in interval notation: [latex]-\\dfrac{3}{4}\\normalsize x\\ge -\\dfrac{5}{8}\\normalsize +\\dfrac{2}{3}\\normalsize x[\/latex].\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q59887\">Show Solution<\/span>\r\n<div id=\"q59887\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nWe begin solving in the same way we do when solving an equation.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{rr}-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x\\hfill &amp; \\hfill \\\\ -\\frac{3}{4}x-\\frac{2}{3}x\\ge -\\frac{5}{8}\\hfill &amp; \\text{Put variable terms on one side}.\\hfill \\\\ -\\frac{9}{12}x-\\frac{8}{12}x\\ge -\\frac{5}{8}\\hfill &amp; \\text{Write fractions with common denominator}.\\hfill \\\\ -\\frac{17}{12}x\\ge -\\frac{5}{8}\\hfill &amp; \\hfill \\\\ x\\le -\\frac{5}{8}\\left(-\\frac{12}{17}\\right)\\hfill &amp; \\text{Multiplying by a negative number reverses the inequality}.\\hfill \\\\ x\\le \\frac{15}{34}\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\r\n<div>The solution set is the interval [latex]\\left(-\\infty ,\\dfrac{15}{34}\\normalsize\\right][\/latex].<\/div>\r\n<\/div>\r\n<\/div>\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<div class=\"wp-nocaption aligncenter wp-image-3710\"><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\" \/><\/div>\r\nIn mathematical terms, consider the inequality [latex]x\\lt6[\/latex] and [latex]x\\gt2[\/latex]. How would we interpret what numbers <em>x<\/em> can be, and what would the interval look like?\r\n\r\nIn words, <em>x<\/em> must be less than [latex]6[\/latex], and at the same time, it must be greater than [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<div class=\"wp-nocaption wp-image-3958 aligncenter\"><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\" \/><\/div>\r\nThe numbers that are shared by both lines on the graph are called the intersection of the two inequalities [latex]x\\lt6[\/latex] and [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. <em>x<\/em> must be less than [latex]6[\/latex] and greater than two\u2014the values for <em>x<\/em> will fall <em>between two numbers.<\/em> In interval notation, this looks like [latex]\\left(2,6\\right)[\/latex]. The graph would look like this:\r\n<div class=\"wp-nocaption aligncenter wp-image-4014\"><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 don\u2019t share any common elements or traits, you can use a 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><\/div>\r\n<div class=\"wp-nocaption aligncenter wp-image-3712\"><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\" \/><\/div>\r\nIn mathematical terms, for example, [latex]x&gt;6[\/latex] <em>or<\/em> [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 [latex]x&lt;2[\/latex] is the same as [latex]\\left(-\\infty, 2\\right)[\/latex]. If we are describing solutions to inequalities, what effect does the <em>or <\/em>have? We are saying that solutions are either real numbers less than two <em>or<\/em> real numbers greater than [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<div class=\"wp-nocaption aligncenter wp-image-3960\"><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\" \/><\/div>\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 [latex]\\left(2,6\\right)[\/latex], where [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 exercises\">\r\n<h3>Example<\/h3>\r\nDraw the graph of the compound inequality [latex]x\\gt3[\/latex] <em>or<\/em> [latex]x\\le4[\/latex] and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q641470\">Show Solution<\/span>\r\n<div id=\"q641470\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nThe graph of [latex]x\\gt3[\/latex] has an open circle on [latex]3[\/latex] and a blue arrow drawn to the right to contain all the numbers greater than [latex]3[\/latex].\r\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\r\nThe graph of [latex]x\\le4[\/latex] has a closed circle at 4 and a red arrow to the left to contain all the numbers less than [latex]4[\/latex].\r\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\r\nWhat do you notice about the graph that combines these two inequalities?\r\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\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 [latex]3[\/latex] and less than or equal to [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<\/div>\r\n<\/div>\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<iframe src=\"https:\/\/www.youtube.com\/embed\/nKarzhZOFIk?