{"id":376,"date":"2016-06-01T20:49:48","date_gmt":"2016-06-01T20:49:48","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/intermediatealgebra\/?post_type=chapter&#038;p=376"},"modified":"2016-10-03T20:56:44","modified_gmt":"2016-10-03T20:56:44","slug":"outcome-solve-single-and-multi-step-inequalities-2","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tallahassee-intermediatealgebra\/chapter\/outcome-solve-single-and-multi-step-inequalities-2\/","title":{"raw":"Single- and Multi-Step Inequalities","rendered":"Single- and Multi-Step Inequalities"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Represent inequalities on a number line<\/li>\r\n \t<li>Represent inequalities using interval notation<\/li>\r\n \t<li>Use the addition and multiplication properties to solve algebraic inequalities<\/li>\r\n \t<li>Express solutions to inequalities graphically, with interval notation, and as an inequality<\/li>\r\n<\/ul>\r\n<\/div>\r\nSometimes there is a range of possible values to describe a situation. When you see a sign that says \u201cSpeed Limit 25,\u201d you know that it doesn\u2019t mean that you have to drive exactly at a speed of 25 miles per hour (mph). This sign means that you are not supposed to go faster than 25 mph, but there are many legal speeds you could drive, such as 22 mph, 24.5 mph or 19 mph. In a situation like this, which has more than one acceptable value, inequalities are used to represent the situation rather than equations.\r\n\r\nSolving multi-step inequalities is very similar to solving equations\u2014what you do to one side you need to do to the other side in order to maintain the \u201cbalance\u201d of the inequality. The <strong>Properties of Inequality<\/strong> can help you understand how to add, subtract, multiply, or divide within an inequality.\r\n<h2>Represent inequalities on a number line<\/h2>\r\nFirst, let's define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right\u2014just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways.\r\n\r\nThe first way you are probably familiar with\u2014the basic inequality. For example:\r\n<ul>\r\n \t<li>[latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than 9. Would you rather write\u00a0[latex]{x}\\lt{9}[\/latex] or try to list all the possible numbers that are less than 9? (hopefully, your answer is no)<\/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\nNote how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.\r\n\r\nFor example:\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 second way is with a graph using the 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\nAnd the third way is with an interval.\r\n\r\nWe will explore the second and third ways in depth in this section. Again, those three ways to write solutions to inequalities are:\r\n<ul>\r\n \t<li>an inequality<\/li>\r\n \t<li>an interval<\/li>\r\n \t<li>a graph<\/li>\r\n<\/ul>\r\n<h3>Inequality Signs<\/h3>\r\nThe box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it's easy to get tangled up in inequalities, just remember to read them from left to right.\r\n<table>\r\n<thead>\r\n<tr>\r\n<th>Symbol<\/th>\r\n<th>Words<\/th>\r\n<th>Example<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]\\neq [\/latex]<\/td>\r\n<td>not equal to<\/td>\r\n<td>[latex]{2}\\neq{8}[\/latex], <i>2<\/i>\u00a0<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\nInequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs. \u00a0Graphs are a very helpful way to visualize information - especially when that information represents an infinite list of numbers!\r\n\r\n[latex]x\\leq -4[\/latex]. This translates to all the real numbers on a number line that are less than or equal to 4.\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 point is part of the solution. An open circle is used for <i>greater than<\/i> (&gt;) or <i>less than<\/i> (&lt;). The point is <i>not <\/i>part of the solution.\r\n\r\nThe graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex] \\displaystyle x\\geq -3[\/latex] shown above, the end point is [latex]\u22123[\/latex], represented with a closed circle since the inequality is <i>greater than or equal to<\/i> [latex]\u22123[\/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]\u22123[\/latex]. The arrow at the end indicates that the solutions continue infinitely.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nGraph the\u00a0inequality [latex]x\\ge 4[\/latex]\r\n[reveal-answer q=\"797241\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"797241\"]\r\n\r\nWe can use a number line as shown. Because the values for <em>x<\/em> include 4, we place a solid dot on the number line at 4.\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 4.\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\" data-media-type=\"image\/jpg\" \/>\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 and inequality describing all the real numbers on the number line that are less than 2, 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]\\left(-\\infty,-2\\right)[\/latex]\r\n\r\nTo draw the graph, place an open dot on the number line first, 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>\u00a0interval notation.\u00a0<\/strong>With this convention, sets are built\u00a0with parentheses or brackets, each having a distinct meaning. The solutions to [latex]x\\geq 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This method is widely used and will be present in other math courses you may take.