{"id":1461,"date":"2016-10-24T23:02:07","date_gmt":"2016-10-24T23:02:07","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=1461"},"modified":"2017-04-10T21:34:10","modified_gmt":"2017-04-10T21:34:10","slug":"write-and-manipulate-inequalities","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-wmopen-collegealgebra\/chapter\/write-and-manipulate-inequalities\/","title":{"raw":"Write and Manipulate Inequalities","rendered":"Write and Manipulate Inequalities"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Objectives<\/h3>\r\n<ul>\r\n \t<li>Use interval notation to express inequalities<\/li>\r\n \t<li>Use properties of inequalities<\/li>\r\n<\/ul>\r\n<\/div>\r\nIndicating the solution to an inequality such as [latex]x\\ge 4[\/latex] can be achieved in several ways.\r\n\r\nWe can use a number line as shown below.\u00a0The blue ray begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225859\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" data-media-type=\"image\/jpg\" \/>\r\n\r\nWe can use <strong>set-builder notation<\/strong>: [latex]\\{x|x\\ge 4\\}[\/latex], which translates to \"all real numbers <em>x <\/em>such that <em>x <\/em>is greater than or equal to 4.\" Notice that braces are used to indicate a set.\r\n\r\nThe third method is <strong>interval notation<\/strong>, in which solution sets are indicated with parentheses or brackets. The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.\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.\r\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\r\n<thead>\r\n<tr>\r\n<th>Set Indicated<\/th>\r\n<th>Set-Builder Notation<\/th>\r\n<th>Interval Notation<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, but not including <em>a <\/em>or <em>b<\/em><\/td>\r\n<td>[latex]\\{x|a&lt;x&lt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|x&gt;a\\}[\/latex]<\/td>\r\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x&lt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|x\\ge a\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\r\n<td>[latex]\\{x|a\\le x&lt;b\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\r\n<td>[latex]\\{x|a&lt;x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\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]\\{x|a\\le x\\le b\\}[\/latex]<\/td>\r\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\r\n<td>[latex]\\{x|x&lt;a\\text{ and }x&gt;b\\}[\/latex]<\/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>[latex]\\{x|x\\text{ is all real numbers}\\}[\/latex]<\/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: Using Interval Notation to Express All Real Numbers Greater Than or Equal to <em>a<\/em><\/h3>\r\nUse interval notation to indicate all real numbers greater than or equal to [latex]-2[\/latex].\r\n\r\n[reveal-answer q=\"143041\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"143041\"]\r\nUse a bracket on the left of [latex]-2[\/latex] and parentheses after infinity: [latex]\\left[-2,\\infty \\right)[\/latex]. The bracket indicates that [latex]-2[\/latex] is included in the set with all real numbers greater than [latex]-2[\/latex] to infinity.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nUse interval notation to indicate all real numbers between and including [latex]-3[\/latex] and [latex]5[\/latex].\r\n\r\n[reveal-answer q=\"814810\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"814810\"]\r\n\r\n[latex]\\left[-3,5\\right][\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=58&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"650\"><\/iframe>\r\n<iframe id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92604&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Using Interval Notation to Express All Real Numbers Less Than or Equal to <em>a <\/em>or Greater Than or Equal to <em>b<\/em><\/h3>\r\nWrite the interval expressing all real numbers less than or equal to [latex]-1[\/latex] or greater than or equal to [latex]1[\/latex].\r\n\r\n[reveal-answer q=\"797079\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"797079\"]\r\nWe have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at [latex]-\\infty [\/latex] and ends at [latex]-1[\/latex], which is written as [latex]\\left(-\\infty ,-1\\right][\/latex].