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDraw a graph of the compound inequality: [latex]x\\lt5[\/latex] <em>and<\/em> [latex]x\\ge\u22121[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q394627\">Show Solution<\/span>\r\n<div id=\"q394627\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nThe graph of each individual inequality is shown in color.\r\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\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<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\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] since the solution is between [latex]\u22121[\/latex] and [latex]5[\/latex], including [latex]\u22121[\/latex]. You read [latex]\u22121\\le x\\lt{5}[\/latex] as \u201c<i>x<\/i> is greater than or equal to [latex]\u22121[\/latex] <i>and<\/i> less than [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<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDraw the graph of the compound inequality [latex]2x+1\\lt{-5}[\/latex] <em>and<\/em> [latex]4x-3\\gt{9}[\/latex], and describe the set of <em>x<\/em>-values that will satisfy it with an interval.\r\n<div class=\"qa-wrapper\" style=\"display: block;\">\r\n\r\n<span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q870500\">Show Solution<\/span>\r\n<div id=\"q870500\" class=\"hidden-answer\" style=\"display: none;\">\r\n\r\nFirst, we must solve each inequality for [latex]x[\/latex]:\r\n[latex]2x+1\\lt{-5}[\/latex] <i>and<\/i> [latex]4x-3\\gt{9}[\/latex]\r\n[latex]2x\\lt{-6}[\/latex] <i>and<\/i> [latex]4x\\gt{12}[\/latex]\r\n[latex]x\\lt{-3}[\/latex] <i>and<\/i> [latex]x\\gt{3}[\/latex]\r\nNow, draw a graph. We are looking for values for <em>x<\/em> that will satisfy <strong>both <\/strong>inequalities since they are joined with the word <em>and<\/em>.\r\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\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 \u201cno solution,\u201d or DNE.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\nThe following video presents two examples of how to draw inequalities involving <em>and<\/em> as well as write the corresponding intervals.\r\n\r\n<iframe src=\"https:\/\/www.youtube.com\/embed\/LP3fsZNjJkc?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Represent inequalities using an inequality symbol and interval notation<\/li>\n<li>Represent inequalities on a number line<\/li>\n<li>Solve one-step inequalities<\/li>\n<li>Solve multi-step inequalities<\/li>\n<li>Use interval notation to describe intersections and unions<\/li>\n<li>Use graphs to describe intersections and unions<\/li>\n<\/ul>\n<\/div>\n<p>An inequality is a mathematical statement that compares two expressions using a phrase such as\u00a0<em>greater than<\/em> or <em>less than<\/em>. Special symbols are used in these statements. In algebra, inequalities are used to describe sets of values, as opposed to single values, of a variable. Sometimes, several numbers will satisfy an inequality, but at other times infinitely many numbers may provide solutions. Rather than try to list a possibly infinitely large set of numbers, mathematicians have developed some efficient ways to describe such large lists.<\/p>\n<h2>Inequality Symbols<\/h2>\n<p>One way to represent such a list of numbers, an inequality, is by using an inequality symbol:<\/p>\n<ul>\n<li>[latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than\u00a0[latex]9[\/latex]. Since this list is infinite, it would be impossilbe to list all numbers less than [latex]9[\/latex].<\/li>\n<li>[latex]-5\\le{t}[\/latex] indicates all the numbers that are greater than or equal to [latex]-5[\/latex].<\/li>\n<\/ul>\n<p>If you were to read the above statement from left to right, it would translate as <em>[latex]-5[\/latex] is less than or equal to t<\/em>. The direction of the symbol is dependent upon the statement you wish to make. For example, the following statements are equivalent. Both represent the list of all numbers less than 9. Note how the open end of the inequality symbol faces the larger value while the smaller, pointy end points to the smaller of the values:<\/p>\n<ul>\n<li>[latex]{x}\\lt{9}[\/latex]<\/li>\n<li>[latex]{9}\\gt{x}[\/latex]<\/li>\n<\/ul>\n<p>Here&#8217;s another way of looking at is:<\/p>\n<ul>\n<li>[latex]x\\lt5[\/latex] means all the real numbers that are less than 5, whereas;<\/li>\n<li>[latex]5\\lt{x}[\/latex] means that 5 is less than x, or we could rewrite this with the x on the left: [latex]x\\gt{5}[\/latex]. Note how the inequality is still pointing the same direction relative to x. This statement represents all the real numbers that are greater than 5 which is easier to interpret than 5 is less than x.