\r\n\r\nThe main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be \"equaled.\" A few examples of an <strong>interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. The table below\u00a0outlines the possibilities. Remember to read inequalities from left to right, just like text.\r\n\r\nThe table below describes all the possible inequalities that can occur and how to write them using interval notation, where <em>a<\/em> and <em>b<\/em> are real numbers.\r\n<table style=\"border: 1px dashed #bbbbbb; min-width: 50%; max-width: 100%; margin-top: 1.5em; margin-bottom: 1.5em; border-collapse: collapse; text-align: left; font-size: 0.9em; color: #333333; font-family: Georgia, 'Times New Roman', 'Bitstream Charter', Times, serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;\" summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th>Inequality<\/th>\r\n<th>Words<\/th>\r\n<th>Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\r\n<td>all real numbers between\u00a0<em>a<\/em> and <em>b<\/em>, not including a and b<\/td>\r\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\gt{a}[\/latex]<\/td>\r\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\r\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\lt{b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\ge{a}[\/latex]<\/td>\r\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\le{b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\r\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\r\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\r\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\r\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers<\/td>\r\n<td>All real numbers<\/td>\r\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nDescribe the inequality [latex]x\\ge 4[\/latex] using interval notation\r\n[reveal-answer q=\"817362\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"817362\"]\r\n\r\nThe solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex].\r\n\r\nNote the use of a bracket on the left because 4 is included in the solution set.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].\r\n[reveal-answer q=\"961990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"961990\"]\r\n\r\nUse a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\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\u00a0asked 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[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"15120\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"15120\"]\r\n\r\nWe will draw the graph first.\r\n\r\nThe interval reads \"all real numbers less\u00a0than 10,\" so we will start by placing an open\u00a0dot on 10 and drawing a line to the left\u00a0with an arrow indicating the solution continues to negative infinity.\r\n\r\n<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\" \/>\r\n\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[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see examples of how to write inequalities in the three ways presented here: as an inequality, in interval notation, and with a graph.\r\n\r\nhttps:\/\/youtu.be\/X0xrHKgbDT0\r\n<h2 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.\u00a0Because 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 similarly to but not exactly as we treat equalities. 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\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\u00a0property 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>\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[reveal-answer q=\"432848\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"432848\"]\r\n<ol>\r\n \t<li>[latex]\\begin{array}{l}3x&lt;6\\hfill \\\\ \\frac{1}{3}\\left(3x\\right)&lt;\\left(6\\right)\\frac{1}{3}\\hfill \\\\ x&lt;2\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}-2x - 1\\ge 5\\hfill &amp; \\hfill \\\\ -2x\\ge 6\\hfill &amp; \\hfill \\\\ \\left(-\\frac{1}{2}\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\frac{1}{2}\\right)\\hfill &amp; \\text{Multiply by }-\\frac{1}{2}.\\hfill \\\\ x\\le -3\\hfill &amp; \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}5-x&gt;10\\hfill &amp; \\hfill \\\\ -x&gt;5\\hfill &amp; \\hfill \\\\ \\left(-1\\right)\\left(-x\\right)&gt;\\left(5\\right)\\left(-1\\right)\\hfill &amp; \\text{Multiply by }-1.\\hfill \\\\ x&lt;-5\\hfill &amp; \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\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[reveal-answer q=\"399605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"399605\"]\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\n<ol style=\"list-style-type: lower-alpha;\">\r\n \t<li>[latex]\\begin{array}{ll}x - 15&lt;4\\hfill &amp; \\hfill \\\\ x - 15+15&lt;4+15 \\hfill &amp; \\text{Add 15 to both sides.}\\hfill \\\\ x&lt;19\\hfill &amp; \\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}6\\ge x - 1\\hfill &amp; \\hfill \\\\ 6+1\\ge x - 1+1\\hfill &amp; \\text{Add 1 to both sides}.\\hfill \\\\ 7\\ge x\\hfill &amp; \\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}x+7&gt;9\\hfill &amp; \\hfill \\\\ x+7 - 7&gt;9 - 7\\hfill &amp; \\text{Subtract 7 from both sides}.