{"id":3407,"date":"2016-05-02T19:41:14","date_gmt":"2016-05-02T19:41:14","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/nrocarithmetic\/?post_type=chapter&#038;p=3407"},"modified":"2018-01-03T23:51:57","modified_gmt":"2018-01-03T23:51:57","slug":"read-solve-single-step-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-beginalgebra\/chapter\/read-solve-single-step-inequalities\/","title":{"raw":"Solve Inequalities","rendered":"Solve Inequalities"},"content":{"raw":"<div class=\"bcc-box bcc-highlight\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Describe solutions to inequalities\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<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<ul>\r\n \t<li>Solve single-step inequalities\r\n<ul>\r\n \t<li>Use the addition and multiplication properties to solve algebraic inequalities and express their solutions graphically and with interval notation<\/li>\r\n \t<li>Solve inequalities that contain absolute value<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Solve multi-step inequalities\r\n<ul>\r\n \t<li>Combine properties of inequality to isolate variables,\u00a0solve algebraic\u00a0inequalities, and express their solutions graphically<\/li>\r\n \t<li>Simplify and solve algebraic inequalities using the distributive property to clear\u00a0parentheses and fractions<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2 id=\"title1\">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\/117\/2016\/05\/09234719\/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\" 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\" 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<img class=\"size-full wp-image-5875 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/08\/01202558\/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 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]x&lt;2[\/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\" 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\nThe following video shows how to write an inequality mathematically when it is given in words. We will then graph it.\r\nhttps:\/\/youtu.be\/E_ZWNVNEvOg\r\n<h2 id=\"title2\">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\nIn the following video we show another example of using interval notation to describe an inequality.\r\nhttps:\/\/youtu.be\/BKhDzNKjVBc\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nUse interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].\r\n[reveal-answer q=\"961990\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"961990\"]\r\n\r\nUse a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video we show another example of translating words into an inequality and writing it in interval notation, as well as drawing the graph.\r\nhttps:\/\/youtu.be\/OYkQ-McI2qg\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=\"size-full wp-image-5876 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/08\/01202657\/4.png\" alt=\"An open circle on 10 and a line going from 10 to all numbers below 10.\" width=\"271\" height=\"76\" \/>\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 draw a graph given an inequality in interval notation.\r\n\r\nhttps:\/\/youtu.be\/lkhILNEPbfk\r\n\r\nAnd finally, one last video that shows how to write inequalities using a graph, with interval notation and as an inequality.\r\n\r\nhttps:\/\/youtu.be\/X0xrHKgbDT0\r\n<h2>Solve Single-Step Inequalities<\/h2>\r\n<h3 id=\"title3\">Solve inequalities with addition and subtraction<\/h3>\r\nYou can solve most inequalities using inverse operations \u00a0as you did for solving equations. \u00a0This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the box below.\r\n<div class=\"textbox shaded\">\r\n<h3>Addition and Subtraction Properties of Inequality<\/h3>\r\nIf [latex]a&gt;b[\/latex],<i>\u00a0<\/i>then [latex]a+c&gt;b+c[\/latex].\r\n\r\nIf\u00a0[latex]a&gt;b[\/latex]<i>, <\/i>then [latex]a\u2212c&gt;b\u2212c[\/latex].\r\n\r\n<\/div>\r\nBecause inequalities have multiple possible solutions, representing the solutions graphically provides a helpful visual of the situation, as we saw in the last section. The example below shows the steps to solve and graph an inequality and express the solution using interval notation.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x.<\/i>\r\n<p style=\"text-align: center;\">[latex] {x}+3\\lt{5}[\/latex]<\/p>\r\n[reveal-answer q=\"952771\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"952771\"]\r\n\r\nIt is helpful to think of this inequality as asking you to find all the values for <em>x<\/em>, including negative numbers, such that when you add three you will get a number less than 5.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}x+3&lt;\\,\\,\\,\\,5\\\\\\underline{\\,\\,\\,\\,\\,-3\\,\\,\\,\\,-3}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,&lt;\\,\\,\\,\\,2\\,\\,\\end{array}[\/latex]<\/p>\r\nIsolate the variable by subtracting 3 from both sides of the inequality.\r\n<h4>Answer<\/h4>\r\nInequality: \u00a0[latex]x&lt;2[\/latex]\r\n\r\nInterval: \u00a0[latex]\\left(-\\infty, 2\\right)[\/latex]\r\n\r\nGraph: <img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image036.jpg#fixme\" alt=\"Number line. Open circle around 2. Shaded line through all numbers less than 2.\" width=\"575\" height=\"31\" \/>[\/hidden-answer]\r\n\r\n<\/div>\r\n<p style=\"text-align: left;\">The line represents <em>all<\/em> the numbers to which\u00a0you can add 3 and get a number that is less than 5. There's a lot of numbers that solve this inequality!<\/p>\r\nJust as you can check the solution to an equation, you can check a solution to an inequality. First, you check the end point by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions. This can also help you check that your graph is correct.\r\n\r\nThe example below shows how you could check that [latex]x&lt;2[\/latex]<i>\u00a0<\/i>is the solution to [latex]x+3&lt;5[\/latex]<i>.<\/i>\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nCheck that [latex]x&lt;2[\/latex]<i>\u00a0<\/i>is the solution to [latex]x+3&lt;5[\/latex].\r\n\r\n[reveal-answer q=\"811564\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811564\"]\r\n\r\nSubstitute the end point 2 into the related equation, [latex]x+3=5[\/latex].\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x+3=5 \\\\ 2+3=5 \\\\ 5=5\\end{array}[\/latex]<\/p>\r\nPick a value less than 2, such as 0, to check into the inequality. (This value will be on the shaded part of the graph.)\r\n<p style=\"text-align: center;\"><img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image036.jpg#fixme\" alt=\"Number line. Open circle around 2. Shaded line through all numbers less than 2.\" width=\"575\" height=\"31\" \/><\/p>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}x+3&lt;5 \\\\ 0+3&lt;5 \\\\ 3&lt;5\\end{array}[\/latex]<\/p>\r\nIt checks!\r\n\r\n[latex]x&lt;2[\/latex] is the solution to [latex]x+3&lt;5[\/latex].[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following examples show inequality problems that include operations with negative numbers. The graph of the solution to the inequality is also shown. Remember to check the solution. This is a good habit to build!\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>:\u00a0[latex]x-10\\leq-12[\/latex]\r\n[reveal-answer q=\"815894\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815894\"]\r\n\r\nIsolate the variable by adding 10 to both sides of the inequality.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}x-10\\le -12\\\\\\underline{\\,\\,\\,+10\\,\\,\\,\\,\\,+10}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,-2\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]x\\leq-2[\/latex]\r\nInterval: [latex]\\left(-\\infty,-2\\right][\/latex]\r\nGraph: Notice that a closed circle is used because the inequality is \u201cless than or equal to\u201d [latex]\\left(\\leq\\right)[\/latex]. The blue arrow is drawn to the left of the point [latex]\u22122[\/latex] because these are the values that are less than [latex]\u22122[\/latex].\r\n<img class=\"aligncenter wp-image-3619\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/06184100\/image038-300x17.jpg\" alt=\"Number line, closed circle on negative 2 and line drawn through all numbers less than negative 2\" width=\"529\" height=\"30\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the\u00a0solution to [latex]x-10\\leq -12[\/latex]\r\n[reveal-answer q=\"268062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"268062\"]\r\n\r\nSubstitute the end point [latex]\u22122[\/latex] into the related equation \u00a0[latex]x-10=\u221212[\/latex]\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}x-10=-12\\,\\,\\,\\\\\\text{Does}\\,\\,\\,-2-10=-12?\\\\-12=-12\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value less than [latex]\u22122[\/latex], such as [latex]\u22125[\/latex], to check in the inequality. (This value will be on the shaded part of the graph.)\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}x-10\\le -12\\,\\,\\,\\\\\\text{ }\\,\\text{ Is}\\,\\,-5-10\\le -12?\\\\-15\\le -12\\,\\,\\,\\\\\\text{It}\\,\\text{checks!}\\end{array}[\/latex]<\/p>\r\n[latex]x\\leq -2[\/latex]\u00a0is the solution to [latex]x-10\\leq -12[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for <em>a<\/em>. [latex]a-17&gt;-17[\/latex]\r\n[reveal-answer q=\"343031\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"343031\"]\r\n\r\nIsolate the variable by adding 17 to both sides of the inequality.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}a-17&gt;-17\\\\\\underline{\\,\\,\\,+17\\,\\,\\,\\,\\,+17}\\\\a\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&gt;\\,\\,\\,\\,\\,\\,0\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality:\u00a0[latex] \\displaystyle a\\,\\,&gt;\\,0[\/latex]\r\n\r\nInterval: [latex]\\left(0,\\infty\\right)[\/latex] \u00a0Note how we use parentheses on the left to show that the solution does not include 0.\r\n\r\nGraph: Note the open circle to show that the solution does not include 0.\r\n<div class=\"bcc-box bcc-info\">\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image044.jpg#fixme\" alt=\"Number line. Open circle on zero. Highlight through all numbers above zero.