<\/li>\n<\/ul>\n<p>The box below shows the symbol, meaning, and an example for each inequality sign, as they would be translated reading 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<strong>is<\/strong> <b>not equal<\/b> to 8<em>.<\/em><\/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<strong>is greater than<\/strong>\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], 2<i>\u00a0<\/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], 4<i>\u00a0<\/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>Another way to represent an inequality is by graphing it on a number line:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3855 size-medium\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182808\/MITE_Lippman_Arithmetic_pdf__page_356_of_417_-300x58.png\" alt=\"A numberline. It is a long horizontal line with evenly spaced points, the middle of which is zero.\" width=\"300\" height=\"58\" \/><\/p>\n<p>Below are three examples of inequalities and their graphs. \u00a0Graphs are often helpful for visualizing information.<\/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 [latex]4[\/latex].<\/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.<\/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 end point is part of the solution. An open circle is used for <i>greater than<\/i> (&gt;) or <i>less than<\/i> (&lt;). The end point is <i>not <\/i>part of the solution. When the end point is not included in the solution, we often say we have <em>strict inequality<\/em> rather than <em>inequality with equality<\/em>.<\/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.<br \/>\n<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 exercises\">\n<h3>Example<\/h3>\n<p>Write an inequality describing all the real numbers on the number line that are strictly less than\u00a0[latex]2[\/latex]. Then draw the corresponding graph.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q867890\">Show Solution<\/span><\/p>\n<div id=\"q867890\" class=\"hidden-answer\" style=\"display: none\">\n<p>We need to start from the left and work right, so we start from negative infinity and end at [latex]2[\/latex]. We will not include either because infinity is not a number, and the inequality does not include [latex]2[\/latex].<\/p>\n<p>Inequality: [latex]x\\lt2[\/latex]<\/p>\n<p>To draw the graph, place an open dot on the number line first, and then draw a line extending to the left. Draw an arrow at the leftmost point of the line to indicate that it continues for infinity.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image033.jpg#fixme#fixme\" alt=\"Number line. Unshaded circle around 2 and shaded line through all numbers less than 2.\" width=\"575\" height=\"31\" \/><\/p>\n<\/div>\n<\/div>\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> interval notation. <\/strong>With this convention, sets are built with 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 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 \u201cequaled.\u201d 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 outlines 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 <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;\">\n<p><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 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;\">\n<p><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<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>In the previous examples you were given an inequality or a description of one with words and asked to draw the corresponding graph and write the interval. In this example you are given an interval and asked to write the inequality and draw the graph.<\/p>\n<p>Given [latex]\\left(-\\infty,10\\right)[\/latex], write the associated inequality and draw the graph.<\/p>\n<p>In the box below, write down whether you think it will be easier to draw the graph first or write the inequality first.