\r\n\r\nThe second interval must show all real numbers greater than or equal to [latex]1[\/latex], which is written as [latex]\\left[1,\\infty \\right)[\/latex]. However, we want to combine these two sets. We accomplish this by inserting the union symbol, [latex]\\cup [\/latex], between the two intervals.\r\n<div style=\"text-align: center;\">[latex]\\left(-\\infty ,-1\\right]\\cup \\left[1,\\infty \\right)[\/latex]<\/div>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nExpress all real numbers less than [latex]-2[\/latex] or greater than or equal to 3 in interval notation.\r\n\r\n[reveal-answer q=\"729196\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"729196\"]\r\n\r\n[latex]\\left(-\\infty ,-2\\right)\\cup \\left[3,\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>try it now<\/h3>\r\nUse the sliders in the graph below to adjust the length of the line.\r\n<ol>\r\n \t<li>Adjust the left endpoint to (-15,0), and the right endpoint to (5,0)<\/li>\r\n \t<li>Write an inequality that represents the line you created.<\/li>\r\n<\/ol>\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n3. Now slide the point labeled b over to 15. What do you think happened to the line?\r\n\r\n[practice-area rows=\"1\"][\/practice-area]\r\n\r\n4. Now adjust the end points and make your own inequality.\r\n\r\nhttps:\/\/www.desmos.com\/calculator\/4529rytfef\r\n[reveal-answer q=\"748650\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"748650\"]\r\n\r\nWith endpoints (-15,0) and (5,0), the values for x on the line are between -15 and 5, so we can write [latex]-15&lt;x&lt;5[\/latex]. We made it a strict inequality because the dots on the endpoints of the lines are open.\r\n\r\nThe line disappeared because the inequality being represented is of the form [latex]b&lt; x&lt; a[\/latex] and when [latex]b&gt;a[\/latex] the inequality is no longer valid.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n&nbsp;\r\n<h2>Using the Properties of Inequalities<\/h2>\r\nWhen we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the <strong>addition property<\/strong> and the <strong>multiplication property<\/strong> to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Properties of Inequalities<\/h3>\r\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\text{Addition Property}\\hfill&amp; \\text{If }a&lt; b,\\text{ then }a+c&lt; b+c.\\hfill \\\\ \\hfill &amp; \\hfill \\\\ \\text{Multiplication Property}\\hfill &amp; \\text{If }a&lt; b\\text{ and }c&gt; 0,\\text{ then }ac&lt; bc.\\hfill \\\\ \\hfill &amp; \\text{If }a&lt; b\\text{ and }c&lt; 0,\\text{ then }ac&gt; bc.\\hfill \\end{array}[\/latex]<\/p>\r\nThese properties also apply to [latex]a\\le b[\/latex], [latex]a&gt;b[\/latex], and [latex]a\\ge b[\/latex].\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Demonstrating the Addition Property<\/h3>\r\nIllustrate the addition property for inequalities by solving each of the following:\r\n<ol>\r\n \t<li>[latex]x - 15&lt;4[\/latex]<\/li>\r\n \t<li>[latex]6\\ge x - 1[\/latex]<\/li>\r\n \t<li>[latex]x+7&gt;9[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"105622\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"105622\"]\r\nThe addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.\r\n<ol>\r\n \t<li>[latex]\\begin{array}{ll}x - 15&lt;4\\hfill &amp; \\hfill \\\\ x - 15+15&lt;4+15 \\hfill &amp; \\text{Add 15 to both sides.}\\hfill \\\\ x&lt;19\\hfill &amp; \\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}6\\ge x - 1\\hfill &amp; \\hfill \\\\ 6+1\\ge x - 1+1\\hfill &amp; \\text{Add 1 to both sides}.\\hfill \\\\ 7\\ge x\\hfill &amp; \\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}x+7&gt;9\\hfill &amp; \\hfill \\\\ x+7 - 7&gt;9 - 7\\hfill &amp; \\text{Subtract 7 from both sides}.\\hfill \\\\ x&gt;2\\hfill &amp; \\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]3x - 2&lt;1[\/latex].\r\n\r\n[reveal-answer q=\"68318\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"68318\"]\r\n\r\n[latex]x&lt;1[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92605&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Demonstrating the Multiplication Property<\/h3>\r\nIllustrate the multiplication property for inequalities by solving each of the following:\r\n<ol>\r\n \t<li>[latex]3x&lt;6[\/latex]<\/li>\r\n \t<li>[latex]-2x - 1\\ge 5[\/latex]<\/li>\r\n \t<li>[latex]5-x&gt;10[\/latex]<\/li>\r\n<\/ol>\r\n[reveal-answer q=\"749552\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"749552\"]\r\n<ol>\r\n \t<li>[latex]\\begin{array}{l}3x&lt;6\\hfill \\\\ \\frac{1}{3}\\left(3x\\right)&lt;\\left(6\\right)\\frac{1}{3}\\hfill \\\\ x&lt;2\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}-2x - 1\\ge 5\\hfill &amp; \\hfill \\\\ -2x\\ge 6\\hfill &amp; \\hfill \\\\ \\left(-\\frac{1}{2}\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\frac{1}{2}\\right)\\hfill &amp; \\text{Multiply by }-\\frac{1}{2}.