\" width=\"575\" height=\"32\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution to\u00a0[latex]a-17&gt;-17[\/latex]\r\n[reveal-answer q=\"653357\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"653357\"]\r\n\r\nIs\u00a0[latex] \\displaystyle a\\,\\,&gt;\\,0[\/latex] the correct solution to\u00a0\u00a0[latex]a-17&gt;-17[\/latex]?\r\n\r\nSubstitute the end point 0 into the related equation.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}a-17=-17\\,\\,\\,\\\\\\text{Does}\\,\\,\\,0-17=-17?\\\\-17=-17\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value greater than 0, such as 20, to check in the inequality. (This value will be on the shaded part of the graph.)\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}a-17&gt;-17\\,\\,\\,\\\\\\text{Is }\\,\\,20-17&gt;-17?\\\\3&gt;-17\\,\\,\\,\\\\\\\\\\text{It checks!}\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex] \\displaystyle a\\,&gt;\\,0[\/latex] is the solution to\u00a0[latex]a-17&gt;-17[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\nThe previous examples showed you how to solve a one-step inequality with the variable on the left hand side. \u00a0The following video provides examples of how to solve the same type of inequality.\r\n\r\nhttps:\/\/youtu.be\/1Z22Xh66VFM\r\n\r\nWhat would you do if the variable were on the right side of the inequality? \u00a0In the following example, you will see how to handle this scenario.\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>:\u00a0[latex]4\\geq{x}+5[\/latex]\r\n[reveal-answer q=\"815893\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"815893\"]\r\n\r\nIsolate the variable by adding 10 to both sides of the inequality.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}4\\geq{x}+5 \\\\\\underline{\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\,\\,\\,x\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Rewrite the inequality with the variable on the left - this makes writing the interval and drawing the graph easier.<\/p>\r\n<p style=\"text-align: center;\">[latex]x\\le{-1}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Note how the the pointy part of the inequality is still directed at the variable, so instead of reading as negative one is greater or equal to x, it now reads as x is less than or equal to negative one.<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]x\\le{-1}[\/latex] This can also be written as\r\nInterval: [latex]\\left(-\\infty,-1\\right][\/latex]\r\nGraph: Notice that a closed circle is used because the inequality is \u201cless than or equal to\u201d . The blue arrow is drawn to the left of the point [latex]\u22121[\/latex] because these are the values that are less than [latex]\u22121[\/latex].\r\n<img class=\" wp-image-4022 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/12012430\/Screen-Shot-2016-05-11-at-6.23.24-PM-300x57.png\" alt=\"(-oo,-1]\" width=\"400\" height=\"76\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the\u00a0solution to [latex]4\\geq{x}+5[\/latex]\r\n[reveal-answer q=\"568062\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"568062\"]\r\n\r\nSubstitute the end point [latex]\u22121[\/latex] into the related equation \u00a0[latex]4=x+5[\/latex]\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}4=x+5\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4=-1+5?\\\\-1=-1\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nPick a value less than [latex]\u22121[\/latex], such as [latex]\u22125[\/latex], to check in the inequality. (This value will be on the shaded part of the graph.)\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}4\\geq{-5}+5\\,\\,\\,\\\\\\text{ }\\,\\text{ Is}\\,\\,4\\ge 0?\\\\\\text{It}\\,\\text{checks!}\\end{array}[\/latex]<\/p>\r\n[latex]x\\le{-1}[\/latex] is the solution to [latex]4\\geq{x}+5[\/latex]<span style=\"line-height: 1.5;\">[\/hidden-answer] <\/span>\r\n\r\n<\/div>\r\nThe following video show examples of solving inequalities with the variable on the right side.\r\nhttps:\/\/youtu.be\/RBonYKvTCLU\r\n<h3 id=\"title4\">Solve inequalities with multiplication and division<\/h3>\r\nSolving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let\u2019s look at what happens to the inequality when you multiply or divide each side by the same number.\r\n<table>\r\n<tbody>\r\n<tr>\r\n<td>Let's start with the true statement:\r\n\r\n[latex]10&gt;5[\/latex]<\/td>\r\n<td>Let's try again by starting with the same true statement:\r\n\r\n[latex]10&gt;5[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>Next, multiply both sides by the same positive number:\r\n\r\n[latex]10\\cdot 2&gt;5\\cdot 2[\/latex]<\/td>\r\n<td>This time, multiply both sides by the same negative number:\r\n\r\n[latex]10\\cdot-2&gt;5 \\\\ \\,\\,\\,\\,\\,\\cdot -2\\,\\cdot-2[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>20 is greater than 10, so you still have a true inequality:\r\n\r\n[latex]20&gt;10[\/latex]<\/td>\r\n<td>Wait a minute! [latex]\u221220[\/latex] is <i>not <\/i>greater than [latex]\u221210[\/latex], so you have an untrue statement.\r\n\r\n[latex]\u221220&gt;\u221210[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>When you multiply by a positive number, leave the inequality sign as it is!<\/td>\r\n<td>You must \u201creverse\u201d the inequality sign to make the statement true:\r\n\r\n[latex]\u221220&lt;\u221210[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"textbox shaded\">\r\n\r\n<img class=\" wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"80\" height=\"70\" \/>Caution! \u00a0When you multiply or divide by a negative number, \u201creverse\u201d the inequality sign. \u00a0 Whenever you multiply or divide both sides of an inequality by a negative\u00a0number, the inequality sign must be reversed in order to keep a true statement. These rules are summarized in the box below.\r\n\r\n<\/div>\r\n\r\n<hr \/>\r\n\r\n<div class=\"textbox shaded\">\r\n<h3>Multiplication and Division Properties of Inequality<\/h3>\r\n<table style=\"height: 162px;\" width=\"419\">\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]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<\/tbody>\r\n<\/table>\r\n<table style=\"height: 77px;\" width=\"418\">\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]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<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<\/div>\r\nKeep in mind that you only change the sign when you are multiplying and dividing by a <i>negative<\/i> number. If you <em>add or subtract<\/em> by a negative\u00a0number, the inequality stays the same.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x.\u00a0<\/i>[latex]3x&gt;12[\/latex]\r\n\r\n[reveal-answer q=\"691711\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"691711\"]Divide both sides by 3 to isolate the variable.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\underline{3x}&gt;\\underline{12}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\\\x&gt;4\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nCheck your solution by first checking the end point 4, and then checking another solution for the inequality.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3\\cdot4=12\\\\12=12\\\\3\\cdot10&gt;12\\\\30&gt;12\\\\\\text{It checks!}\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\n<p style=\"text-align: left;\">Inequality: [latex] \\displaystyle x&gt;4[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Interval: [latex]\\left(4,\\infty\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Graph: <img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image050.jpg#fixme\" alt=\"Number line. Open circle on 4. Highlight through all numbers greater than 4.\" width=\"575\" height=\"31\" \/><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThere was no need to make any changes to the inequality sign because both sides of the inequality were divided by <i>positive<\/i> 3. In the next example, there is division by a negative number, so there is an additional step in the solution!\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>. [latex]\u22122x&gt;6[\/latex]\r\n\r\n[reveal-answer q=\"604033\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"604033\"]Divide each side of the inequality by [latex]\u22122[\/latex] to isolate the variable, and change the direction of the inequality sign because of the division by a negative number.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\underline{-2x}&lt;\\underline{\\,6\\,}\\\\-2\\,\\,\\,\\,-2\\,\\\\x&lt;-3\\end{array}[\/latex]<\/p>\r\nCheck your solution by first checking the end point [latex]\u22123[\/latex], and then checking another solution for the inequality.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-2\\left(-3\\right)=6 \\\\6=6\\\\ -2\\left(-6\\right)&gt;6 \\\\ 12&gt;6\\end{array}[\/latex]<\/p>\r\nIt checks!\r\n<h4>Answer<\/h4>\r\nInequality: [latex] \\displaystyle x&lt;-3[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty, -3\\right)[\/latex]\r\n\r\nGraph: <img class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image051.jpg#fixme\" alt=\"Number line. Open circle on negative 3. Highlight on all numbers less than negative 3.\" width=\"575\" height=\"31\" \/>\r\nBecause both sides of the inequality were divided by a negative number, [latex]\u22122[\/latex], the inequality symbol was switched from &gt; to &lt;.\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video shows examples of solving one step inequalities using the multiplication property of equality where the variable is on the left hand side.\r\n\r\nhttps:\/\/youtu.be\/IajiD3R7U-0\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think About It<\/h3>\r\nBefore you read the solution to the next example, think about what properties of inequalities you may need to use to solve the inequality. What is different about this example from the previous one? Write your ideas in the box below.