<\/p>\n<p>&nbsp;<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q15120\">Show Solution<\/span><\/p>\n<div id=\"q15120\" class=\"hidden-answer\" style=\"display: none;\">\n<p>We will draw the graph first.<\/p>\n<p>The interval reads \u201call real numbers less than 10,\u201d so we will start by placing an open dot on 10 and drawing a line to the left with an arrow indicating the solution continues to negative infinity.<\/p>\n<div class=\"wp-nocaption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/121\/2016\/06\/01182811\/4.png\" alt=\"An open circle on 10 and a line going from 10 to all numbers below 10.\" width=\"339\" height=\"95\" \/><\/div>\n<p>To write the inequality, we will use &lt; since the parentheses indicate that 10 is not included. [latex]x<10[\/latex]\n\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\" src=\"https:\/\/www.youtube.com\/embed\/X0xrHKgbDT0?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p><!-- .entry-content --><\/p>\n<p><!-- #post-## --><\/p>\n<h2 style=\"text-align: left;\">Multiplication and Division Properties of Inequality<\/h2>\n<p>Solving inequalities is very similar to solving equations, except you have to reverse the inequality symbols when you multiply or divide both sides of an inequality by a negative number. There are three ways to represent solutions to inequalities: an interval, a graph, and an inequality. Because there is usually more than one solution to an inequality, when you check your answer, you should check the end point and one other value to check the direction of the inequality. When we work with inequalities, we can usually treat them similar to but not exactly as we treat equations. We can use the <strong>addition property<\/strong> and the <strong>multiplication property<\/strong> to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.<\/p>\n<p>The following table illustrates how the multiplication property is applied to inequalities, and how multiplication by a negative reverses the inequality:<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td><strong>Start With<\/strong><\/td>\n<td><strong>Multiply By<\/strong><\/td>\n<td><strong>Final Inequality<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]ac>bc[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]15>9[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]-ac<-bc[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]-15<-9[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The following table illustrates how the division property is applied to inequalities, and how dividing by a negative reverses the inequality:<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td><strong>Start With<\/strong><\/td>\n<td><strong>Divide By<\/strong><\/td>\n<td><strong>Final Inequality<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{a}{c}>\\frac{b}{c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4>2[\/latex]<\/td>\n<td>[latex]2[\/latex]<\/td>\n<td>[latex]\\displaystyle \\frac{4}{2}>\\frac{2}{2}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]\\displaystyle -\\frac{a}{c}<-\\frac{b}{c}[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]4>2[\/latex]<\/td>\n<td>[latex]-2[\/latex]<\/td>\n<td>[latex]\\displaystyle -\\frac{4}{2}<-\\frac{2}{2}[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>In the first example, we will show how to apply the multiplication and division properties of equality to solve some inequalities.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Illustrate the multiplication property for inequalities by solving each of the following:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]3x<6[\/latex]<\/li>\n<li>[latex]-2x - 1\\ge 5[\/latex]<\/li>\n<li>[latex]5-x>10[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q432848\">Show Solution<\/span><\/p>\n<div id=\"q432848\" class=\"hidden-answer\" style=\"display: none;\">\n<p>a.<br \/>\n[latex]\\begin{array}{cc}\\hfill3x<6 \\hfill\\\\\\dfrac{1}{3}\\normalsize\\left(3x\\right)<\\left(6\\right)\\dfrac{1}{3} \\\\ \\hfill{x}<2 \\hfill\\end{array}[\/latex]\n\n&nbsp;\n\nb.\n[latex]\\begin{array}{rr}-2x - 1\\ge 5\\\\ \\hfill\\hfill-2x\\ge 6\\end{array}[\/latex]\n\nMultiply both sides by [latex]-\\dfrac{1}{2}[\/latex].\n\n[latex]\\begin{array}{ll}\\hfill\\hfill\\left(-\\dfrac{1}{2}\\normalsize\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\dfrac{1}{2}\\normalsize\\right)\\end{array}[\/latex]\n\nReverse the inequality.