\\hfill \\\\ x\\le -3\\hfill &amp; \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/li>\r\n \t<li>[latex]\\begin{array}{ll}5-x&gt;10\\hfill &amp; \\hfill \\\\ -x&gt;5\\hfill &amp; \\hfill \\\\ \\left(-1\\right)\\left(-x\\right)&gt;\\left(5\\right)\\left(-1\\right)\\hfill &amp; \\text{Multiply by }-1.\\hfill \\\\ x&lt;-5\\hfill &amp; \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/li>\r\n<\/ol>\r\n<div>[\/hidden-answer]<\/div>\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve [latex]4x+7\\ge 2x - 3[\/latex].\r\n\r\n[reveal-answer q=\"32307\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"32307\"][latex]x\\ge -5[\/latex][\/hidden-answer]\r\n<iframe id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92606&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<h2 data-type=\"title\">Solving Inequalities in One Variable Algebraically<\/h2>\r\nAs the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Inequality Algebraically<\/h3>\r\nSolve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].\r\n\r\n[reveal-answer q=\"453286\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"453286\"]\r\nSolving this inequality is similar to solving an equation up until the last step.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}13 - 7x\\ge 10x - 4\\hfill &amp; \\hfill \\\\ 13 - 17x\\ge -4\\hfill &amp; \\text{Move variable terms to one side of the inequality}.\\hfill \\\\ -17x\\ge -17\\hfill &amp; \\text{Isolate the variable term}.\\hfill \\\\ x\\le 1\\hfill &amp; \\text{Dividing both sides by }-17\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is given by the interval [latex]\\left(-\\infty ,1\\right][\/latex], or all real numbers less than and including 1.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the inequality and write the answer using interval notation: [latex]-x+4&lt;\\frac{1}{2}x+1[\/latex].\r\n\r\n[reveal-answer q=\"703883\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"703883\"][latex]\\left(2,\\infty \\right)[\/latex][\/hidden-answer]\r\n<iframe id=\"mom6\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92607&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Solving an Inequality with Fractions<\/h3>\r\nSolve the following inequality and write the answer in interval notation: [latex]-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x[\/latex].\r\n\r\n[reveal-answer q=\"37354\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"37354\"]\r\nWe begin solving in the same way we do when solving an equation.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x\\hfill &amp; \\hfill \\\\ -\\frac{3}{4}x-\\frac{2}{3}x\\ge -\\frac{5}{8}\\hfill &amp; \\text{Put variable terms on one side}.\\hfill \\\\ -\\frac{9}{12}x-\\frac{8}{12}x\\ge -\\frac{5}{8}\\hfill &amp; \\text{Write fractions with common denominator}.\\hfill \\\\ -\\frac{17}{12}x\\ge -\\frac{5}{8}\\hfill &amp; \\hfill \\\\ x\\le -\\frac{5}{8}\\left(-\\frac{12}{17}\\right)\\hfill &amp; \\text{Multiplying by a negative number reverses the inequality}.\\hfill \\\\ x\\le \\frac{15}{34}\\hfill &amp; \\hfill \\end{array}[\/latex]<\/div>\r\nThe solution set is the interval [latex]\\left(-\\infty ,\\frac{15}{34}\\right][\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nSolve the inequality and write the answer in interval notation: [latex]-\\frac{5}{6}x\\le \\frac{3}{4}+\\frac{8}{3}x[\/latex].\r\n\r\n[reveal-answer q=\"524889\"]Solution[\/reveal-answer]\r\n[hidden-answer a=\"524889\"]\r\n\r\n[latex]\\left[-\\frac{3}{14},\\infty \\right)[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom7\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=72891&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"450\"><\/iframe>\r\n\r\n<\/div>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Objectives<\/h3>\n<ul>\n<li>Use interval notation to express inequalities<\/li>\n<li>Use properties of inequalities<\/li>\n<\/ul>\n<\/div>\n<p>Indicating the solution to an inequality such as [latex]x\\ge 4[\/latex] can be achieved in several ways.