\\hfill \\\\ x&gt;2\\hfill &amp; \\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows examples of solving single-step inequalities using the multiplication and addition properties.\r\nhttps:\/\/youtu.be\/1Z22Xh66VFM\r\nThe following video show examples of solving inequalities with the variable on the right side.\r\nhttps:\/\/youtu.be\/RBonYKvTCLU\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,\u00a0we 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\r\n[reveal-answer q=\"532189\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"532189\"]Solving this inequality is similar to solving an equation up until the last step.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}13 - 7x\\ge 10x - 4\\hfill &amp; \\hfill \\\\ 13 - 17x\\ge -4\\hfill &amp; \\text{Move variable terms to one side of the inequality}.\\hfill \\\\ -17x\\ge -17\\hfill &amp; \\text{Isolate the variable term}.\\hfill \\\\ 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[\/hidden-answer]\r\n\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]-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x[\/latex].\r\n\r\n[reveal-answer q=\"59887\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"59887\"]\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}{ll}-\\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 ,\\frac{15}{34}\\right][\/latex].<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/div>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Represent inequalities on a number line<\/li>\n<li>Represent inequalities using interval notation<\/li>\n<li>Use the addition and multiplication properties to solve algebraic inequalities<\/li>\n<li>Express solutions to inequalities graphically, with interval notation, and as an inequality<\/li>\n<\/ul>\n<\/div>\n<p>Sometimes there is a range of possible values to describe a situation. When you see a sign that says \u201cSpeed Limit 25,\u201d you know that it doesn\u2019t mean that you have to drive exactly at a speed of 25 miles per hour (mph). This sign means that you are not supposed to go faster than 25 mph, but there are many legal speeds you could drive, such as 22 mph, 24.5 mph or 19 mph. In a situation like this, which has more than one acceptable value, inequalities are used to represent the situation rather than equations.<\/p>\n<p>Solving multi-step inequalities is very similar to solving equations\u2014what you do to one side you need to do to the other side in order to maintain the \u201cbalance\u201d of the inequality. The <strong>Properties of Inequality<\/strong> can help you understand how to add, subtract, multiply, or divide within an inequality.<\/p>\n<h2>Represent inequalities on a number line<\/h2>\n<p>First, let&#8217;s define some important terminology. An inequality is a mathematical statement that compares two expressions using the ideas of greater than or less than. Special symbols are used in these statements. When you read an inequality, read it from left to right\u2014just like reading text on a page. In algebra, inequalities are used to describe large sets of solutions. Sometimes there are an infinite amount of numbers that will satisfy an inequality, so rather than try to list off an infinite amount of numbers, we have developed some ways to describe very large lists in succinct ways.<\/p>\n<p>The first way you are probably familiar with\u2014the basic inequality. For example:<\/p>\n<ul>\n<li>[latex]{x}\\lt{9}[\/latex] indicates the list of numbers that are less than 9. Would you rather write\u00a0[latex]{x}\\lt{9}[\/latex] or try to list all the possible numbers that are less than 9? (hopefully, your answer is no)<\/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>Note how placing the variable on the left or right of the inequality sign can change whether you are looking for greater than or less than.<\/p>\n<p>For example:<\/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 second way is with a graph using the 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>And the third way is with an interval.<\/p>\n<p>We will explore the second and third ways in depth in this section. Again, those three ways to write solutions to inequalities are:<\/p>\n<ul>\n<li>an inequality<\/li>\n<li>an interval<\/li>\n<li>a graph<\/li>\n<\/ul>\n<h3>Inequality Signs<\/h3>\n<p>The box below shows the symbol, meaning, and an example for each inequality sign. Sometimes it&#8217;s easy to get tangled up in inequalities, just remember to read them from left to right.<\/p>\n<table>\n<thead>\n<tr>\n<th>Symbol<\/th>\n<th>Words<\/th>\n<th>Example<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]\\neq[\/latex]<\/td>\n<td>not equal to<\/td>\n<td>[latex]{2}\\neq{8}[\/latex], <i>2<\/i>\u00a0<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>Inequalities can also be graphed on a number line. Below are three examples of inequalities and their graphs. \u00a0Graphs are a very helpful way to visualize information &#8211; especially when that information represents an infinite list of numbers!<\/p>\n<p>[latex]x\\leq -4[\/latex]. This translates to all the real numbers on a number line that are less than or equal to 4.<\/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 point is part of the solution. An open circle is used for <i>greater than<\/i> (&gt;) or <i>less than<\/i> (&lt;). The point is <i>not <\/i>part of the solution.