\n\n[latex]\\begin{array}{l}\\hfill&\\hfill&\\hfill&\\hfill&\\hfill x\\le -3\\end{array}[\/latex]\n\nc.\n[latex]\\begin{array}{ll}5-x>10\\\\ -x>5\\hfill &\\hfill\\end{array}[\/latex]<\/p>\n<p>Multiply both sides by [latex]-1[\/latex].<\/p>\n<p>[latex]\\left(-1\\right)\\left(-x\\right)>\\left(5\\right)\\left(-1\\right)[\/latex]<\/p>\n<p>Reverse the inequality<\/p>\n<p>[latex]x<-5[\/latex]\n\n<\/div>\n<\/div>\n<\/div>\n<h2 style=\"text-align: left;\">Solve Inequalities Using the Addition Property<\/h2>\n<p>When we solve equations we may need to add or subtract in order to isolate the variable, the same is true for inequalities. There are no special behaviors to watch out for when using the addition property to solve inequalities.<\/p>\n<p>The following table illustrates how the addition property applies to inequalities.<\/p>\n<table style=\"width: 20%;\">\n<tbody>\n<tr>\n<td><strong>Start With<\/strong><\/td>\n<td><strong>Add<\/strong><\/td>\n<td><strong>Final Inequality<\/strong><\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]c[\/latex]<\/td>\n<td>[latex]a+c>b+c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]3[\/latex]<\/td>\n<td>[latex]8>6[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]a-c>b-c[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]5>3[\/latex]<\/td>\n<td>[latex]-3[\/latex]<\/td>\n<td>[latex]2>0[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>These properties also apply to [latex]a\\le b[\/latex], [latex]a>b[\/latex], and [latex]a\\ge b[\/latex].<\/p>\n<p>In our next example we will use the addition property to solve inequalities.<\/p>\n<div class=\"textbox exercises\" style=\"text-align: left;\">\n<h3>Example<\/h3>\n<p>Illustrate the addition property for inequalities by solving each of the following:<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]x - 15<4[\/latex]<\/li>\n<li>[latex]6\\ge x - 1[\/latex]<\/li>\n<li>[latex]x+7>9[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q399605\">Show Solution<\/span><\/p>\n<div id=\"q399605\" class=\"hidden-answer\" style=\"display: none;\">\n<p>The addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.<br \/>\na.<br \/>\n[latex]\\begin{array}{rr}\\hfill x - 15<4\\hfill\\hfill \\\\ \\hfill x - 15+15<4+15\\hfill& \\text{Add 15 to both sides.}\\hfill\\\\\\hfill\\quad x<19 \\hfill\\end{array}[\/latex]\n\nb.\n[latex]\\begin{array}{rr}\\hfill 6\u2265 x - 1\\hfill\\hfill \\\\\\hfill 6+1\\ge x - 1+1\\hfill & \\text{Add 1 to both sides}.\\hfill \\\\\\quad\\quad 7\u2265 x\\hfill \\end{array}[\/latex]\n\nc.\n[latex]\\begin{array}{rr}\\hfill x+7>9\\hfill\\hfill\\\\\\hfill x+7 - 7>9 - 7\\hfill & \\text{Subtract 7 from both sides}.\\hfill\\quad \\\\\\hfill x>2\\hfill \\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows examples of solving single-step inequalities using the multiplication and addition properties.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/1Z22Xh66VFM?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>The following video shows examples of solving inequalities with the variable on the right side.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/RBonYKvTCLU?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 style=\"text-align: left;\">Solve Multi-Step Inequalities<\/h2>\n<p>As the previous examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations. To isolate the variable and solve, we combine like terms and perform operations with the multiplication and addition properties.<\/p>\n<div class=\"textbox exercises\" style=\"text-align: left;\">\n<h3>Example<\/h3>\n<p>Solve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q532189\">Show Solution<\/span><\/p>\n<div id=\"q532189\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Solving this inequality is similar to solving an equation up until the last step.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rr}13 - 7x\\ge 10x - 4\\hfill & \\\\ 13 - 17x\\ge -4\\hfill & \\text{Move variable terms to one side of the inequality}.\\hfill&\\quad \\\\-17x\\ge -17\\hfill&\\text{Isolate the variable term}.