<\/p>\n<p>We can use a number line as shown below.\u00a0The blue ray begins at [latex]x=4[\/latex] and, as indicated by the arrowhead, continues to infinity, which illustrates that the solution set includes all real numbers greater than or equal to 4.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/24225859\/CNX_CAT_Figure_02_07_002.jpg\" alt=\"A number line starting at zero with the last tick mark being labeled 11. There is a dot at the number 4 and an arrow extends toward the right.\" width=\"487\" height=\"49\" data-media-type=\"image\/jpg\" \/><\/p>\n<p>We can use <strong>set-builder notation<\/strong>: [latex]\\{x|x\\ge 4\\}[\/latex], which translates to &#8220;all real numbers <em>x <\/em>such that <em>x <\/em>is greater than or equal to 4.&#8221; Notice that braces are used to indicate a set.<\/p>\n<p>The third method is <strong>interval notation<\/strong>, in which solution sets are indicated with parentheses or brackets. The solutions to [latex]x\\ge 4[\/latex] are represented as [latex]\\left[4,\\infty \\right)[\/latex]. This is perhaps the most useful method, as it applies to concepts studied later in this course and to other higher-level math courses.<\/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.<\/p>\n<table summary=\"A table with 11 rows and 3 columns. The entries in the first row are: Set Indicated, Set-Builder Notation, Interval Notation. The entries in the second row are: All real numbers between a and b, but not including a and b; {x| a &lt; x &lt; b}; (a,b). The entries in the third row are: All real numbers greater than a, but not including a; {x| x &gt; a}; (a , infinity). The entries in the fourth row are: All real numbers less than b, but not including b; {x| x &lt; b}; (negative infinity, b). The entries in the fifth row are: All real numbers greater than a, including a; {x| x a}; [a, infinity). The entries in the sixth row are: All real numbers less than b, including b; {x| x b}; (negative infinity, b]. The entries in the seventh row are: All real numbers between a and b, including a; {x| a x &lt; b}; [a, b). The entries in the eighth row are: All real numbers between a and b, including b; {x| a &lt; x b}; (a, b]. The entries in the ninth row are: All real numbers between a and b, including a and b; {x| a x b}; [a, b]. The entries in the tenth row are: all real numbers less than a and greater than b; {x| x &lt; a and x &gt; b}; (negative infinity, a) union (b, infinity). The entries in the eleventh row are: All real numbers; {x| x is all real numbers}; (negative infinity, infinity).\">\n<thead>\n<tr>\n<th>Set Indicated<\/th>\n<th>Set-Builder Notation<\/th>\n<th>Interval Notation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>All real numbers between <em>a <\/em>and <em>b<\/em>, but not including <em>a <\/em>or <em>b<\/em><\/td>\n<td>[latex]\\{x|a<x<b\\}[\/latex]<\/td>\n<td>[latex]\\left(a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers greater than <em>a<\/em>, but not including <em>a<\/em><\/td>\n<td>[latex]\\{x|x>a\\}[\/latex]<\/td>\n<td>[latex]\\left(a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>b<\/em>, but not including <em>b<\/em><\/td>\n<td>[latex]\\{x|x<b\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers greater than <em>a<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\{x|x\\ge a\\}[\/latex]<\/td>\n<td>[latex]\\left[a,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\{x|x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em>a <\/em>and<em> b<\/em>, including <em>a<\/em><\/td>\n<td>[latex]\\{x|a\\le x<b\\}[\/latex]<\/td>\n<td>[latex]\\left[a,b\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers between <em>a<\/em> and <em>b<\/em>, including <em>b<\/em><\/td>\n<td>[latex]\\{x|a<x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left(a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\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]\\{x|a\\le x\\le b\\}[\/latex]<\/td>\n<td>[latex]\\left[a,b\\right][\/latex]<\/td>\n<\/tr>\n<tr>\n<td>All real numbers less than <em>a<\/em> or greater than <em>b<\/em><\/td>\n<td>[latex]\\{x|x<a\\text{ and }x>b\\}[\/latex]<\/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>[latex]\\{x|x\\text{ is all real numbers}\\}[\/latex]<\/td>\n<td>[latex]\\left(-\\infty ,\\infty \\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"textbox exercises\">\n<h3>Example: Using Interval Notation to Express All Real Numbers Greater Than or Equal to <em>a<\/em><\/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=\"q143041\">Solution<\/span><\/p>\n<div id=\"q143041\" class=\"hidden-answer\" style=\"display: none\">\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.