\r\n\r\nSolve for <em>x<\/em>. [latex]-\\frac{1}{2}&gt;-12x[\/latex]\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n[reveal-answer q=\"811465\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"811465\"]\r\n\r\nThis inequality has the variable on the right hand side, which is different from the previous examples. Start the solution process as before, and at the end, you can move the variable to the left to write the final solution.\r\n\r\nDivide both sides by [latex]-12[\/latex] to isolate the variable. Since you are dividing by a negative number, you need to change the direction of the inequality sign.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\begin{array}{l}-\\frac{1}{2}\\gt{-12x}\\\\\\\\\\frac{-\\frac{1}{2}}{-12}\\gt\\frac{-12x}{-12}\\\\\\end{array}[\/latex]<\/p>\r\nDividing a fraction by an integer requires you to multiply by the reciprocal, and the reciprocal of [latex]-12[\/latex] is [latex]\\frac{1}{-12}[\/latex]\r\n<p style=\"text-align: center;\">[latex]\\displaystyle\\begin{array}{r}\\left(-\\frac{1}{12}\\right)\\left(-\\frac{1}{2}\\right)\\lt\\frac{-12x}{-12}\\,\\,\\\\\\\\ \\frac{1}{24}\\lt\\frac{\\cancel{-12}x}{\\cancel{-12}}\\\\\\\\ \\frac{1}{24}\\lt{x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n\r\n<h4 style=\"text-align: left;\">Answer<\/h4>\r\n<p style=\"text-align: left;\">Inequality: [latex]\\frac{1}{24}\\lt{x}[\/latex] \u00a0This can also be written with the variable on the left as [latex]x\\gt\\frac{1}{24}[\/latex]. \u00a0Writing the inequality with the variable on the left requires a little thinking, but helps you write the interval and draw the graph correctly.<\/p>\r\n<p style=\"text-align: left;\">Interval: [latex]\\left(\\frac{1}{24},\\infty\\right)[\/latex]<\/p>\r\n<p style=\"text-align: left;\">Graph:\u00a0<img class=\"aligncenter wp-image-3950\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10211010\/Screen-Shot-2016-05-10-at-2.09.52-PM-300x57.png\" alt=\"Open dot on zero with a line through all numbers greater than zero.\" width=\"389\" height=\"74\" \/><\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nThe following video gives examples of how to solve an inequality with the multiplication property of equality where the variable is on the right hand side.\r\n\r\nhttps:\/\/youtu.be\/s9fJOnVTHhs\r\n<h2 id=\"title1\">Combine properties of inequality to \u00a0solve algebraic\u00a0inequalities<\/h2>\r\nA popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and\/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one-step inequalities, the solutions to multi-step inequalities can be graphed on a number line.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>p<\/i>. [latex]4p+5&lt;29[\/latex]\r\n\r\n[reveal-answer q=\"211828\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"211828\"]\r\n\r\nBegin to isolate the variable by subtracting 5 from both sides of the inequality.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}4p+5&lt;\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\4p\\,\\,\\,\\,\\,\\,\\,\\,\\,&lt;\\,\\,24\\,\\,\\end{array}[\/latex]<\/p>\r\nDivide both sides of the inequality by 4 to express the variable with a coefficient of 1.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\underline{4p}\\,&lt;\\,\\,\\underline{24}\\,\\,\\\\\\,4\\,\\,\\,\\,&lt;\\,\\,4\\\\\\,\\,\\,\\,\\,p&lt;6\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality:\u00a0[latex]p&lt;6[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,6\\right)[\/latex]\r\n\r\nGraph: Note the\u00a0open circle at the end point 6 to show that solutions to the inequality do not include 6.\u00a0The values where <i>p<\/i> is less than 6 are found all along the number line to the left of 6.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04063949\/image057.jpg\" alt=\"Number line. Open circle on 6. Highlight on every number less than 6.\" width=\"575\" height=\"31\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"291597\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"291597\"]\r\n\r\nCheck the end point 6 in the related equation.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}4p+5=29\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4(6)+5=29?\\\\24+5=29\\,\\,\\,\\\\29=29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nTry another value to check the inequality. Let\u2019s use [latex]p=0[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}4p+5&lt;29\\,\\,\\,\\\\\\text{Is}\\,\\,\\,4(0)+5&lt;29?\\\\0+5&lt;29\\,\\,\\,\\\\5&lt;29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex]p&lt;6[\/latex] is the solution to\u00a0[latex]4p+5&lt;29[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example<\/h3>\r\nSolve for <i>x<\/i>: \u00a0[latex]3x\u20137\\ge 41[\/latex]\r\n[reveal-answer q=\"238157\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"238157\"]\r\n\r\nBegin to isolate the variable by adding 7 to both sides of the inequality, then divide both sides of the inequality by 3 to express the variable with a coefficient of 1.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}3x-7\\ge 41\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+7\\,\\,\\,\\,+7}\\\\\\frac{3x}{3}\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\frac{48}{3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 16\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]x\\ge 16[\/latex]\r\n\r\nInterval: [latex]\\left[16,\\infty\\right)[\/latex]\r\n\r\nGraph:\u00a0To graph this inequality, you draw a closed circle at the end point 16 on the number line\u00a0to show that solutions include the value 16. The line continues to the right from 16 because all the numbers greater than 16 will also make the inequality\u00a0[latex]3x\u20137\\ge 41[\/latex] true.\r\n<img class=\"aligncenter wp-image-3956\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10233054\/Screen-Shot-2016-05-10-at-4.28.03-PM-300x48.png\" alt=\"Closed dot on 16, line through all numbers greater than 16.\" width=\"425\" height=\"68\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"437341\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"437341\"]\r\n\r\nFirst, check the end point 16 in the related equation.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}3x-7=41\\,\\,\\,\\\\\\text{Does}\\,\\,\\,3(16)-7=41?\\\\48-7=41\\,\\,\\,\\\\41=41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen, try another value to check the inequality. Let\u2019s use [latex]x = 20[\/latex].\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\,\\,\\,\\,3x-7\\ge 41\\,\\,\\,\\\\\\text{Is}\\,\\,\\,\\,\\,3(20)-7\\ge 41?\\\\60-7\\ge 41\\,\\,\\,\\\\53\\ge 41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nWhen solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <i>p<\/i>. [latex]\u221258&gt;14\u22126p[\/latex]\r\n\r\n[reveal-answer q=\"424351\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"424351\"]\r\n\r\nNote how the variable is on the right hand side of the inequality, the method for solving does not change in this case.\r\n\r\nBegin to isolate the variable by subtracting 14 from both sides of the inequality.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}\u221258\\,\\,&gt;14\u22126p\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-14\\,\\,\\,\\,\\,\\,\\,-14}\\\\-72\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,&gt;-6p\\end{array}[\/latex]<\/p>\r\nDivide both sides of the inequality by [latex]\u22126[\/latex] to express the variable with a coefficient of 1.\u00a0Dividing by a negative number results in reversing the inequality sign.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\underline{-72}&gt;\\underline{-6p}\\\\-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\\\\\,\\,\\,\\,\\,\\,12\\lt{p}\\end{array}[\/latex]<\/p>\r\n<p style=\"text-align: left;\">We can also write this as [latex]p&gt;12[\/latex]. \u00a0 Notice how the inequality sign is still opening up toward the variable p.<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]p&gt;12[\/latex]\r\nInterval: [latex]\\left(12,\\infty\\right)[\/latex]\r\nGraph: The graph of the inequality <i>p <\/i>&gt; 12 has an open circle at 12 with an arrow stretching to the right.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04063950\/image059.jpg\" alt=\"Number line. Open circle on 12. Highlight on all numbers over 12.\" width=\"575\" height=\"31\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"500309\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"500309\"]\r\n\r\nFirst, check the end point 12 in the related equation.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-58=14-6p\\\\-58=14-6\\left(12\\right)\\\\-58=14-72\\\\-58=-58\\end{array}[\/latex]<\/p>\r\nThen, try another value to check the inequality. Try 100.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-58&gt;14-6p\\\\-58&gt;14-6\\left(100\\right)\\\\-58&gt;14-600\\\\-58&gt;-586\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you will see an example of solving a linear inequality with the variable on the\u00a0left side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.\r\n\r\nhttps:\/\/youtu.be\/RB9wvIogoEM\r\n\r\nIn the following video, you will see an example of solving a linear inequality with the variable on the right side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.\r\n\r\nhttps:\/\/youtu.be\/9D2g_FaNBkY\r\n<h2>Simplify and solve algebraic inequalities using the distributive property<\/h2>\r\nAs with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.\r\n<div class=\"bcc-box bcc-info\">\r\n<h3>Example<\/h3>\r\nSolve for <em>x<\/em>. [latex]2\\left(3x\u20135\\right)\\leq 4x+6[\/latex]\r\n\r\n[reveal-answer q=\"587737\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"587737\"]\r\n\r\nDistribute to clear the parentheses.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\,2(3x-5)\\leq 4x+6\\\\\\,\\,\\,\\,6x-10\\leq 4x+6\\end{array}[\/latex]<\/p>\r\nSubtract 4<i>x <\/i>from both sides to get the variable term on one side only.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}6x-10\\le 4x+6\\\\\\underline{-4x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-4x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,2x-10\\,\\,\\leq \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6\\end{array}[\/latex]<\/p>\r\nAdd 10 to both sides to isolate the variable.