\\hfill&\\quad \\\\x\\le 1\\hfill & \\text{Dividing both sides by }-17\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is given by the interval [latex]\\left(-\\infty ,1\\right][\/latex], or all real numbers less than and including 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\" style=\"text-align: left;\">\n<h3>Example<\/h3>\n<p>Solve the inequality. Write the inequality in interval notation. [latex]8(x+6)-9x\\lt 6(2x+1)-7x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q532190\">Show Solution<\/span><\/p>\n<div id=\"q532190\" class=\"hidden-answer\" style=\"display: none;\">\n<p>Solving this inequality is similar to solving an equation up until the last step.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rr}8(x+6)-9x\\lt 6(2x+1)-7x\\hfill & \\\\ 8x+48-9x\\lt 12x+6-7x\\hfill & \\text{Distribute}.\\hfill&\\quad \\\\ 48-x\\lt 5x+6\\hfill & \\text{Add like terms}.\\hfill&\\quad \\\\ 48 - 6x\\lt 6\\hfill & \\text{Move variable terms to one side of the inequality}.\\hfill&\\quad \\\\-6x\\lt -42\\hfill&\\text{Isolate the variable term}.\\hfill&\\quad \\\\x\\gt 7\\hfill & \\text{Dividing both sides by }-6\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is given by the interval [latex]\\left[7,\\infty\\right)[\/latex], or all real numbers greater than 7.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the next example we solve an inequality that contains fractions, not how we need to reverse the inequality sign at the end because we multiply by a negative.<\/p>\n<div class=\"textbox exercises\" style=\"text-align: left;\">\n<h3>Example<\/h3>\n<p>Solve the following inequality and write the answer in interval notation: [latex]-\\dfrac{3}{4}\\normalsize x\\ge -\\dfrac{5}{8}\\normalsize +\\dfrac{2}{3}\\normalsize x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block;\">\n<p><span class=\"show-answer collapsed\" style=\"cursor: pointer;\" data-target=\"q59887\">Show Solution<\/span><\/p>\n<div id=\"q59887\" class=\"hidden-answer\" style=\"display: none;\">\n<p>We begin solving in the same way we do when solving an equation.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{rr}-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x\\hfill & \\hfill \\\\ -\\frac{3}{4}x-\\frac{2}{3}x\\ge -\\frac{5}{8}\\hfill & \\text{Put variable terms on one side}.\\hfill \\\\ -\\frac{9}{12}x-\\frac{8}{12}x\\ge -\\frac{5}{8}\\hfill & \\text{Write fractions with common denominator}.\\hfill \\\\ -\\frac{17}{12}x\\ge -\\frac{5}{8}\\hfill & \\hfill \\\\ x\\le -\\frac{5}{8}\\left(-\\frac{12}{17}\\right)\\hfill & \\text{Multiplying by a negative number reverses the inequality}.\\hfill \\\\ x\\le \\frac{15}{34}\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<div>The solution set is the interval [latex]\\left(-\\infty ,\\dfrac{15}{34}\\normalsize\\right][\/latex].<\/div>\n<\/div>\n<\/div>\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<div class=\"wp-nocaption aligncenter wp-image-3710\"><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\" \/><\/div>\n<p>In mathematical terms, consider the inequality [latex]x\\lt6[\/latex] and [latex]x\\gt2[\/latex]. How would we interpret what numbers <em>x<\/em> can be, and what would the interval look like?<\/p>\n<p>In words, <em>x<\/em> must be less than [latex]6[\/latex], and at the same time, it must be greater than [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<div class=\"wp-nocaption wp-image-3958 aligncenter\"><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\" \/><\/div>\n<p>The numbers that are shared by both lines on the graph are called the intersection of the two inequalities [latex]x\\lt6[\/latex] and [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. <em>x<\/em> must be less than [latex]6[\/latex] and greater than two\u2014the values for <em>x<\/em> will fall <em>between two numbers.<\/em> In interval notation, this looks like [latex]\\left(2,6\\right)[\/latex]. The graph would look like this:<\/p>\n<div class=\"wp-nocaption aligncenter wp-image-4014\"><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 don\u2019t share any common elements or traits, you can use a 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><\/div>\n<div class=\"wp-nocaption aligncenter wp-image-3712\"><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\" \/><\/div>\n<p>In mathematical terms, for example, [latex]x>6[\/latex] <em>or<\/em> [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 [latex]x<2[\/latex] is the same as [latex]\\left(-\\infty, 2\\right)[\/latex]. If we are describing solutions to inequalities, what effect does the <em>or <\/em>have? We are saying that solutions are either real numbers less than two <em>or<\/em> real