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Use interval notation to indicate all real numbers between and including [latex]-3[\/latex] and [latex]5[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814810\">Solution<\/span><\/p>\n<div id=\"q814810\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left[-3,5\\right][\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom10\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=58&amp;theme=oea&amp;iframe_resize_id=mom10\" width=\"100%\" height=\"650\"><\/iframe><br \/>\n<iframe loading=\"lazy\" id=\"mom1\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92604&amp;theme=oea&amp;iframe_resize_id=mom1\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Using Interval Notation to Express All Real Numbers Less Than or Equal to <em>a <\/em>or Greater Than or Equal to <em>b<\/em><\/h3>\n<p>Write the interval expressing all real numbers less than or equal to [latex]-1[\/latex] or greater than or equal to [latex]1[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q797079\">Solution<\/span><\/p>\n<div id=\"q797079\" class=\"hidden-answer\" style=\"display: none\">\nWe have to write two intervals for this example. The first interval must indicate all real numbers less than or equal to 1. So, this interval begins at [latex]-\\infty[\/latex] and ends at [latex]-1[\/latex], which is written as [latex]\\left(-\\infty ,-1\\right][\/latex].<\/p>\n<p>The second interval must show all real numbers greater than or equal to [latex]1[\/latex], which is written as [latex]\\left[1,\\infty \\right)[\/latex]. However, we want to combine these two sets. We accomplish this by inserting the union symbol, [latex]\\cup[\/latex], between the two intervals.<\/p>\n<div style=\"text-align: center;\">[latex]\\left(-\\infty ,-1\\right]\\cup \\left[1,\\infty \\right)[\/latex]<\/div>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Express all real numbers less than [latex]-2[\/latex] or greater than or equal to 3 in interval notation.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q729196\">Solution<\/span><\/p>\n<div id=\"q729196\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left(-\\infty ,-2\\right)\\cup \\left[3,\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>try it now<\/h3>\n<p>Use the sliders in the graph below to adjust the length of the line.<\/p>\n<ol>\n<li>Adjust the left endpoint to (-15,0), and the right endpoint to (5,0)<\/li>\n<li>Write an inequality that represents the line you created.<\/li>\n<\/ol>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<p>3. Now slide the point labeled b over to 15. What do you think happened to the line?<\/p>\n<p><textarea aria-label=\"Your Answer\" rows=\"1\"><\/textarea><\/p>\n<p>4. Now adjust the end points and make your own inequality.<\/p>\n<p>https:\/\/www.desmos.com\/calculator\/4529rytfef<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q748650\">Show Answer<\/span><\/p>\n<div id=\"q748650\" class=\"hidden-answer\" style=\"display: none\">\n<p>With endpoints (-15,0) and (5,0), the values for x on the line are between -15 and 5, so we can write [latex]-15<x<5[\/latex]. We made it a strict inequality because the dots on the endpoints of the lines are open.\n\nThe line disappeared because the inequality being represented is of the form [latex]b< x< a[\/latex] and when [latex]b>a[\/latex] the inequality is no longer valid.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<p>&nbsp;<\/p>\n<h2>Using the Properties of Inequalities<\/h2>\n<p>When we work with inequalities, we can usually treat them similarly to but not exactly as we treat equalities. We can use the <strong>addition property<\/strong> and the <strong>multiplication property<\/strong> to help us solve them. The one exception is when we multiply or divide by a negative number; doing so reverses the inequality symbol.