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\\\\\,\\,\\,2x-10\\,\\,\\le \\,\\,\\,\\,\\,\\,\\,\\,6\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,+10\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\\\,\\,\\,2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,\\,\\,16\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nDivide both sides by 2 to express the variable with a coefficient of 1.\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{2x}\\le \\,\\,\\,\\underline{16}\\\\\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\le \\,\\,\\,\\,\\,8\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]x\\le8[\/latex]\r\nInterval: [latex]\\left(-\\infty,8\\right][\/latex]\r\nGraph: The graph of this solution set includes 8 and everything left of 8 on the number line.\r\n\r\n<img class=\"wp-image-3947 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10205137\/Screen-Shot-2016-05-10-at-1.51.18-PM-300x40.png\" alt=\"Number line with the interval (-oo,8] graphed\" width=\"443\" height=\"59\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"808701\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"808701\"]\r\n\r\nFirst, check the end point 8 in the related equation.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}2(3x-5)=4x+6\\,\\,\\,\\,\\,\\,\\\\2(3\\,\\cdot \\,8-5)=4\\,\\cdot \\,8+6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(24-5)=32+6\\,\\,\\,\\,\\,\\,\\\\2(19)=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\38=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen, choose another solution and evaluate the inequality for that value to make sure it is a true statement.\u00a0Try 0.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{l}2(3\\,\\cdot \\,0-5)\\le 4\\,\\cdot \\,0+6?\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(-5)\\le 6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10\\le 6\\,\\,\\end{array}[\/latex]<\/p>\r\n[latex]x\\le8[\/latex] is the solution to\u00a0[latex]\\left(-\\infty,8\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\nIn the following video, you are given an example of how to solve a multi-step inequality that requires using the distributive property.\r\nhttps:\/\/youtu.be\/vjZ3rQFVkh8\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Think\u00a0About It<\/h3>\r\nIn the next example, you are given an inequality with a term that looks complicated. If you pause and think about how to use the order of operations to solve the inequality, it will hopefully seem like a straightforward problem. Use the textbox to write down what you think is the best first step to take.\r\n\r\nSolve for a. [latex] \\displaystyle\\frac{{2}{a}-{4}}{{6}}{&lt;2}[\/latex]\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n[reveal-answer q=\"701072\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"701072\"]\r\n\r\nClear the fraction by multiplying both sides of the equation by 6.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\frac{{2}{a}-{4}}{{6}}{&lt;2}\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\6\\,\\cdot \\,\\frac{2a-4}{6}&lt;2\\,\\cdot \\,6\\\\\\\\{2a-4}&lt;12\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nAdd 4 to both sides to isolate the variable.\r\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}2a-4&lt;12\\\\\\underline{\\,\\,\\,+4\\,\\,\\,\\,+4}\\\\2a&lt;16\\end{array}[\/latex]<\/p>\r\n<span style=\"line-height: 1.5;\">Divide both sides by 2 to express the variable with a coefficient of 1.<\/span>\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{c}\\frac{2a}{2}&lt;\\,\\frac{16}{2}\\\\\\\\a&lt;8\\end{array}[\/latex]<\/p>\r\n\r\n<h4>Answer<\/h4>\r\nInequality: [latex]a&lt;8[\/latex]\r\n\r\nInterval: [latex]\\left(-\\infty,8\\right)[\/latex]\r\n\r\nGraph: The graph of this solution contains a solid dot at 8 to show that 8 is included in the solution set. The line continues to the left to show that values less than 8 are also included in the solution set.<img class=\"aligncenter wp-image-3948\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10205449\/Screen-Shot-2016-05-10-at-1.54.23-PM-300x32.png\" alt=\"Open circle on 8 and line through all numbers less than 8.\" width=\"469\" height=\"50\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\nCheck the solution.\r\n[reveal-answer q=\"905072\"]Show Solution[\/reveal-answer]\r\n\r\n[hidden-answer a=\"905072\"]\r\nFirst, check the end point 8 in the related equation.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\frac{2a-4}{6}=2\\,\\,\\,\\,\\\\\\\\\\text{Does}\\,\\,\\,\\frac{2(8)-4}{6}=2?\\\\\\\\\\frac{16-4}{6}=2\\,\\,\\,\\,\\\\\\\\\\frac{12}{6}=2\\,\\,\\,\\,\\\\\\\\2=2\\,\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\nThen choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 5.\r\n<p style=\"text-align: center;\">[latex] \\displaystyle \\begin{array}{r}\\text{Is}\\,\\,\\,\\frac{2(5)-4}{6}&lt;2?\\\\\\\\\\frac{10-4}{6}&lt;2\\,\\,\\,\\\\\\\\\\,\\,\\,\\,\\frac{6}{6}&lt;2\\,\\,\\,\\\\\\\\1&lt;2\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<h2><\/h2>\r\n<h2>Summary<\/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.\r\n\r\nInequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract <i>either positive or negative<\/i> numbers to both sides of the inequality.\r\n\r\n&nbsp;\r\n<h2><\/h2>","rendered":"<div class=\"bcc-box bcc-highlight\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Describe solutions to inequalities\n<ul>\n<li>Represent inequalities on a number line<\/li>\n<li>Represent inequalities using interval notation<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<ul>\n<li>Solve single-step inequalities\n<ul>\n<li>Use the addition and multiplication properties to solve algebraic inequalities and express their solutions graphically and with interval notation<\/li>\n<li>Solve inequalities that contain absolute value<\/li>\n<\/ul>\n<\/li>\n<li>Solve multi-step inequalities\n<ul>\n<li>Combine properties of inequality to isolate variables,\u00a0solve algebraic\u00a0inequalities, and express their solutions graphically<\/li>\n<li>Simplify and solve algebraic inequalities using the distributive property to clear\u00a0parentheses and fractions<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/div>\n<h2 id=\"title1\">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\/117\/2016\/05\/09234719\/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\" 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\" 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.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-5875 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/08\/01202558\/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 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]x<2[\/latex]\n\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.\n\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image033.jpg#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<p>The following video shows how to write an inequality mathematically when it is given in words. We will then graph it.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-2\" title=\"Given Interval in Words, Graph and Give Inequality\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/E_ZWNVNEvOg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title2\">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<p>In the following video we show another example of using interval notation to describe an inequality.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-3\" title=\"Given an Inequality, Graph and Give Interval Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/BKhDzNKjVBc?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Use interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q961990\">Show Solution<\/span><\/p>\n<div id=\"q961990\" class=\"hidden-answer\" style=\"display: none\">\n<p>Use a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video we show another example of translating words into an inequality and writing it in interval notation, as well as drawing the graph.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-4\" title=\"Given Interval in Words, Graph and Give Interval Notation\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/OYkQ-McI2qg?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\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=\"size-full wp-image-5876 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/08\/01202657\/4.png\" alt=\"An open circle on 10 and a line going from 10 to all numbers below 10.\" width=\"271\" height=\"76\" \/><\/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 draw a graph given an inequality in interval notation.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-5\" title=\"Given Interval Notation, Graph and Give Inequality\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/lkhILNEPbfk?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>And finally, one last video that shows how to write inequalities using a graph, with interval notation and as an inequality.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-6\" 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>Solve Single-Step Inequalities<\/h2>\n<h3 id=\"title3\">Solve inequalities with addition and subtraction<\/h3>\n<p>You can solve most inequalities using inverse operations \u00a0as you did for solving equations. \u00a0This is because when you add or subtract the same value from both sides of an inequality, you have maintained the inequality. These properties are outlined in the box below.<\/p>\n<div class=\"textbox shaded\">\n<h3>Addition and Subtraction Properties of Inequality<\/h3>\n<p>If [latex]a>b[\/latex],<i>\u00a0<\/i>then [latex]a+c>b+c[\/latex].<\/p>\n<p>If\u00a0[latex]a>b[\/latex]<i>, <\/i>then [latex]a\u2212c>b\u2212c[\/latex].<\/p>\n<\/div>\n<p>Because inequalities have multiple possible solutions, representing the solutions graphically provides a helpful visual of the situation, as we saw in the last section. The example below shows the steps to solve and graph an inequality and express the solution using interval notation.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x.