<\/p>\n<p>The graph then extends endlessly in one direction. This is shown by a line with an arrow at the end. For example, notice that for the graph of [latex]\\displaystyle x\\geq -3[\/latex] shown above, the end point is [latex]\u22123[\/latex], represented with a closed circle since the inequality is <i>greater than or equal to<\/i> [latex]\u22123[\/latex]. The blue line is drawn to the right on the number line because the values in this area are greater than [latex]\u22123[\/latex]. The arrow at the end indicates that the solutions continue infinitely.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Graph the\u00a0inequality [latex]x\\ge 4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797241\">Show Solution<\/span><\/p>\n<div id=\"q797241\" class=\"hidden-answer\" style=\"display: none\">\n<p>We can use a number line as shown. Because the values for <em>x<\/em> include 4, we place a solid dot on the number line at 4.<\/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 4.<\/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\" data-media-type=\"image\/jpg\" \/><\/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 and inequality describing all the real numbers on the number line that are less than 2, 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]\\left(-\\infty,-2\\right)[\/latex]<\/p>\n<p>To draw the graph, place an open dot on the number line first, 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>\u00a0interval notation.\u00a0<\/strong>With this convention, sets are built\u00a0with parentheses or brackets, each having a distinct meaning. The solutions to [latex]x\\geq 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This method is widely used and will be present in other math courses you may take.<\/p>\n<p>The main concept to remember is that parentheses represent solutions greater or less than the number, and brackets represent solutions that are greater than or equal to or less than or equal to the number. Use parentheses to represent infinity or negative infinity, since positive and negative infinity are not numbers in the usual sense of the word and, therefore, cannot be &#8220;equaled.&#8221; A few examples of an <strong>interval<\/strong>, or a set of numbers in which a solution falls, are [latex]\\left[-2,6\\right)[\/latex], or all numbers between [latex]-2[\/latex] and [latex]6[\/latex], including [latex]-2[\/latex], but not including [latex]6[\/latex]; [latex]\\left(-1,0\\right)[\/latex], all real numbers between, but not including [latex]-1[\/latex] and [latex]0[\/latex]; and [latex]\\left(-\\infty,1\\right][\/latex], all real numbers less than and including [latex]1[\/latex]. The table below\u00a0outlines the possibilities. Remember to read inequalities from left to right, just like text.<\/p>\n<p>The table below describes all the possible inequalities that can occur and how to write them using interval notation, where <em>a<\/em> and <em>b<\/em> are real numbers.<\/p>\n<table style=\"border: 1px dashed #bbbbbb; min-width: 50%; max-width: 100%; margin-top: 1.5em; margin-bottom: 1.5em; border-collapse: collapse; text-align: left; font-size: 0.9em; color: #333333; font-family: Georgia, 'Times New Roman', 'Bitstream Charter', Times, serif; font-style: normal; font-variant-ligatures: normal; font-variant-caps: normal; font-weight: normal; letter-spacing: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-stroke-width: 0px;\" summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\n<thead>\n<tr>\n<th>Inequality<\/th>\n<th>Words<\/th>\n<th>Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex]{a}\\lt{x}\\lt{ b}[\/latex]<\/td>\n<td>all real numbers between\u00a0<em>a<\/em> and <em>b<\/em>, not including a and b<\/td>\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\gt{a}[\/latex]<\/td>\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\lt{b}[\/latex]<\/td>\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\ge{a}[\/latex]<\/td>\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\le{b}[\/latex]<\/td>\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\le{x}\\lt{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\lt{x}\\le{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{a}\\le{x}\\le{ b}[\/latex]<\/td>\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, including <em>a <\/em>and <em>b<\/em><\/td>\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex]{x}\\lt{a}\\text{ or }{x}\\gt{ b}[\/latex]<\/td>\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\n<td>[latex]\\left(-\\infty ,a\\right)\\cup \\left(b,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers<\/td>\n<td>All real numbers<\/td>\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Describe the inequality [latex]x\\ge 4[\/latex] using interval notation<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q817362\">Show Solution<\/span><\/p>\n<div id=\"q817362\" class=\"hidden-answer\" style=\"display: none\">\n<p>The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex].<\/p>\n<p>Note the use of a bracket on the left because 4 is included in the solution set.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q961990\">Show Solution<\/span><\/p>\n<div id=\"q961990\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<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\u00a0asked 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><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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 &#8220;all real numbers less\u00a0than 10,&#8221; so we will start by placing an open\u00a0dot on 10 and drawing a line to the left\u00a0with an arrow indicating the solution continues to negative infinity.