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Properties of Inequalities<\/h3>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ll}\\text{Addition Property}\\hfill& \\text{If }a< b,\\text{ then }a+c< b+c.\\hfill \\\\ \\hfill & \\hfill \\\\ \\text{Multiplication Property}\\hfill & \\text{If }a< b\\text{ and }c> 0,\\text{ then }ac< bc.\\hfill \\\\ \\hfill & \\text{If }a< b\\text{ and }c< 0,\\text{ then }ac> bc.\\hfill \\end{array}[\/latex]<\/p>\n<p>These properties also apply to [latex]a\\le b[\/latex], [latex]a>b[\/latex], and [latex]a\\ge b[\/latex].<\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Demonstrating the Addition Property<\/h3>\n<p>Illustrate the addition property for inequalities by solving each of the following:<\/p>\n<ol>\n<li>[latex]x - 15<4[\/latex]<\/li>\n<li>[latex]6\\ge x - 1[\/latex]<\/li>\n<li>[latex]x+7>9[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q105622\">Solution<\/span><\/p>\n<div id=\"q105622\" class=\"hidden-answer\" style=\"display: none\">\nThe addition property for inequalities states that if an inequality exists, adding or subtracting the same number on both sides does not change the inequality.<\/p>\n<ol>\n<li>[latex]\\begin{array}{ll}x - 15<4\\hfill & \\hfill \\\\ x - 15+15<4+15 \\hfill & \\text{Add 15 to both sides.}\\hfill \\\\ x<19\\hfill & \\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}6\\ge x - 1\\hfill & \\hfill \\\\ 6+1\\ge x - 1+1\\hfill & \\text{Add 1 to both sides}.\\hfill \\\\ 7\\ge x\\hfill & \\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}x+7>9\\hfill & \\hfill \\\\ x+7 - 7>9 - 7\\hfill & \\text{Subtract 7 from both sides}.\\hfill \\\\ x>2\\hfill & \\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]3x - 2<1[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q68318\">Solution<\/span><\/p>\n<div id=\"q68318\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]x<1[\/latex]\n\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom3\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92605&amp;theme=oea&amp;iframe_resize_id=mom3\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Demonstrating the Multiplication Property<\/h3>\n<p>Illustrate the multiplication property for inequalities by solving each of the following:<\/p>\n<ol>\n<li>[latex]3x<6[\/latex]<\/li>\n<li>[latex]-2x - 1\\ge 5[\/latex]<\/li>\n<li>[latex]5-x>10[\/latex]<\/li>\n<\/ol>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q749552\">Solution<\/span><\/p>\n<div id=\"q749552\" class=\"hidden-answer\" style=\"display: none\">\n<ol>\n<li>[latex]\\begin{array}{l}3x<6\\hfill \\\\ \\frac{1}{3}\\left(3x\\right)<\\left(6\\right)\\frac{1}{3}\\hfill \\\\ x<2\\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}-2x - 1\\ge 5\\hfill & \\hfill \\\\ -2x\\ge 6\\hfill & \\hfill \\\\ \\left(-\\frac{1}{2}\\right)\\left(-2x\\right)\\ge \\left(6\\right)\\left(-\\frac{1}{2}\\right)\\hfill & \\text{Multiply by }-\\frac{1}{2}.\\hfill \\\\ x\\le -3\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/li>\n<li>[latex]\\begin{array}{ll}5-x>10\\hfill & \\hfill \\\\ -x>5\\hfill & \\hfill \\\\ \\left(-1\\right)\\left(-x\\right)>\\left(5\\right)\\left(-1\\right)\\hfill & \\text{Multiply by }-1.\\hfill \\\\ x<-5\\hfill & \\text{Reverse the inequality}.\\hfill \\end{array}[\/latex]<\/li>\n<\/ol>\n<div><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve [latex]4x+7\\ge 2x - 3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q32307\">Solution<\/span><\/p>\n<div id=\"q32307\" class=\"hidden-answer\" style=\"display: none\">[latex]x\\ge -5[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom5\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92606&amp;theme=oea&amp;iframe_resize_id=mom5\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<h2 data-type=\"title\">Solving Inequalities in One Variable Algebraically<\/h2>\n<p>As the examples have shown, we can perform the same operations on both sides of an inequality, just as we do with equations; we combine like terms and perform operations. To solve, we isolate the variable.