<\/i><\/p>\n<p style=\"text-align: center;\">[latex]{x}+3\\lt{5}[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q952771\">Show Solution<\/span><\/p>\n<div id=\"q952771\" class=\"hidden-answer\" style=\"display: none\">\n<p>It is helpful to think of this inequality as asking you to find all the values for <em>x<\/em>, including negative numbers, such that when you add three you will get a number less than 5.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}x+3<\\,\\,\\,\\,5\\\\\\underline{\\,\\,\\,\\,\\,-3\\,\\,\\,\\,-3}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,\\,\\,2\\,\\,\\end{array}[\/latex]<\/p>\n<p>Isolate the variable by subtracting 3 from both sides of the inequality.<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: \u00a0[latex]x<2[\/latex]\n\nInterval: \u00a0[latex]\\left(-\\infty, 2\\right)[\/latex]\n\nGraph: <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image036.jpg#fixme\" alt=\"Number line. Open circle around 2. Shaded line through all numbers less than 2.\" width=\"575\" height=\"31\" \/><\/div>\n<\/div>\n<\/div>\n<p style=\"text-align: left;\">The line represents <em>all<\/em> the numbers to which\u00a0you can add 3 and get a number that is less than 5. There&#8217;s a lot of numbers that solve this inequality!<\/p>\n<p>Just as you can check the solution to an equation, you can check a solution to an inequality. First, you check the end point by substituting it in the related equation. Then you check to see if the inequality is correct by substituting any other solution to see if it is one of the solutions. Because there are multiple solutions, it is a good practice to check more than one of the possible solutions. This can also help you check that your graph is correct.<\/p>\n<p>The example below shows how you could check that [latex]x<2[\/latex]<i>\u00a0<\/i>is the solution to [latex]x+3<5[\/latex]<i>.<\/i><\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Check that [latex]x<2[\/latex]<i>\u00a0<\/i>is the solution to [latex]x+3<5[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q811564\">Show Solution<\/span><\/p>\n<div id=\"q811564\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute the end point 2 into the related equation, [latex]x+3=5[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}x+3=5 \\\\ 2+3=5 \\\\ 5=5\\end{array}[\/latex]<\/p>\n<p>Pick a value less than 2, such as 0, to check into the inequality. (This value will be on the shaded part of the graph.)<\/p>\n<p style=\"text-align: center;\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image036.jpg#fixme\" alt=\"Number line. Open circle around 2. Shaded line through all numbers less than 2.\" width=\"575\" height=\"31\" \/><\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}x+3<5 \\\\ 0+3<5 \\\\ 3<5\\end{array}[\/latex]<\/p>\n<p>It checks!<\/p>\n<p>[latex]x<2[\/latex] is the solution to [latex]x+3<5[\/latex].<\/div>\n<\/div>\n<\/div>\n<p>The following examples show inequality problems that include operations with negative numbers. The graph of the solution to the inequality is also shown. Remember to check the solution. This is a good habit to build!<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>:\u00a0[latex]x-10\\leq-12[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q815894\">Show Solution<\/span><\/p>\n<div id=\"q815894\" class=\"hidden-answer\" style=\"display: none\">\n<p>Isolate the variable by adding 10 to both sides of the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}x-10\\le -12\\\\\\underline{\\,\\,\\,+10\\,\\,\\,\\,\\,+10}\\\\x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,-2\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]x\\leq-2[\/latex]<br \/>\nInterval: [latex]\\left(-\\infty,-2\\right][\/latex]<br \/>\nGraph: Notice that a closed circle is used because the inequality is \u201cless than or equal to\u201d [latex]\\left(\\leq\\right)[\/latex]. The blue arrow is drawn to the left of the point [latex]\u22122[\/latex] because these are the values that are less than [latex]\u22122[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3619\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/06184100\/image038-300x17.jpg\" alt=\"Number line, closed circle on negative 2 and line drawn through all numbers less than negative 2\" width=\"529\" height=\"30\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the\u00a0solution to [latex]x-10\\leq -12[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q268062\">Show Solution<\/span><\/p>\n<div id=\"q268062\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute the end point [latex]\u22122[\/latex] into the related equation \u00a0[latex]x-10=\u221212[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}x-10=-12\\,\\,\\,\\\\\\text{Does}\\,\\,\\,-2-10=-12?\\\\-12=-12\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Pick a value less than [latex]\u22122[\/latex], such as [latex]\u22125[\/latex], to check in the inequality. (This value will be on the shaded part of the graph.)<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}x-10\\le -12\\,\\,\\,\\\\\\text{ }\\,\\text{ Is}\\,\\,-5-10\\le -12?\\\\-15\\le -12\\,\\,\\,\\\\\\text{It}\\,\\text{checks!}\\end{array}[\/latex]<\/p>\n<p>[latex]x\\leq -2[\/latex]\u00a0is the solution to [latex]x-10\\leq -12[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for <em>a<\/em>. [latex]a-17>-17[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q343031\">Show Solution<\/span><\/p>\n<div id=\"q343031\" class=\"hidden-answer\" style=\"display: none\">\n<p>Isolate the variable by adding 17 to both sides of the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}a-17>-17\\\\\\underline{\\,\\,\\,+17\\,\\,\\,\\,\\,+17}\\\\a\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>\\,\\,\\,\\,\\,\\,0\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality:\u00a0[latex]\\displaystyle a\\,\\,>\\,0[\/latex]<\/p>\n<p>Interval: [latex]\\left(0,\\infty\\right)[\/latex] \u00a0Note how we use parentheses on the left to show that the solution does not include 0.<\/p>\n<p>Graph: Note the open circle to show that the solution does not include 0.<\/p>\n<div class=\"bcc-box bcc-info\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image044.jpg#fixme\" alt=\"Number line. Open circle on zero. Highlight through all numbers above zero.\" width=\"575\" height=\"32\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the solution to\u00a0[latex]a-17>-17[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q653357\">Show Solution<\/span><\/p>\n<div id=\"q653357\" class=\"hidden-answer\" style=\"display: none\">\n<p>Is\u00a0[latex]\\displaystyle a\\,\\,>\\,0[\/latex] the correct solution to\u00a0\u00a0[latex]a-17>-17[\/latex]?<\/p>\n<p>Substitute the end point 0 into the related equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}a-17=-17\\,\\,\\,\\\\\\text{Does}\\,\\,\\,0-17=-17?\\\\-17=-17\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Pick a value greater than 0, such as 20, to check in the inequality. (This value will be on the shaded part of the graph.)<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}a-17>-17\\,\\,\\,\\\\\\text{Is }\\,\\,20-17>-17?\\\\3>-17\\,\\,\\,\\\\\\\\\\text{It checks!}\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>[latex]\\displaystyle a\\,>\\,0[\/latex] is the solution to\u00a0[latex]a-17>-17[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<p>The previous examples showed you how to solve a one-step inequality with the variable on the left hand side. \u00a0The following video provides examples of how to solve the same type of inequality.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-7\" 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><\/p>\n<p>What would you do if the variable were on the right side of the inequality? \u00a0In the following example, you will see how to handle this scenario.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>:\u00a0[latex]4\\geq{x}+5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q815893\">Show Solution<\/span><\/p>\n<div id=\"q815893\" class=\"hidden-answer\" style=\"display: none\">\n<p>Isolate the variable by adding 10 to both sides of the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}4\\geq{x}+5 \\\\\\underline{\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\-1\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\,\\,\\,x\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">Rewrite the inequality with the variable on the left &#8211; this makes writing the interval and drawing the graph easier.<\/p>\n<p style=\"text-align: center;\">[latex]x\\le{-1}[\/latex]<\/p>\n<p style=\"text-align: left;\">Note how the the pointy part of the inequality is still directed at the variable, so instead of reading as negative one is greater or equal to x, it now reads as x is less than or equal to negative one.<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]x\\le{-1}[\/latex] This can also be written as<br \/>\nInterval: [latex]\\left(-\\infty,-1\\right][\/latex]<br \/>\nGraph: Notice that a closed circle is used because the inequality is \u201cless than or equal to\u201d . The blue arrow is drawn to the left of the point [latex]\u22121[\/latex] because these are the values that are less than [latex]\u22121[\/latex].<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"wp-image-4022 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/12012430\/Screen-Shot-2016-05-11-at-6.23.24-PM-300x57.png\" alt=\"(-oo,-1&#093;\" width=\"400\" height=\"76\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the\u00a0solution to [latex]4\\geq{x}+5[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q568062\">Show Solution<\/span><\/p>\n<div id=\"q568062\" class=\"hidden-answer\" style=\"display: none\">\n<p>Substitute the end point [latex]\u22121[\/latex] into the related equation \u00a0[latex]4=x+5[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}4=x+5\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4=-1+5?\\\\-1=-1\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Pick a value less than [latex]\u22121[\/latex], such as [latex]\u22125[\/latex], to check in the inequality. (This value will be on the shaded part of the graph.)