numbers greater than [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<div class=\"wp-nocaption aligncenter wp-image-3960\"><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\" \/><\/div>\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 [latex]\\left(2,6\\right)[\/latex], where [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 exercises\">\n<h3>Example<\/h3>\n<p>Draw the graph of the compound inequality [latex]x\\gt3[\/latex] <em>or<\/em> [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;\">\n<p><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] has an open circle on [latex]3[\/latex] and a blue arrow drawn to the right to contain all the numbers greater than [latex]3[\/latex].<\/p>\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\n<p>The graph of [latex]x\\le4[\/latex] has a closed circle at 4 and a red arrow to the left to contain all the numbers less than [latex]4[\/latex].<\/p>\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\n<p>What do you notice about the graph that combines these two inequalities?<\/p>\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\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 [latex]3[\/latex] and less than or equal to [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\" src=\"https:\/\/www.youtube.com\/embed\/nKarzhZOFIk?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Draw a graph of the compound inequality: [latex]x\\lt5[\/latex] <em>and<\/em> [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;\">\n<p><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<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\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<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\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] since the solution is between [latex]\u22121[\/latex] and [latex]5[\/latex], including [latex]\u22121[\/latex]. You read [latex]\u22121\\le x\\lt{5}[\/latex] as \u201c<i>x<\/i> is greater than or equal to [latex]\u22121[\/latex] <i>and<\/i> less than [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>Example<\/h3>\n<p>Draw the graph of the compound inequality [latex]2x+1\\lt{-5}[\/latex] <em>and<\/em> [latex]4x-3\\gt{9}[\/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;\">\n<p><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, we must solve each inequality for [latex]x[\/latex]:<br \/>\n[latex]2x+1\\lt{-5}[\/latex] <i>and<\/i> [latex]4x-3\\gt{9}[\/latex]<br \/>\n[latex]2x\\lt{-6}[\/latex] <i>and<\/i> [latex]4x\\gt{12}[\/latex]<br \/>\n[latex]x\\lt{-3}[\/latex] <i>and<\/i> [latex]x\\gt{3}[\/latex]<br \/>\nNow, draw a graph. We are looking for values for <em>x<\/em> that will satisfy <strong>both <\/strong>inequalities since they are joined with the word <em>and<\/em>.<\/p>\n<div class=\"wp-nocaption aligncenter\"><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\" \/><\/div>\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 \u201cno solution,\u201d 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> as well as write the corresponding intervals.<\/p>\n<p><iframe loading=\"lazy\" src=\"https:\/\/www.youtube.com\/embed\/LP3fsZNjJkc?feature=oembed&amp;rel=0\" width=\"500\" height=\"281\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n","protected":false},"author":264444,"menu_order":25,"template":"","meta":{"_candela_citation":"[]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-18649","chapter","type-chapter","status-publish","hentry"],"part":18142,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18649","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/users\/264444"}],"version-history":[{"count":24,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18649\/revisions"}],"predecessor-version":[{"id":18720,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18649\/revisions\/18720"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/parts\/18142"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapters\/18649\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/media?parent=18649"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/pressbooks\/v2\/chapter-type?post=18649"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/contributor?post=18649"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/wp-json\/wp\/v2\/license?post=18649"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}