<\/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\/01182811\/4.png\" alt=\"An open circle on 10 and a line going from 10 to all numbers below 10.\" width=\"339\" height=\"95\" \/><\/p>\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\" id=\"oembed-2\" title=\"Ex: Graph Basic Inequalities and Express Using Interval Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/X0xrHKgbDT0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 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.\u00a0Because 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 similarly to but not exactly as we treat equalities. 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.<br \/>\nThe 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\u00a0property 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>\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\"><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<ol>\n<li>[latex]\\begin{array}{l}3x<6\\hfill \\\\ \\frac{1}{3}\\left(3x\\right)<\\left(6\\right)\\frac{1}{3}\\hfill \\\\ x<2\\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}-2x - 1\\ge 5\\hfill & \\hfill \\\\ -2x\\ge 6\\hfill & \\hfill \\\\ \\left(-\\frac{1}{2}\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\frac{1}{2}\\right)\\hfill & \\text{Multiply by }-\\frac{1}{2}.\\hfill \\\\ x\\le -3\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}5-x>10\\hfill & \\hfill \\\\ -x>5\\hfill & \\hfill \\\\ \\left(-1\\right)\\left(-x\\right)>\\left(5\\right)\\left(-1\\right)\\hfill & \\text{Multiply by }-1.\\hfill \\\\ x<-5\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\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\"><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\">\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.<\/p>\n<ol style=\"list-style-type: lower-alpha;\">\n<li>[latex]\\begin{array}{ll}x - 15<4\\hfill & \\hfill \\\\ x - 15+15<4+15 \\hfill & \\text{Add 15 to both sides.}\\hfill \\\\ x<19\\hfill & \\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}6\\ge x - 1\\hfill & \\hfill \\\\ 6+1\\ge x - 1+1\\hfill & \\text{Add 1 to both sides}.\\hfill \\\\ 7\\ge x\\hfill & \\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}x+7>9\\hfill & \\hfill \\\\ x+7 - 7>9 - 7\\hfill & \\text{Subtract 7 from both sides}.\\hfill \\\\ x>2\\hfill & \\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows examples of solving single-step inequalities using the multiplication and addition properties.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-3\" title=\"Ex:  Solving One Step Inequalities by Adding and Subtracting (Variable Left Side)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/1Z22Xh66VFM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\nThe following video show examples of solving inequalities with the variable on the right side.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-4\" title=\"Ex:  Solving One Step Inequalities by Adding and Subtracting (Variable Right Side)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/RBonYKvTCLU?feature=oembed&#38;rel=0\" 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,\u00a0we 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\"><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\">Solving this inequality is similar to solving an equation up until the last step.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}13 - 7x\\ge 10x - 4\\hfill & \\hfill \\\\ 13 - 17x\\ge -4\\hfill & \\text{Move variable terms to one side of the inequality}.\\hfill \\\\ -17x\\ge -17\\hfill & \\text{Isolate the variable term}.\\hfill \\\\ 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<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]-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><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}{ll}-\\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 ,\\frac{15}{34}\\right][\/latex].<\/div>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-376\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Revision and Adaptation. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Graph Linear Inequalities in One Variable (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-kiAeGbSe5c\">https:\/\/youtu.be\/-kiAeGbSe5c<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><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><li>Ex: Graph Basic Inequalities and Express Using Interval Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/X0xrHKgbDT0\">https:\/\/youtu.be\/X0xrHKgbDT0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>College Algebra. <strong>Authored by<\/strong>: Jay Abramson, et al.. <strong>Provided by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/courses.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>Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/1Z22Xh66VFM\">https:\/\/youtu.be\/1Z22Xh66VFM<\/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>Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Right Side). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) 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