<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}4\\geq{-5}+5\\,\\,\\,\\\\\\text{ }\\,\\text{ Is}\\,\\,4\\ge 0?\\\\\\text{It}\\,\\text{checks!}\\end{array}[\/latex]<\/p>\n<p>[latex]x\\le{-1}[\/latex] is the solution to [latex]4\\geq{x}+5[\/latex]<span style=\"line-height: 1.5;\"><\/div>\n<\/div>\n<p> <\/span><\/p>\n<\/div>\n<p>The following video show examples of solving inequalities with the variable on the right side.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-8\" 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<h3 id=\"title4\">Solve inequalities with multiplication and division<\/h3>\n<p>Solving an inequality with a variable that has a coefficient other than 1 usually involves multiplication or division. The steps are like solving one-step equations involving multiplication or division EXCEPT for the inequality sign. Let\u2019s look at what happens to the inequality when you multiply or divide each side by the same number.<\/p>\n<table>\n<tbody>\n<tr>\n<td>Let&#8217;s start with the true statement:<\/p>\n<p>[latex]10>5[\/latex]<\/td>\n<td>Let&#8217;s try again by starting with the same true statement:<\/p>\n<p>[latex]10>5[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>Next, multiply both sides by the same positive number:<\/p>\n<p>[latex]10\\cdot 2>5\\cdot 2[\/latex]<\/td>\n<td>This time, multiply both sides by the same negative number:<\/p>\n<p>[latex]10\\cdot-2>5 \\\\ \\,\\,\\,\\,\\,\\cdot -2\\,\\cdot-2[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>20 is greater than 10, so you still have a true inequality:<\/p>\n<p>[latex]20>10[\/latex]<\/td>\n<td>Wait a minute! [latex]\u221220[\/latex] is <i>not <\/i>greater than [latex]\u221210[\/latex], so you have an untrue statement.<\/p>\n<p>[latex]\u221220>\u221210[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>When you multiply by a positive number, leave the inequality sign as it is!<\/td>\n<td>You must \u201creverse\u201d the inequality sign to make the statement true:<\/p>\n<p>[latex]\u221220<\u221210[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox shaded\">\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-2132 alignleft\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/03\/22011815\/traffic-sign-160659-300x265.png\" alt=\"Caution\" width=\"80\" height=\"70\" \/>Caution! \u00a0When you multiply or divide by a negative number, \u201creverse\u201d the inequality sign. \u00a0 Whenever you multiply or divide both sides of an inequality by a negative\u00a0number, the inequality sign must be reversed in order to keep a true statement. These rules are summarized in the box below.<\/p>\n<\/div>\n<hr \/>\n<div class=\"textbox shaded\">\n<h3>Multiplication and Division Properties of Inequality<\/h3>\n<table style=\"height: 162px; width: 419px;\">\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]a>b[\/latex]<\/td>\n<td>[latex]-c[\/latex]<\/td>\n<td>[latex]ac<bc[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<table style=\"height: 77px; width: 418px;\">\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]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<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<\/div>\n<p>Keep in mind that you only change the sign when you are multiplying and dividing by a <i>negative<\/i> number. If you <em>add or subtract<\/em> by a negative\u00a0number, the inequality stays the same.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>x.\u00a0<\/i>[latex]3x>12[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q691711\">Show Solution<\/span><\/p>\n<div id=\"q691711\" class=\"hidden-answer\" style=\"display: none\">Divide both sides by 3 to isolate the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\underline{3x}>\\underline{12}\\\\3\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,3\\\\x>4\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Check your solution by first checking the end point 4, and then checking another solution for the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}3\\cdot4=12\\\\12=12\\\\3\\cdot10>12\\\\30>12\\\\\\text{It checks!}\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p style=\"text-align: left;\">Inequality: [latex]\\displaystyle x>4[\/latex]<\/p>\n<p style=\"text-align: left;\">Interval: [latex]\\left(4,\\infty\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">Graph: <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image050.jpg#fixme\" alt=\"Number line. Open circle on 4. Highlight through all numbers greater than 4.\" width=\"575\" height=\"31\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>There was no need to make any changes to the inequality sign because both sides of the inequality were divided by <i>positive<\/i> 3. In the next example, there is division by a negative number, so there is an additional step in the solution!<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>. [latex]\u22122x>6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q604033\">Show Solution<\/span><\/p>\n<div id=\"q604033\" class=\"hidden-answer\" style=\"display: none\">Divide each side of the inequality by [latex]\u22122[\/latex] to isolate the variable, and change the direction of the inequality sign because of the division by a negative number.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\underline{-2x}<\\underline{\\,6\\,}\\\\-2\\,\\,\\,\\,-2\\,\\\\x<-3\\end{array}[\/latex]<\/p>\n<p>Check your solution by first checking the end point [latex]\u22123[\/latex], and then checking another solution for the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-2\\left(-3\\right)=6 \\\\6=6\\\\ -2\\left(-6\\right)>6 \\\\ 12>6\\end{array}[\/latex]<\/p>\n<p>It checks!<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]\\displaystyle x<-3[\/latex]\n\nInterval: [latex]\\left(-\\infty, -3\\right)[\/latex]\n\nGraph: <img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/textimgs.s3.amazonaws.com\/MITEdevmath\/NROCUnit10_files\/image051.jpg#fixme\" alt=\"Number line. Open circle on negative 3. Highlight on all numbers less than negative 3.\" width=\"575\" height=\"31\" \/><br \/>\nBecause both sides of the inequality were divided by a negative number, [latex]\u22122[\/latex], the inequality symbol was switched from &gt; to &lt;.\n<\/div>\n<\/div>\n<\/div>\n<p>The following video shows examples of solving one step inequalities using the multiplication property of equality where the variable is on the left hand side.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-9\" title=\"Ex:  Solve One Step Linear Inequality by Dividing (Variable Left)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/IajiD3R7U-0?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think About It<\/h3>\n<p>Before you read the solution to the next example, think about what properties of inequalities you may need to use to solve the inequality. What is different about this example from the previous one? Write your ideas in the box below.<\/p>\n<p>Solve for <em>x<\/em>. [latex]-\\frac{1}{2}>-12x[\/latex]<\/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=\"q811465\">Show Solution<\/span><\/p>\n<div id=\"q811465\" class=\"hidden-answer\" style=\"display: none\">\n<p>This inequality has the variable on the right hand side, which is different from the previous examples. Start the solution process as before, and at the end, you can move the variable to the left to write the final solution.<\/p>\n<p>Divide both sides by [latex]-12[\/latex] to isolate the variable. Since you are dividing by a negative number, you need to change the direction of the inequality sign.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\begin{array}{l}-\\frac{1}{2}\\gt{-12x}\\\\\\\\\\frac{-\\frac{1}{2}}{-12}\\gt\\frac{-12x}{-12}\\\\\\end{array}[\/latex]<\/p>\n<p>Dividing a fraction by an integer requires you to multiply by the reciprocal, and the reciprocal of [latex]-12[\/latex] is [latex]\\frac{1}{-12}[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle\\begin{array}{r}\\left(-\\frac{1}{12}\\right)\\left(-\\frac{1}{2}\\right)\\lt\\frac{-12x}{-12}\\,\\,\\\\\\\\ \\frac{1}{24}\\lt\\frac{\\cancel{-12}x}{\\cancel{-12}}\\\\\\\\ \\frac{1}{24}\\lt{x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<h4 style=\"text-align: left;\">Answer<\/h4>\n<p style=\"text-align: left;\">Inequality: [latex]\\frac{1}{24}\\lt{x}[\/latex] \u00a0This can also be written with the variable on the left as [latex]x\\gt\\frac{1}{24}[\/latex]. \u00a0Writing the inequality with the variable on the left requires a little thinking, but helps you write the interval and draw the graph correctly.<\/p>\n<p style=\"text-align: left;\">Interval: [latex]\\left(\\frac{1}{24},\\infty\\right)[\/latex]<\/p>\n<p style=\"text-align: left;\">Graph:\u00a0<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3950\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10211010\/Screen-Shot-2016-05-10-at-2.09.52-PM-300x57.png\" alt=\"Open dot on zero with a line through all numbers greater than zero.\" width=\"389\" height=\"74\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>The following video gives examples of how to solve an inequality with the multiplication property of equality where the variable is on the right hand side.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-10\" title=\"Ex:  Solve One Step Linear Inequality by Dividing (Variable Right)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/s9fJOnVTHhs?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2 id=\"title1\">Combine properties of inequality to \u00a0solve algebraic\u00a0inequalities<\/h2>\n<p>A popular strategy for solving equations, isolating the variable, also applies to solving inequalities. By adding, subtracting, multiplying and\/or dividing, you can rewrite the inequality so that the variable is on one side and everything else is on the other. As with one-step inequalities, the solutions to multi-step inequalities can be graphed on a number line.