<\/p>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inequality Algebraically<\/h3>\n<p>Solve the inequality: [latex]13 - 7x\\ge 10x - 4[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q453286\">Solution<\/span><\/p>\n<div id=\"q453286\" class=\"hidden-answer\" style=\"display: none\">\nSolving this inequality is similar to solving an equation up until the last step.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}13 - 7x\\ge 10x - 4\\hfill & \\hfill \\\\ 13 - 17x\\ge -4\\hfill & \\text{Move variable terms to one side of the inequality}.\\hfill \\\\ -17x\\ge -17\\hfill & \\text{Isolate the variable term}.\\hfill \\\\ x\\le 1\\hfill & \\text{Dividing both sides by }-17\\text{ reverses the inequality}.\\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is given by the interval [latex]\\left(-\\infty ,1\\right][\/latex], or all real numbers less than and including 1.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the inequality and write the answer using interval notation: [latex]-x+4<\\frac{1}{2}x+1[\/latex].\n\n\n\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q703883\">Solution<\/span><\/p>\n<div id=\"q703883\" class=\"hidden-answer\" style=\"display: none\">[latex]\\left(2,\\infty \\right)[\/latex]<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom6\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=92607&amp;theme=oea&amp;iframe_resize_id=mom6\" width=\"100%\" height=\"250\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Solving an Inequality with Fractions<\/h3>\n<p>Solve the following inequality and write the answer in interval notation: [latex]-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q37354\">Solution<\/span><\/p>\n<div id=\"q37354\" class=\"hidden-answer\" style=\"display: none\">\nWe begin solving in the same way we do when solving an equation.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{ll}-\\frac{3}{4}x\\ge -\\frac{5}{8}+\\frac{2}{3}x\\hfill & \\hfill \\\\ -\\frac{3}{4}x-\\frac{2}{3}x\\ge -\\frac{5}{8}\\hfill & \\text{Put variable terms on one side}.\\hfill \\\\ -\\frac{9}{12}x-\\frac{8}{12}x\\ge -\\frac{5}{8}\\hfill & \\text{Write fractions with common denominator}.\\hfill \\\\ -\\frac{17}{12}x\\ge -\\frac{5}{8}\\hfill & \\hfill \\\\ x\\le -\\frac{5}{8}\\left(-\\frac{12}{17}\\right)\\hfill & \\text{Multiplying by a negative number reverses the inequality}.\\hfill \\\\ x\\le \\frac{15}{34}\\hfill & \\hfill \\end{array}[\/latex]<\/div>\n<p>The solution set is the interval [latex]\\left(-\\infty ,\\frac{15}{34}\\right][\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Solve the inequality and write the answer in interval notation: [latex]-\\frac{5}{6}x\\le \\frac{3}{4}+\\frac{8}{3}x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q524889\">Solution<\/span><\/p>\n<div id=\"q524889\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]\\left[-\\frac{3}{14},\\infty \\right)[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom7\" class=\"resizable\" src=\"https:\/\/www.myopenmath.com\/multiembedq.php?id=72891&amp;theme=oea&amp;iframe_resize_id=mom7\" width=\"100%\" height=\"450\"><\/iframe><\/p>\n<\/div>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1461\">\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>Graphing Inequalities. <strong>Authored by<\/strong>: Lumen Learning. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/www.desmos.com\/calculator\/4529rytfef\">https:\/\/www.desmos.com\/calculator\/4529rytfef<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 92604, 92605, 92606, 92607. <strong>Authored by<\/strong>: Michael Jenck. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 72891. <strong>Authored by<\/strong>: Alyson Day. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 58. <strong>Authored by<\/strong>: Lippman, David. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: OpenStax College Algebra. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface<\/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":15,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"College Algebra\",\"author\":\"OpenStax College Algebra\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@3.278:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"original\",\"description\":\"Revision and Adaptation\",\"author\":\"\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"},{\"type\":\"cc\",\"description\":\"College Algebra\",\"author\":\"Abramson, Jay et 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