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>p<\/i>. [latex]4p+5<29[\/latex]\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q211828\">Show Solution<\/span><\/p>\n<div id=\"q211828\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin to isolate the variable by subtracting 5 from both sides of the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}4p+5<\\,\\,\\,29\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,-5\\,\\,\\,\\,\\,-5}\\\\4p\\,\\,\\,\\,\\,\\,\\,\\,\\,<\\,\\,24\\,\\,\\end{array}[\/latex]<\/p>\n<p>Divide both sides of the inequality by 4 to express the variable with a coefficient of 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\underline{4p}\\,<\\,\\,\\underline{24}\\,\\,\\\\\\,4\\,\\,\\,\\,<\\,\\,4\\\\\\,\\,\\,\\,\\,p<6\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality:\u00a0[latex]p<6[\/latex]\n\nInterval: [latex]\\left(-\\infty,6\\right)[\/latex]\n\nGraph: Note the\u00a0open circle at the end point 6 to show that solutions to the inequality do not include 6.\u00a0The values where <i>p<\/i> is less than 6 are found all along the number line to the left of 6.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04063949\/image057.jpg\" alt=\"Number line. Open circle on 6. Highlight on every number less than 6.\" width=\"575\" height=\"31\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the solution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q291597\">Show Solution<\/span><\/p>\n<div id=\"q291597\" class=\"hidden-answer\" style=\"display: none\">\n<p>Check the end point 6 in the related equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}4p+5=29\\,\\,\\,\\\\\\text{Does}\\,\\,\\,4(6)+5=29?\\\\24+5=29\\,\\,\\,\\\\29=29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Try another value to check the inequality. Let\u2019s use [latex]p=0[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}4p+5<29\\,\\,\\,\\\\\\text{Is}\\,\\,\\,4(0)+5<29?\\\\0+5<29\\,\\,\\,\\\\5<29\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>[latex]p<6[\/latex] is the solution to\u00a0[latex]4p+5<29[\/latex]\n\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example<\/h3>\n<p>Solve for <i>x<\/i>: \u00a0[latex]3x\u20137\\ge 41[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q238157\">Show Solution<\/span><\/p>\n<div id=\"q238157\" class=\"hidden-answer\" style=\"display: none\">\n<p>Begin to isolate the variable by adding 7 to both sides of the inequality, then divide both sides of the inequality by 3 to express the variable with a coefficient of 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}3x-7\\ge 41\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,+7\\,\\,\\,\\,+7}\\\\\\frac{3x}{3}\\,\\,\\,\\,\\,\\,\\,\\,\\ge \\frac{48}{3}\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\ge 16\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]x\\ge 16[\/latex]<\/p>\n<p>Interval: [latex]\\left[16,\\infty\\right)[\/latex]<\/p>\n<p>Graph:\u00a0To graph this inequality, you draw a closed circle at the end point 16 on the number line\u00a0to show that solutions include the value 16. The line continues to the right from 16 because all the numbers greater than 16 will also make the inequality\u00a0[latex]3x\u20137\\ge 41[\/latex] true.<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3956\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10233054\/Screen-Shot-2016-05-10-at-4.28.03-PM-300x48.png\" alt=\"Closed dot on 16, line through all numbers greater than 16.\" width=\"425\" height=\"68\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the solution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q437341\">Show Solution<\/span><\/p>\n<div id=\"q437341\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, check the end point 16 in the related equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}3x-7=41\\,\\,\\,\\\\\\text{Does}\\,\\,\\,3(16)-7=41?\\\\48-7=41\\,\\,\\,\\\\41=41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Then, try another value to check the inequality. Let\u2019s use [latex]x = 20[\/latex].<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,\\,\\,\\,3x-7\\ge 41\\,\\,\\,\\\\\\text{Is}\\,\\,\\,\\,\\,3(20)-7\\ge 41?\\\\60-7\\ge 41\\,\\,\\,\\\\53\\ge 41\\,\\,\\,\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>When solving multi-step equations, pay attention to situations in which you multiply or divide by a negative number. In these cases, you must reverse the inequality sign.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <i>p<\/i>. [latex]\u221258>14\u22126p[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q424351\">Show Solution<\/span><\/p>\n<div id=\"q424351\" class=\"hidden-answer\" style=\"display: none\">\n<p>Note how the variable is on the right hand side of the inequality, the method for solving does not change in this case.<\/p>\n<p>Begin to isolate the variable by subtracting 14 from both sides of the inequality.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}\u221258\\,\\,>14\u22126p\\\\\\underline{\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-14\\,\\,\\,\\,\\,\\,\\,-14}\\\\-72\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,>-6p\\end{array}[\/latex]<\/p>\n<p>Divide both sides of the inequality by [latex]\u22126[\/latex] to express the variable with a coefficient of 1.\u00a0Dividing by a negative number results in reversing the inequality sign.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{l}\\underline{-72}>\\underline{-6p}\\\\-6\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-6\\\\\\,\\,\\,\\,\\,\\,12\\lt{p}\\end{array}[\/latex]<\/p>\n<p style=\"text-align: left;\">We can also write this as [latex]p>12[\/latex]. \u00a0 Notice how the inequality sign is still opening up toward the variable p.<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]p>12[\/latex]<br \/>\nInterval: [latex]\\left(12,\\infty\\right)[\/latex]<br \/>\nGraph: The graph of the inequality <i>p <\/i>&gt; 12 has an open circle at 12 with an arrow stretching to the right.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/1468\/2016\/02\/04063950\/image059.jpg\" alt=\"Number line. Open circle on 12. Highlight on all numbers over 12.\" width=\"575\" height=\"31\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the solution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q500309\">Show Solution<\/span><\/p>\n<div id=\"q500309\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, check the end point 12 in the related equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-58=14-6p\\\\-58=14-6\\left(12\\right)\\\\-58=14-72\\\\-58=-58\\end{array}[\/latex]<\/p>\n<p>Then, try another value to check the inequality. Try 100.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}-58>14-6p\\\\-58>14-6\\left(100\\right)\\\\-58>14-600\\\\-58>-586\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you will see an example of solving a linear inequality with the variable on the\u00a0left side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-11\" title=\"Ex:  Solve a Two Step Linear Inequality  (Variable Left)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/RB9wvIogoEM?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<p>In the following video, you will see an example of solving a linear inequality with the variable on the right side of the inequality, and an example of switching the direction of the inequality after dividing by a negative number.<\/p>\n<p><iframe loading=\"lazy\" id=\"oembed-12\" title=\"Ex:  Solve a Two Step Linear Inequality  (Variable Right)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/9D2g_FaNBkY?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<h2>Simplify and solve algebraic inequalities using the distributive property<\/h2>\n<p>As with equations, the distributive property can be applied to simplify expressions that are part of an inequality. Once the parentheses have been cleared, solving the inequality will be straightforward.<\/p>\n<div class=\"bcc-box bcc-info\">\n<h3>Example<\/h3>\n<p>Solve for <em>x<\/em>. [latex]2\\left(3x\u20135\\right)\\leq 4x+6[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q587737\">Show Solution<\/span><\/p>\n<div id=\"q587737\" class=\"hidden-answer\" style=\"display: none\">\n<p>Distribute to clear the parentheses.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\,2(3x-5)\\leq 4x+6\\\\\\,\\,\\,\\,6x-10\\leq 4x+6\\end{array}[\/latex]<\/p>\n<p>Subtract 4<i>x <\/i>from both sides to get the variable term on one side only.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}6x-10\\le 4x+6\\\\\\underline{-4x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-4x}\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\,\\,\\,2x-10\\,\\,\\leq \\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,6\\end{array}[\/latex]<\/p>\n<p>Add 10 to both sides to isolate the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\\\\\,\\,\\,2x-10\\,\\,\\le \\,\\,\\,\\,\\,\\,\\,\\,6\\,\\,\\,\\\\\\underline{\\,\\,\\,\\,\\,\\,+10\\,\\,\\,\\,\\,\\,\\,\\,\\,+10}\\\\\\,\\,\\,2x\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\le \\,\\,\\,\\,\\,16\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Divide both sides by 2 to express the variable with a coefficient of 1.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{r}\\underline{2x}\\le \\,\\,\\,\\underline{16}\\\\\\,\\,\\,2\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2\\,\\,\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,x\\,\\,\\,\\le \\,\\,\\,\\,\\,8\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]x\\le8[\/latex]<br \/>\nInterval: [latex]\\left(-\\infty,8\\right][\/latex]<br \/>\nGraph: The graph of this solution set includes 8 and everything left of 8 on the number line.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-3947 aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10205137\/Screen-Shot-2016-05-10-at-1.51.18-PM-300x40.png\" alt=\"Number line with the interval (-oo,8&#093; graphed\" width=\"443\" height=\"59\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the solution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q808701\">Show Solution<\/span><\/p>\n<div id=\"q808701\" class=\"hidden-answer\" style=\"display: none\">\n<p>First, check the end point 8 in the related equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}2(3x-5)=4x+6\\,\\,\\,\\,\\,\\,\\\\2(3\\,\\cdot \\,8-5)=4\\,\\cdot \\,8+6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(24-5)=32+6\\,\\,\\,\\,\\,\\,\\\\2(19)=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\\\38=38\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Then, choose another solution and evaluate the inequality for that value to make sure it is a true statement.\u00a0Try 0.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{l}2(3\\,\\cdot \\,0-5)\\le 4\\,\\cdot \\,0+6?\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,2(-5)\\le 6\\\\\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,-10\\le 6\\,\\,\\end{array}[\/latex]<\/p>\n<p>[latex]x\\le8[\/latex] is the solution to\u00a0[latex]\\left(-\\infty,8\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>In the following video, you are given an example of how to solve a multi-step inequality that requires using the distributive property.<br \/>\n<iframe loading=\"lazy\" id=\"oembed-13\" title=\"Ex:  Solve a Linear Inequality Requiring Multiple Steps (One Var)\" width=\"500\" height=\"281\" src=\"https:\/\/www.youtube.com\/embed\/vjZ3rQFVkh8?feature=oembed&#38;rel=0\" frameborder=\"0\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/p>\n<div class=\"textbox key-takeaways\">\n<h3>Think\u00a0About It<\/h3>\n<p>In the next example, you are given an inequality with a term that looks complicated. If you pause and think about how to use the order of operations to solve the inequality, it will hopefully seem like a straightforward problem. Use the textbox to write down what you think is the best first step to take.<\/p>\n<p>Solve for a. [latex]\\displaystyle\\frac{{2}{a}-{4}}{{6}}{<2}[\/latex]\n\n<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=\"q701072\">Show Solution<\/span><\/p>\n<div id=\"q701072\" class=\"hidden-answer\" style=\"display: none\">\n<p>Clear the fraction by multiplying both sides of the equation by 6.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\frac{{2}{a}-{4}}{{6}}{<2}\\,\\,\\,\\,\\,\\,\\,\\,\\\\\\\\6\\,\\cdot \\,\\frac{2a-4}{6}<2\\,\\cdot \\,6\\\\\\\\{2a-4}<12\\,\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Add 4 to both sides to isolate the variable.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}2a-4<12\\\\\\underline{\\,\\,\\,+4\\,\\,\\,\\,+4}\\\\2a<16\\end{array}[\/latex]<\/p>\n<p><span style=\"line-height: 1.5;\">Divide both sides by 2 to express the variable with a coefficient of 1.<\/span><\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{c}\\frac{2a}{2}<\\,\\frac{16}{2}\\\\\\\\a<8\\end{array}[\/latex]<\/p>\n<h4>Answer<\/h4>\n<p>Inequality: [latex]a<8[\/latex]\n\nInterval: [latex]\\left(-\\infty,8\\right)[\/latex]\n\nGraph: The graph of this solution contains a solid dot at 8 to show that 8 is included in the solution set. The line continues to the left to show that values less than 8 are also included in the solution set.<img loading=\"lazy\" decoding=\"async\" class=\"aligncenter wp-image-3948\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/117\/2016\/05\/10205449\/Screen-Shot-2016-05-10-at-1.54.23-PM-300x32.png\" alt=\"Open circle on 8 and line through all numbers less than 8.\" width=\"469\" height=\"50\" \/><\/p>\n<\/div>\n<\/div>\n<p>Check the solution.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q905072\">Show Solution<\/span><\/p>\n<div id=\"q905072\" class=\"hidden-answer\" style=\"display: none\">\nFirst, check the end point 8 in the related equation.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\frac{2a-4}{6}=2\\,\\,\\,\\,\\\\\\\\\\text{Does}\\,\\,\\,\\frac{2(8)-4}{6}=2?\\\\\\\\\\frac{16-4}{6}=2\\,\\,\\,\\,\\\\\\\\\\frac{12}{6}=2\\,\\,\\,\\,\\\\\\\\2=2\\,\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<p>Then choose another solution and evaluate the inequality for that value to make sure it is a true statement. Try 5.<\/p>\n<p style=\"text-align: center;\">[latex]\\displaystyle \\begin{array}{r}\\text{Is}\\,\\,\\,\\frac{2(5)-4}{6}<2?\\\\\\\\\\frac{10-4}{6}<2\\,\\,\\,\\\\\\\\\\,\\,\\,\\,\\frac{6}{6}<2\\,\\,\\,\\\\\\\\1<2\\,\\,\\,\\\\\\\\\\text{Yes!}\\,\\,\\,\\,\\,\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2><\/h2>\n<h2>Summary<\/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.<\/p>\n<p>Inequalities can have a range of answers. The solutions are often graphed on a number line in order to visualize all of the solutions. Multi-step inequalities are solved using the same processes that work for solving equations with one exception. When you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality symbol. The inequality symbols stay the same whenever you add or subtract <i>either positive or negative<\/i> numbers to both sides of the inequality.<\/p>\n<p>&nbsp;<\/p>\n<h2><\/h2>\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-3407\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Graph Linear Inequalities in One Variable (Basic). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/-kiAeGbSe5c\">https:\/\/youtu.be\/-kiAeGbSe5c<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Given Interval in Words, Graph and Give Inequality. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/E_ZWNVNEvOg\">https:\/\/youtu.be\/E_ZWNVNEvOg<\/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>Given an Inequality, Graph and Give Interval Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/BKhDzNKjVBc\">https:\/\/youtu.be\/BKhDzNKjVBc<\/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>Given Interval in Words, Graph and Give Interval Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/OYkQ-McI2qg\">https:\/\/youtu.be\/OYkQ-McI2qg<\/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>Given Interval Notation, Graph and Give Inequality. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/lkhILNEPbfk\">https:\/\/youtu.be\/lkhILNEPbfk<\/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>Screenshot: Cecilia Venn Diagram. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Screenshot: Internet Privacy. <strong>Authored by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solutions to Basic OR Compound Inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/nKarzhZOFIk\">https:\/\/youtu.be\/nKarzhZOFIk<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Solutions to Basic AND Compound Inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/LP3fsZNjJkc\">https:\/\/youtu.be\/LP3fsZNjJkc<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>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\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\">http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/<\/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) for Lumen Learning. <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) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/RBonYKvTCLU\">https:\/\/youtu.be\/RBonYKvTCLU<\/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: Solve One Step Linear Inequality by Dividing (Variable Left). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/IajiD3R7U-0\">https:\/\/youtu.be\/IajiD3R7U-0<\/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: Solve One Step Linear Inequality by Dividing (Variable Right). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/s9fJOnVTHhs\">https:\/\/youtu.be\/s9fJOnVTHhs<\/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>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: Graph Basic Inequalities and Express Using Interval Notation. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/X0xrHKgbDT0\">https:\/\/youtu.be\/X0xrHKgbDT0<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><li>Ex: Solve a Two Step Linear Inequality (Variable Left). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/RB9wvIogoEM\">https:\/\/youtu.be\/RB9wvIogoEM<\/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: Solve a Two Step Linear Inequality (Variable Right). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/9D2g_FaNBkY\">https:\/\/youtu.be\/9D2g_FaNBkY<\/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: Solve a Linear Inequality Requiring Multiple Steps (One Var). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/vjZ3rQFVkh8\">https:\/\/youtu.be\/vjZ3rQFVkh8<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/about\/pdm\">Public Domain: No Known Copyright<\/a><\/em><\/li><li>Ex: Solve a Compound Inequality Involving OR (Union). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/oRlJ8G7trR8\">https:\/\/youtu.be\/oRlJ8G7trR8<\/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 1: Solve and Graph Basic Absolute Value inequalities. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/0cXxATY2S-k\">https:\/\/youtu.be\/0cXxATY2S-k<\/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 2: Solve and Graph Absolute Value inequalities Mathispower4u . <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) for Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/d-hUviSkmqE\">https:\/\/youtu.be\/d-hUviSkmqE<\/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 3: Solve and Graph Absolute Value inequalitie. <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/ttUaRf-GzpM\">https:\/\/youtu.be\/ttUaRf-GzpM<\/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 4: Solve and Graph Absolute Value inequalities (Requires Isolating Abs. Value). <strong>Authored by<\/strong>: James Sousa (Mathispower4u.com) . <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/youtu.be\/5jRUuiMUxWQ\">https:\/\/youtu.be\/5jRUuiMUxWQ<\/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>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><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":5,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Unit 10: Solving Equations and Inequalities, from Developmental Math: An Open Program\",\"author\":\"\",\"organization\":\"Monterey Institute of Technology and Education\",\"url\":\" http:\/\/nrocnetwork.org\/resources\/downloads\/nroc-math-open-textbook-units-1-12-pdf-and-word-formats\/\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"Ex: Solving One Step Inequalities by Adding and Subtracting (Variable Left Side)\",\"author\":\"James Sousa (Mathispower4u.com) for Lumen 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